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Most recent proofs    These are the 100 (Unicode, GIF) or 1000 (Unicode, GIF) most recent proofs in the iset.mm database for the Intuitionistic Logic Explorer. The iset.mm database is maintained on GitHub with master (stable) and develop (development) versions. This page was created from the commit given on the MPE Most Recent Proofs page. The database from that commit is also available here: iset.mm.

See the MPE Most Recent Proofs page for news and some useful links.

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Last updated on 25-Oct-2021 at 2:47 PM ET.
Recent Additions to the Intuitionistic Logic Explorer
DateLabelDescription
Theorem
 
24-Oct-2021ax-ddkcomp 10114 Axiom of Dedekind completeness for Dedekind real numbers: every nonempty upper-bounded located set of reals has a real upper bound. Ideally, this axiom should be "proved" as "axddkcomp" for the real numbers constructed from IZF, and then the axiom ax-ddkcomp 10114 should be used in place of construction specific results. In particular, axcaucvg 6974 should be proved from it. (Contributed by BJ, 24-Oct-2021.)
(((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 < 𝑦))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ∧ ((𝐵 ∈ 𝑅 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝐵) → 𝑥 ≤ 𝐵)))
 
20-Oct-2021onunsnss 6355 Adding a singleton to create an ordinal. (Contributed by Jim Kingdon, 20-Oct-2021.)
((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → 𝐵 ⊆ 𝐴)
 
19-Oct-2021snon0 6356 An ordinal which is a singleton is {∅}. (Contributed by Jim Kingdon, 19-Oct-2021.)
((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → 𝐴 = ∅)
 
18-Oct-2021qdencn 10124 The set of complex numbers whose real and imaginary parts are rational is dense in the complex plane. This is a two dimensional analogue to qdenre 9798 (and also would hold for ℝ × ℝ with the usual metric; this is not about complex numbers in particular). (Contributed by Jim Kingdon, 18-Oct-2021.)
𝑄 = {𝑧 ∈ ℂ ∣ ((ℜ‘𝑧) ∈ ℚ ∧ (ℑ‘𝑧) ∈ ℚ)}    ⇒   ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ 𝑄 (abs‘(𝑥 − 𝐴)) < 𝐵)
 
18-Oct-2021modqmulnn 9184 Move a positive integer in and out of a floor in the first argument of a modulo operation. (Contributed by Jim Kingdon, 18-Oct-2021.)
((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → ((𝑁 · (⌊‘𝐴)) mod (𝑁 · 𝑀)) ≤ ((⌊‘(𝑁 · 𝐴)) mod (𝑁 · 𝑀)))
 
18-Oct-2021intqfrac 9181 Break a number into its integer part and its fractional part. (Contributed by Jim Kingdon, 18-Oct-2021.)
(𝐴 ∈ ℚ → 𝐴 = ((⌊‘𝐴) + (𝐴 mod 1)))
 
18-Oct-2021flqmod 9180 The floor function expressed in terms of the modulo operation. (Contributed by Jim Kingdon, 18-Oct-2021.)
(𝐴 ∈ ℚ → (⌊‘𝐴) = (𝐴 − (𝐴 mod 1)))
 
18-Oct-2021modqfrac 9179 The fractional part of a number is the number modulo 1. (Contributed by Jim Kingdon, 18-Oct-2021.)
(𝐴 ∈ ℚ → (𝐴 mod 1) = (𝐴 − (⌊‘𝐴)))
 
18-Oct-2021modqdifz 9178 The modulo operation differs from 𝐴 by an integer multiple of 𝐵. (Contributed by Jim Kingdon, 18-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) ∈ ℤ)
 
18-Oct-2021modqdiffl 9177 The modulo operation differs from 𝐴 by an integer multiple of 𝐵. (Contributed by Jim Kingdon, 18-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 − (𝐴 mod 𝐵)) / 𝐵) = (⌊‘(𝐴 / 𝐵)))
 
18-Oct-2021modqelico 9176 Modular reduction produces a half-open interval. (Contributed by Jim Kingdon, 18-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) ∈ (0[,)𝐵))
 
18-Oct-2021modqlt 9175 The modulo operation is less than its second argument. (Contributed by Jim Kingdon, 18-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) < 𝐵)
 
18-Oct-2021modqge0 9174 The modulo operation is nonnegative. (Contributed by Jim Kingdon, 18-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → 0 ≤ (𝐴 mod 𝐵))
 
18-Oct-2021negqmod0 9173 𝐴 is divisible by 𝐵 iff its negative is. (Contributed by Jim Kingdon, 18-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 mod 𝐵) = 0 ↔ (-𝐴 mod 𝐵) = 0))
 
18-Oct-2021mulqmod0 9172 The product of an integer and a positive rational number is 0 modulo the positive real number. (Contributed by Jim Kingdon, 18-Oct-2021.)
((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) → ((𝐴 · 𝑀) mod 𝑀) = 0)
 
18-Oct-2021flqdiv 9163 Cancellation of the embedded floor of a real divided by an integer. (Contributed by Jim Kingdon, 18-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ) → (⌊‘((⌊‘𝐴) / 𝑁)) = (⌊‘(𝐴 / 𝑁)))
 
18-Oct-2021intqfrac2 9161 Decompose a real into integer and fractional parts. (Contributed by Jim Kingdon, 18-Oct-2021.)
𝑍 = (⌊‘𝐴)    &   𝐹 = (𝐴 − 𝑍)    ⇒   (𝐴 ∈ ℚ → (0 ≤ 𝐹 ∧ 𝐹 < 1 ∧ 𝐴 = (𝑍 + 𝐹)))
 
17-Oct-2021modq0 9171 𝐴 mod 𝐵 is zero iff 𝐴 is evenly divisible by 𝐵. (Contributed by Jim Kingdon, 17-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → ((𝐴 mod 𝐵) = 0 ↔ (𝐴 / 𝐵) ∈ ℤ))
 
16-Oct-2021modqcld 9170 Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
(𝜑 → 𝐴 ∈ ℚ)    &   (𝜑 → 𝐵 ∈ ℚ)    &   (𝜑 → 0 < 𝐵)    ⇒   (𝜑 → (𝐴 mod 𝐵) ∈ ℚ)
 
16-Oct-2021flqpmodeq 9169 Partition of a division into its integer part and the remainder. (Contributed by Jim Kingdon, 16-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (((⌊‘(𝐴 / 𝐵)) · 𝐵) + (𝐴 mod 𝐵)) = 𝐴)
 
16-Oct-2021modqcl 9168 Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) ∈ ℚ)
 
16-Oct-2021modqvalr 9167 The value of the modulo operation (multiplication in reversed order). (Contributed by Jim Kingdon, 16-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) = (𝐴 − ((⌊‘(𝐴 / 𝐵)) · 𝐵)))
 
16-Oct-2021modqval 9166 The value of the modulo operation. The modulo congruence notation of number theory, 𝐽≡𝐾 (modulo 𝑁), can be expressed in our notation as (𝐽 mod 𝑁) = (𝐾 mod 𝑁). Definition 1 in Knuth, The Art of Computer Programming, Vol. I (1972), p. 38. Knuth uses "mod" for the operation and "modulo" for the congruence. Unlike Knuth, we restrict the second argument to positive numbers to simplify certain theorems. (This also gives us future flexibility to extend it to any one of several different conventions for a zero or negative second argument, should there be an advantage in doing so.) As with flqcl 9117 we only prove this for rationals although other particular kinds of real numbers may be possible. (Contributed by Jim Kingdon, 16-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 0 < 𝐵) → (𝐴 mod 𝐵) = (𝐴 − (𝐵 · (⌊‘(𝐴 / 𝐵)))))
 
15-Oct-2021qdenre 9798 The rational numbers are dense in ℝ: any real number can be approximated with arbitrary precision by a rational number. For order theoretic density, see qbtwnre 9111. (Contributed by BJ, 15-Oct-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℚ (abs‘(𝑥 − 𝐴)) < 𝐵)
 
14-Oct-2021qbtwnrelemcalc 9110 Lemma for qbtwnre 9111. Calculations involved in showing the constructed rational number is less than 𝐵. (Contributed by Jim Kingdon, 14-Oct-2021.)
(𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝑀 < (𝐴 · (2 · 𝑁)))    &   (𝜑 → (1 / 𝑁) < (𝐵 − 𝐴))    ⇒   (𝜑 → ((𝑀 + 2) / (2 · 𝑁)) < 𝐵)
 
13-Oct-2021rebtwn2z 9109 A real number can be bounded by integers above and below which are two apart.

The proof starts by finding two integers which are less than and greater than the given real number. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on weak linearity, and iterating until the range consists of integers which are two apart. (Contributed by Jim Kingdon, 13-Oct-2021.)

(𝐴 ∈ ℝ → ∃𝑥 ∈ ℤ (𝑥 < 𝐴 ∧ 𝐴 < (𝑥 + 2)))
 
13-Oct-2021rebtwn2zlemshrink 9108 Lemma for rebtwn2z 9109. Shrinking the range around the given real number. (Contributed by Jim Kingdon, 13-Oct-2021.)
((𝐴 ∈ ℝ ∧ 𝐽 ∈ (ℤ≥‘2) ∧ ∃𝑚 ∈ ℤ (𝑚 < 𝐴 ∧ 𝐴 < (𝑚 + 𝐽))) → ∃𝑥 ∈ ℤ (𝑥 < 𝐴 ∧ 𝐴 < (𝑥 + 2)))
 
13-Oct-2021rebtwn2zlemstep 9107 Lemma for rebtwn2z 9109. Induction step. (Contributed by Jim Kingdon, 13-Oct-2021.)
((𝐾 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ ℝ ∧ ∃𝑚 ∈ ℤ (𝑚 < 𝐴 ∧ 𝐴 < (𝑚 + (𝐾 + 1)))) → ∃𝑚 ∈ ℤ (𝑚 < 𝐴 ∧ 𝐴 < (𝑚 + 𝐾)))
 
11-Oct-2021flqeqceilz 9160 A rational number is an integer iff its floor equals its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → (𝐴 ∈ ℤ ↔ (⌊‘𝐴) = (⌈‘𝐴)))
 
11-Oct-2021flqleceil 9159 The floor of a rational number is less than or equal to its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → (⌊‘𝐴) ≤ (⌈‘𝐴))
 
11-Oct-2021ceilqidz 9158 A rational number equals its ceiling iff it is an integer. (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → (𝐴 ∈ ℤ ↔ (⌈‘𝐴) = 𝐴))
 
11-Oct-2021ceilqle 9156 The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jim Kingdon, 11-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≤ 𝐵) → (⌈‘𝐴) ≤ 𝐵)
 
11-Oct-2021ceiqle 9155 The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jim Kingdon, 11-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≤ 𝐵) → -(⌊‘-𝐴) ≤ 𝐵)
 
11-Oct-2021ceilqm1lt 9154 One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → ((⌈‘𝐴) − 1) < 𝐴)
 
11-Oct-2021ceiqm1l 9153 One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → (-(⌊‘-𝐴) − 1) < 𝐴)
 
11-Oct-2021ceilqge 9152 The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → 𝐴 ≤ (⌈‘𝐴))
 
11-Oct-2021ceiqge 9151 The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → 𝐴 ≤ -(⌊‘-𝐴))
 
11-Oct-2021ceilqcl 9150 Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → (⌈‘𝐴) ∈ ℤ)
 
11-Oct-2021ceiqcl 9149 The ceiling function returns an integer (closure law). (Contributed by Jim Kingdon, 11-Oct-2021.)
(𝐴 ∈ ℚ → -(⌊‘-𝐴) ∈ ℤ)
 
11-Oct-2021qdceq 9102 Equality of rationals is decidable. (Contributed by Jim Kingdon, 11-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → DECID 𝐴 = 𝐵)
 
11-Oct-2021qltlen 8575 Rational 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 7621 which is a similar result for real numbers. (Contributed by Jim Kingdon, 11-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 < 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴)))
 
10-Oct-2021ceilqval 9148 The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.)
(𝐴 ∈ ℚ → (⌈‘𝐴) = -(⌊‘-𝐴))
 
10-Oct-2021flqmulnn0 9141 Move a nonnegative integer in and out of a floor. (Contributed by Jim Kingdon, 10-Oct-2021.)
((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℚ) → (𝑁 · (⌊‘𝐴)) ≤ (⌊‘(𝑁 · 𝐴)))
 
10-Oct-2021flqzadd 9140 An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.)
((𝑁 ∈ ℤ ∧ 𝐴 ∈ ℚ) → (⌊‘(𝑁 + 𝐴)) = (𝑁 + (⌊‘𝐴)))
 
10-Oct-2021flqaddz 9139 An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℤ) → (⌊‘(𝐴 + 𝑁)) = ((⌊‘𝐴) + 𝑁))
 
10-Oct-2021flqge1nn 9136 The floor of a number greater than or equal to 1 is a positive integer. (Contributed by Jim Kingdon, 10-Oct-2021.)
((𝐴 ∈ ℚ ∧ 1 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ)
 
10-Oct-2021flqge0nn0 9135 The floor of a number greater than or equal to 0 is a nonnegative integer. (Contributed by Jim Kingdon, 10-Oct-2021.)
((𝐴 ∈ ℚ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ0)
 
10-Oct-20214ap0 8015 The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
4 # 0
 
10-Oct-20213ap0 8012 The number 3 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
3 # 0
 
9-Oct-2021flqbi2 9133 A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
((𝑁 ∈ ℤ ∧ 𝐹 ∈ ℚ) → ((⌊‘(𝑁 + 𝐹)) = 𝑁 ↔ (0 ≤ 𝐹 ∧ 𝐹 < 1)))
 
9-Oct-2021flqbi 9132 A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) = 𝐵 ↔ (𝐵 ≤ 𝐴 ∧ 𝐴 < (𝐵 + 1))))
 
9-Oct-2021flqword2 9131 Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐴 ≤ 𝐵) → (⌊‘𝐵) ∈ (ℤ≥‘(⌊‘𝐴)))
 
9-Oct-2021flqwordi 9130 Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐴 ≤ 𝐵) → (⌊‘𝐴) ≤ (⌊‘𝐵))
 
9-Oct-2021flqltnz 9129 If A is not an integer, then the floor of A is less than A. (Contributed by Jim Kingdon, 9-Oct-2021.)
((𝐴 ∈ ℚ ∧ ¬ 𝐴 ∈ ℤ) → (⌊‘𝐴) < 𝐴)
 
9-Oct-2021flqidz 9128 A rational number equals its floor iff it is an integer. (Contributed by Jim Kingdon, 9-Oct-2021.)
(𝐴 ∈ ℚ → ((⌊‘𝐴) = 𝐴 ↔ 𝐴 ∈ ℤ))
 
8-Oct-2021flqidm 9127 The floor function is idempotent. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → (⌊‘(⌊‘𝐴)) = (⌊‘𝐴))
 
8-Oct-2021flqlt 9125 The floor function value is less than the next integer. (Contributed by Jim Kingdon, 8-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (⌊‘𝐴) < 𝐵))
 
8-Oct-2021flqge 9124 The floor function value is the greatest integer less than or equal to its argument. (Contributed by Jim Kingdon, 8-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ) → (𝐵 ≤ 𝐴 ↔ 𝐵 ≤ (⌊‘𝐴)))
 
8-Oct-2021qfracge0 9123 The fractional part of a rational number is nonnegative. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → 0 ≤ (𝐴 − (⌊‘𝐴)))
 
8-Oct-2021qfraclt1 9122 The fractional part of a rational number is less than one. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → (𝐴 − (⌊‘𝐴)) < 1)
 
8-Oct-2021flqltp1 9121 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → 𝐴 < ((⌊‘𝐴) + 1))
 
8-Oct-2021flqle 9120 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → (⌊‘𝐴) ≤ 𝐴)
 
8-Oct-2021flqcld 9119 The floor (greatest integer) function is an integer (closure law). (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝜑 → 𝐴 ∈ ℚ)    ⇒   (𝜑 → (⌊‘𝐴) ∈ ℤ)
 
8-Oct-2021flqlelt 9118 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → ((⌊‘𝐴) ≤ 𝐴 ∧ 𝐴 < ((⌊‘𝐴) + 1)))
 
8-Oct-2021flqcl 9117 The floor (greatest integer) function yields an integer when applied to a rational (closure law). It would presumably be possible to prove a similar result for some real numbers (for example, those apart from any integer), but not real numbers in general. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℤ)
 
8-Oct-2021qbtwnz 9106 There is a unique greatest integer less than or equal to a rational number. (Contributed by Jim Kingdon, 8-Oct-2021.)
(𝐴 ∈ ℚ → ∃!𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))
 
8-Oct-2021qbtwnzlemex 9105 Lemma for qbtwnz 9106. Existence of the integer.

The proof starts by finding two integers which are less than and greater than the given rational number. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on rational number trichotomy, and iterating until the range consists of two consecutive integers. (Contributed by Jim Kingdon, 8-Oct-2021.)

(𝐴 ∈ ℚ → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))
 
8-Oct-2021qbtwnzlemshrink 9104 Lemma for qbtwnz 9106. Shrinking the range around the given rational number. (Contributed by Jim Kingdon, 8-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐽 ∈ ℕ ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝐽))) → ∃𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))
 
8-Oct-2021qbtwnzlemstep 9103 Lemma for qbtwnz 9106. Induction step. (Contributed by Jim Kingdon, 8-Oct-2021.)
((𝐾 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + (𝐾 + 1)))) → ∃𝑚 ∈ ℤ (𝑚 ≤ 𝐴 ∧ 𝐴 < (𝑚 + 𝐾)))
 
8-Oct-2021qltnle 9101 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 8-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
 
7-Oct-2021qlelttric 9100 Rational trichotomy. (Contributed by Jim Kingdon, 7-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴))
 
6-Oct-2021qletric 9099 Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))
 
6-Oct-2021qtri3or 9098 Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
((𝑀 ∈ ℚ ∧ 𝑁 ∈ ℚ) → (𝑀 < 𝑁 ∨ 𝑀 = 𝑁 ∨ 𝑁 < 𝑀))
 
3-Oct-2021reg3exmid 4304 If any inhabited set satisfying df-wetr 4071 for E has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.)
(( E We 𝑧 ∧ ∃𝑤 𝑤 ∈ 𝑧) → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦)    ⇒   (𝜑 ∨ ¬ 𝜑)
 
3-Oct-2021reg3exmidlemwe 4303 Lemma for reg3exmid 4304. Our counterexample 𝐴 satisfies We. (Contributed by Jim Kingdon, 3-Oct-2021.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}    ⇒    E We 𝐴
 
2-Oct-2021reg2exmid 4261 If any inhabited set has a minimal element (when expressed by ⊆), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
∀𝑧(∃𝑤 𝑤 ∈ 𝑧 → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦)    ⇒   (𝜑 ∨ ¬ 𝜑)
 
2-Oct-2021reg2exmidlema 4259 Lemma for reg2exmid 4261. If 𝐴 has a minimal element (expressed by ⊆), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}    ⇒   (∃𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣 → (𝜑 ∨ ¬ 𝜑))
 
30-Sep-2021fin0or 6343 A finite set is either empty or inhabited. (Contributed by Jim Kingdon, 30-Sep-2021.)
(𝐴 ∈ Fin → (𝐴 = ∅ ∨ ∃𝑥 𝑥 ∈ 𝐴))
 
30-Sep-2021wessep 4302 A subset of a set well-ordered by set membership is well-ordered by set membership. (Contributed by Jim Kingdon, 30-Sep-2021.)
(( E We 𝐴 ∧ 𝐵 ⊆ 𝐴) → E We 𝐵)
 
28-Sep-2021frind 4089 Induction over a well-founded set. (Contributed by Jim Kingdon, 28-Sep-2021.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ((𝜒 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝜓) → 𝜑))    &   (𝜒 → 𝑅 Fr 𝐴)    &   (𝜒 → 𝐴 ∈ 𝑉)    ⇒   ((𝜒 ∧ 𝑥 ∈ 𝐴) → 𝜑)
 
26-Sep-2021wetriext 4301 A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.)
(𝜑 → 𝑅 We 𝐴)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎))    &   (𝜑 → 𝐵 ∈ 𝐴)    &   (𝜑 → 𝐶 ∈ 𝐴)    &   (𝜑 → ∀𝑧 ∈ 𝐴 (𝑧𝑅𝐵 ↔ 𝑧𝑅𝐶))    ⇒   (𝜑 → 𝐵 = 𝐶)
 
25-Sep-2021nnwetri 6354 A natural number is well-ordered by E. More specifically, this order both satisfies We and is trichotomous. (Contributed by Jim Kingdon, 25-Sep-2021.)
(𝐴 ∈ ω → ( E We 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)))
 
23-Sep-2021wepo 4096 A well-ordering is a partial ordering. (Contributed by Jim Kingdon, 23-Sep-2021.)
((𝑅 We 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝑅 Po 𝐴)
 
23-Sep-2021df-wetr 4071 Define the well-ordering predicate. It is unusual to define "well-ordering" in the absence of excluded middle, but we mean an ordering which is like the ordering which we have for ordinals (for example, it does not entail trichotomy because ordinals don't have that as seen at ordtriexmid 4247). Given excluded middle, well-ordering is usually defined to require trichotomy (and the defintion of Fr is typically also different). (Contributed by Mario Carneiro and Jim Kingdon, 23-Sep-2021.)
(𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
 
22-Sep-2021frforeq3 4084 Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
(𝑆 = 𝑇 → ( FrFor 𝑅𝐴𝑆 ↔ FrFor 𝑅𝐴𝑇))
 
22-Sep-2021frforeq2 4082 Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
(𝐴 = 𝐵 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑅𝐵𝑇))
 
22-Sep-2021frforeq1 4080 Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
(𝑅 = 𝑆 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑆𝐴𝑇))
 
22-Sep-2021df-frfor 4068 Define the well-founded relation predicate where 𝐴 might be a proper class. By passing in 𝑆 we allow it potentially to be a proper class rather than a set. (Contributed by Jim Kingdon and Mario Carneiro, 22-Sep-2021.)
( FrFor 𝑅𝐴𝑆 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) → 𝐴 ⊆ 𝑆))
 
21-Sep-2021frirrg 4087 A well-founded relation is irreflexive. This is the case where 𝐴 exists. (Contributed by Jim Kingdon, 21-Sep-2021.)
((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)
 
21-Sep-2021df-frind 4069 Define the well-founded relation predicate. In the presence of excluded middle, there are a variety of equivalent ways to define this. In our case, this definition, in terms of an inductive principle, works better than one along the lines of "there is an element which is minimal when A is ordered by R". Because 𝑠 is constrained to be a set (not a proper class) here, sometimes it may be necessary to use FrFor directly rather than via Fr. (Contributed by Jim Kingdon and Mario Carneiro, 21-Sep-2021.)
(𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)
 
17-Sep-2021ontr2exmid 4250 An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.)
∀𝑥 ∈ On ∀𝑦∀𝑧 ∈ On ((𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧)    ⇒   (𝜑 ∨ ¬ 𝜑)
 
16-Sep-2021nnsseleq 6079 For natural numbers, inclusion is equivalent to membership or equality. (Contributed by Jim Kingdon, 16-Sep-2021.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
 
15-Sep-2021fientri3 6353 Trichotomy of dominance for finite sets. (Contributed by Jim Kingdon, 15-Sep-2021.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ≼ 𝐵 ∨ 𝐵 ≼ 𝐴))
 
15-Sep-2021nntri2or2 6076 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-Sep-2021.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
 
14-Sep-2021findcard2sd 6349 Deduction form of finite set induction . (Contributed by Jim Kingdon, 14-Sep-2021.)
(𝑥 = ∅ → (𝜓 ↔ 𝜒))    &   (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))    &   (𝑥 = (𝑦 ∪ {𝑧}) → (𝜓 ↔ 𝜏))    &   (𝑥 = 𝐴 → (𝜓 ↔ 𝜂))    &   (𝜑 → 𝜒)    &   (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝜃 → 𝜏))    &   (𝜑 → 𝐴 ∈ Fin)    ⇒   (𝜑 → 𝜂)
 
13-Sep-2021php5fin 6339 A finite set is not equinumerous to a set which adds one element. (Contributed by Jim Kingdon, 13-Sep-2021.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ (V ∖ 𝐴)) → ¬ 𝐴 ≈ (𝐴 ∪ {𝐵}))
 
13-Sep-2021fiunsnnn 6338 Adding one element to a finite set which is equinumerous to a natural number. (Contributed by Jim Kingdon, 13-Sep-2021.)
(((𝐴 ∈ Fin ∧ 𝐵 ∈ (V ∖ 𝐴)) ∧ (𝑁 ∈ ω ∧ 𝐴 ≈ 𝑁)) → (𝐴 ∪ {𝐵}) ≈ suc 𝑁)
 
12-Sep-2021fisbth 6340 Schroeder-Bernstein Theorem for finite sets. (Contributed by Jim Kingdon, 12-Sep-2021.)
(((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)) → 𝐴 ≈ 𝐵)
 
11-Sep-2021diffisn 6350 Subtracting a singleton from a finite set produces a finite set. (Contributed by Jim Kingdon, 11-Sep-2021.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) → (𝐴 ∖ {𝐵}) ∈ Fin)
 
10-Sep-2021fin0 6342 A nonempty finite set has at least one element. (Contributed by Jim Kingdon, 10-Sep-2021.)
(𝐴 ∈ Fin → (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴))
 
9-Sep-2021fidifsnid 6332 If we remove a single element from a finite set then put it back in, we end up with the original finite set. This strengthens difsnss 3510 from subset to equality when the set is finite. (Contributed by Jim Kingdon, 9-Sep-2021.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)
 
9-Sep-2021fidifsnen 6331 All decrements of a finite set are equinumerous. (Contributed by Jim Kingdon, 9-Sep-2021.)
((𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))
 
8-Sep-2021diffifi 6351 Subtracting one finite set from another produces a finite set. (Contributed by Jim Kingdon, 8-Sep-2021.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → (𝐴 ∖ 𝐵) ∈ Fin)
 
8-Sep-2021diffitest 6344 If subtracting any set from a finite set gives a finite set, any proposition of the form ¬ 𝜑 is decidable. This is not a proof of full excluded middle, but it is close enough to show we won't be able to prove 𝐴 ∈ Fin → (𝐴 ∖ 𝐵) ∈ Fin. (Contributed by Jim Kingdon, 8-Sep-2021.)
∀𝑎 ∈ Fin ∀𝑏(𝑎 ∖ 𝑏) ∈ Fin    ⇒   (¬ 𝜑 ∨ ¬ ¬ 𝜑)
 
6-Sep-2021phpelm 6328 Pigeonhole Principle. A natural number is not equinumerous to an element of itself. (Contributed by Jim Kingdon, 6-Sep-2021.)
((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ¬ 𝐴 ≈ 𝐵)
 
5-Sep-2021fidceq 6330 Equality of members of a finite set is decidable. This may be counterintuitive: cannot any two sets be elements of a finite set? Well, to show, for example, that {𝐵, 𝐶} is finite would require showing it is equinumerous to 1𝑜 or to 2𝑜 but to show that you'd need to know 𝐵 = 𝐶 or ¬ 𝐵 = 𝐶, respectively. (Contributed by Jim Kingdon, 5-Sep-2021.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → DECID 𝐵 = 𝐶)
 
5-Sep-2021phplem4on 6329 Equinumerosity of successors of an ordinal and a natural number implies equinumerosity of the originals. (Contributed by Jim Kingdon, 5-Sep-2021.)
((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵 → 𝐴 ≈ 𝐵))
 
4-Sep-2021ex-ceil 9896 Example for df-ceil 9115. (Contributed by AV, 4-Sep-2021.)
((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1)
 
3-Sep-20210elsucexmid 4289 If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.)
∀𝑥 ∈ On ∅ ∈ suc 𝑥    ⇒   (𝜑 ∨ ¬ 𝜑)
 
1-Sep-2021php5dom 6325 A natural number does not dominate its successor. (Contributed by Jim Kingdon, 1-Sep-2021.)
(𝐴 ∈ ω → ¬ suc 𝐴 ≼ 𝐴)
 
1-Sep-2021phplem4dom 6324 Dominance of successors implies dominance of the original natural numbers. (Contributed by Jim Kingdon, 1-Sep-2021.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≼ suc 𝐵 → 𝐴 ≼ 𝐵))
 
1-Sep-2021snnen2oprc 6323 A singleton {𝐴} is never equinumerous with the ordinal number 2. If 𝐴 is a set, see snnen2og 6322. (Contributed by Jim Kingdon, 1-Sep-2021.)
(¬ 𝐴 ∈ V → ¬ {𝐴} ≈ 2𝑜)
 
1-Sep-2021snnen2og 6322 A singleton {𝐴} is never equinumerous with the ordinal number 2. If 𝐴 is a proper class, see snnen2oprc 6323. (Contributed by Jim Kingdon, 1-Sep-2021.)
(𝐴 ∈ 𝑉 → ¬ {𝐴} ≈ 2𝑜)
 
1-Sep-2021phplem3g 6319 A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 6317 with unnecessary hypotheses removed. (Contributed by Jim Kingdon, 1-Sep-2021.)
((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))
 
31-Aug-2021nndifsnid 6080 If we remove a single element from a natural number then put it back in, we end up with the original natural number. This strengthens difsnss 3510 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 31-Aug-2021.)
((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)
 
30-Aug-2021carden2bex 6369 If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.)
((𝐴 ≈ 𝐵 ∧ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) → (card‘𝐴) = (card‘𝐵))
 
30-Aug-2021cardval3ex 6365 The value of (card‘𝐴). (Contributed by Jim Kingdon, 30-Aug-2021.)
(∃𝑥 ∈ On 𝑥 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
 
30-Aug-2021cardcl 6361 The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.)
(∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (card‘𝐴) ∈ On)
 
30-Aug-2021onintrab2im 4244 An existence condition which implies an intersection is an ordinal number. (Contributed by Jim Kingdon, 30-Aug-2021.)
(∃𝑥 ∈ On 𝜑 → ∩ {𝑥 ∈ On ∣ 𝜑} ∈ On)
 
30-Aug-2021onintonm 4243 The intersection of an inhabited collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by Mario Carneiro and Jim Kingdon, 30-Aug-2021.)
((𝐴 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ∈ On)
 
29-Aug-2021onintexmid 4297 If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis would be provable given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.)
((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) → ∩ 𝑦 ∈ 𝑦)    ⇒   (𝜑 ∨ ¬ 𝜑)
 
29-Aug-2021ordtri2or2exmid 4296 Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.)
∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)    ⇒   (𝜑 ∨ ¬ 𝜑)
 
29-Aug-2021ordtri2or2exmidlem 4251 A set which is 2𝑜 if 𝜑 or ∅ if ¬ 𝜑 is an ordinal. (Contributed by Jim Kingdon, 29-Aug-2021.)
{𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On
 
29-Aug-20212ordpr 4249 Version of 2on 6009 with the definition of 2𝑜 expanded and expressed in terms of Ord. (Contributed by Jim Kingdon, 29-Aug-2021.)
Ord {∅, {∅}}
 
25-Aug-2021nnap0d 7959 A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.)
(𝜑 → 𝐴 ∈ ℕ)    ⇒   (𝜑 → 𝐴 # 0)
 
24-Aug-2021climcaucn 9870 A converging sequence of complex numbers is a Cauchy sequence. This is like climcau 9866 but adds the part that (𝐹‘𝑘) is complex. (Contributed by Jim Kingdon, 24-Aug-2021.)
𝑍 = (ℤ≥‘𝑀)    ⇒   ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))
 
24-Aug-2021climcvg1nlem 9868 Lemma for climcvg1n 9869. We construct sequences of the real and imaginary parts of each term of 𝐹, show those converge, and use that to show that 𝐹 converges. (Contributed by Jim Kingdon, 24-Aug-2021.)
(𝜑 → 𝐹:ℕ⟶ℂ)    &   (𝜑 → 𝐶 ∈ ℝ+)    &   (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘((𝐹‘𝑘) − (𝐹‘𝑛))) < (𝐶 / 𝑛))    &   𝐺 = (𝑥 ∈ ℕ ↦ (ℜ‘(𝐹‘𝑥)))    &   𝐻 = (𝑥 ∈ ℕ ↦ (ℑ‘(𝐹‘𝑥)))    &   𝐽 = (𝑥 ∈ ℕ ↦ (i · (𝐻‘𝑥)))    ⇒   (𝜑 → 𝐹 ∈ dom ⇝ )
 
23-Aug-2021climcvg1n 9869 A Cauchy sequence of complex numbers converges, existence version. The rate of convergence is fixed: all terms after the nth term must be within 𝐶 / 𝑛 of the nth term, where 𝐶 is a constant multiplier. (Contributed by Jim Kingdon, 23-Aug-2021.)
(𝜑 → 𝐹:ℕ⟶ℂ)    &   (𝜑 → 𝐶 ∈ ℝ+)    &   (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘((𝐹‘𝑘) − (𝐹‘𝑛))) < (𝐶 / 𝑛))    ⇒   (𝜑 → 𝐹 ∈ dom ⇝ )
 
23-Aug-2021climrecvg1n 9867 A Cauchy sequence of real numbers converges, existence version. The rate of convergence is fixed: all terms after the nth term must be within 𝐶 / 𝑛 of the nth term, where 𝐶 is a constant multiplier. (Contributed by Jim Kingdon, 23-Aug-2021.)
(𝜑 → 𝐹:ℕ⟶ℝ)    &   (𝜑 → 𝐶 ∈ ℝ+)    &   (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘((𝐹‘𝑘) − (𝐹‘𝑛))) < (𝐶 / 𝑛))    ⇒   (𝜑 → 𝐹 ∈ dom ⇝ )
 
22-Aug-2021climserile 9865 The partial sums of a converging infinite series with nonnegative terms are bounded by its limit. (Contributed by Jim Kingdon, 22-Aug-2021.)
𝑍 = (ℤ≥‘𝑀)    &   (𝜑 → 𝑁 ∈ 𝑍)    &   (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ 𝐴)    &   ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    &   ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘))    ⇒   (𝜑 → (seq𝑀( + , 𝐹, ℂ)‘𝑁) ≤ 𝐴)
 
22-Aug-2021iserige0 9863 The limit of an infinite series of nonnegative reals is nonnegative. (Contributed by Jim Kingdon, 22-Aug-2021.)
𝑍 = (ℤ≥‘𝑀)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ 𝐴)    &   ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    &   ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘))    ⇒   (𝜑 → 0 ≤ 𝐴)
 
22-Aug-2021iserile 9862 Comparison of the limits of two infinite series. (Contributed by Jim Kingdon, 22-Aug-2021.)
𝑍 = (ℤ≥‘𝑀)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ 𝐴)    &   (𝜑 → seq𝑀( + , 𝐺, ℂ) ⇝ 𝐵)    &   ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    &   ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)    &   ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐺‘𝑘))    ⇒   (𝜑 → 𝐴 ≤ 𝐵)
 
22-Aug-2021serile 9253 Comparison of partial sums of two infinite series of reals. (Contributed by Jim Kingdon, 22-Aug-2021.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℝ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ≤ (𝐺‘𝑘))    ⇒   (𝜑 → (seq𝑀( + , 𝐹, ℂ)‘𝑁) ≤ (seq𝑀( + , 𝐺, ℂ)‘𝑁))
 
22-Aug-2021serige0 9252 A finite sum of nonnegative terms is nonnegative. (Contributed by Jim Kingdon, 22-Aug-2021.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 0 ≤ (𝐹‘𝑘))    ⇒   (𝜑 → 0 ≤ (seq𝑀( + , 𝐹, ℂ)‘𝑁))
 
21-Aug-2021clim2iser2 9858 The limit of an infinite series with an initial segment added. (Contributed by Jim Kingdon, 21-Aug-2021.)
𝑍 = (ℤ≥‘𝑀)    &   (𝜑 → 𝑁 ∈ 𝑍)    &   ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   (𝜑 → seq(𝑁 + 1)( + , 𝐹, ℂ) ⇝ 𝐴)    ⇒   (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ (𝐴 + (seq𝑀( + , 𝐹, ℂ)‘𝑁)))
 
21-Aug-2021iseqdistr 9249 The distributive property for series. (Contributed by Jim Kingdon, 21-Aug-2021.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇(𝑥 + 𝑦)) = ((𝐶𝑇𝑥) + (𝐶𝑇𝑦)))    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) = (𝐶𝑇(𝐺‘𝑥)))    &   (𝜑 → 𝑆 ∈ 𝑉)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑇𝑦) ∈ 𝑆)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   (𝜑 → 𝐶 ∈ 𝑆)    ⇒   (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (𝐶𝑇(seq𝑀( + , 𝐺, 𝑆)‘𝑁)))
 
21-Aug-2021iseqhomo 9248 Apply a homomorphism to a sequence. (Contributed by Jim Kingdon, 21-Aug-2021.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦)))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)    ⇒   (𝜑 → (𝐻‘(seq𝑀( + , 𝐹, 𝑆)‘𝑁)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑁))
 
20-Aug-2021clim2iser 9857 The limit of an infinite series with an initial segment removed. (Contributed by Jim Kingdon, 20-Aug-2021.)
𝑍 = (ℤ≥‘𝑀)    &   (𝜑 → 𝑁 ∈ 𝑍)    &   ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹, ℂ) ⇝ 𝐴)    ⇒   (𝜑 → seq(𝑁 + 1)( + , 𝐹, ℂ) ⇝ (𝐴 − (seq𝑀( + , 𝐹, ℂ)‘𝑁)))
 
20-Aug-2021iseqex 9213 Existence of the sequence builder operation. (Contributed by Jim Kingdon, 20-Aug-2021.)
seq𝑀( + , 𝐹, 𝑆) ∈ V
 
20-Aug-2021frecex 5981 Finite recursion produces a set. (Contributed by Jim Kingdon, 20-Aug-2021.)
frec(𝐹, 𝐴) ∈ V
 
20-Aug-2021tfrfun 5955 Transfinite recursion produces a function. (Contributed by Jim Kingdon, 20-Aug-2021.)
Fun recs(𝐹)
 
19-Aug-2021iserclim0 9826 The zero series converges to zero. (Contributed by Jim Kingdon, 19-Aug-2021.)
(𝑀 ∈ ℤ → seq𝑀( + , ((ℤ≥‘𝑀) × {0}), ℂ) ⇝ 0)
 
19-Aug-2021shftval4g 9438 Value of a sequence shifted by -𝐴. (Contributed by Jim Kingdon, 19-Aug-2021.)
((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift -𝐴)‘𝐵) = (𝐹‘(𝐴 + 𝐵)))
 
19-Aug-2021iser0f 9251 A zero-valued infinite series is equal to the constant zero function. (Contributed by Jim Kingdon, 19-Aug-2021.)
𝑍 = (ℤ≥‘𝑀)    ⇒   (𝑀 ∈ ℤ → seq𝑀( + , (𝑍 × {0}), ℂ) = (𝑍 × {0}))
 
19-Aug-2021iser0 9250 The value of the partial sums in a zero-valued infinite series. (Contributed by Jim Kingdon, 19-Aug-2021.)
𝑍 = (ℤ≥‘𝑀)    ⇒   (𝑁 ∈ 𝑍 → (seq𝑀( + , (𝑍 × {0}), ℂ)‘𝑁) = 0)
 
19-Aug-2021ovexg 5539 Evaluating a set operation at two sets gives a set. (Contributed by Jim Kingdon, 19-Aug-2021.)
((𝐴 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐹𝐵) ∈ V)
 
18-Aug-2021iseqid3s 9246 A sequence that consists of zeroes up to 𝑁 sums to zero at 𝑁. In this case by "zero" we mean whatever the identity 𝑍 is for the operation +). (Contributed by Jim Kingdon, 18-Aug-2021.)
(𝜑 → (𝑍 + 𝑍) = 𝑍)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = 𝑍)    &   (𝜑 → 𝑍 ∈ 𝑆)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = 𝑍)
 
18-Aug-2021iseqid3 9245 A sequence that consists entirely of zeroes (or whatever the identity 𝑍 is for operation +) sums to zero. (Contributed by Jim Kingdon, 18-Aug-2021.)
(𝜑 → (𝑍 + 𝑍) = 𝑍)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) = 𝑍)    &   (𝜑 → 𝑍 ∈ 𝑉)    ⇒   (𝜑 → (seq𝑀( + , 𝐹, {𝑍})‘𝑁) = 𝑍)
 
18-Aug-2021iseqss 9226 Specifying a larger universe for seq. As long as 𝐹 and + are closed over 𝑆, then any set which contains 𝑆 can be used as the last argument to seq. This theorem does not allow 𝑇 to be a proper class, however. It also currently requires that + be closed over 𝑇 (as well as 𝑆). (Contributed by Jim Kingdon, 18-Aug-2021.)
(𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑇 ∈ 𝑉)    &   (𝜑 → 𝑆 ⊆ 𝑇)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑇)) → (𝑥 + 𝑦) ∈ 𝑇)    ⇒   (𝜑 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐹, 𝑇))
 
17-Aug-2021iseqcaopr 9242 The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Jim Kingdon, 17-Aug-2021.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ 𝑆)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ 𝑆)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘)))    &   (𝜑 → 𝑆 ∈ 𝑉)    ⇒   (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑁) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁) + (seq𝑀( + , 𝐺, 𝑆)‘𝑁)))
 
16-Aug-2021iseqcaopr3 9240 Lemma for iseqcaopr2 . (Contributed by Jim Kingdon, 16-Aug-2021.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ 𝑆)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ 𝑆)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)))    &   ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐹, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + (𝐺‘(𝑛 + 1)))))    &   (𝜑 → 𝑆 ∈ 𝑉)    ⇒   (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑁) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑁)))
 
16-Aug-2021iseq1p 9239 Removing the first term from a sequence. (Contributed by Jim Kingdon, 16-Aug-2021.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑀 ∈ ℤ)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    ⇒   (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = ((𝐹‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁)))
 
16-Aug-2021iseqsplit 9238 Split a sequence into two sequences. (Contributed by Jim Kingdon, 16-Aug-2021.)
((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑀 ∈ (ℤ≥‘𝐾))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑥) ∈ 𝑆)    ⇒   (𝜑 → (seq𝐾( + , 𝐹, 𝑆)‘𝑁) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁)))
 
15-Aug-2021shftfibg 9421 Value of a fiber of the relation 𝐹. (Contributed by Jim Kingdon, 15-Aug-2021.)
((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴) “ {𝐵}) = (𝐹 “ {(𝐵 − 𝐴)}))
 
15-Aug-2021ovshftex 9420 Existence of the result of applying shift. (Contributed by Jim Kingdon, 15-Aug-2021.)
((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴) ∈ V)
 
15-Aug-2021isermono 9237 The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by Jim Kingdon, 15-Aug-2021.)
(𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾))    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ ℝ)    &   ((𝜑 ∧ 𝑥 ∈ ((𝐾 + 1)...𝑁)) → 0 ≤ (𝐹‘𝑥))    ⇒   (𝜑 → (seq𝑀( + , 𝐹, ℝ)‘𝐾) ≤ (seq𝑀( + , 𝐹, ℝ)‘𝑁))
 
15-Aug-2021iserf 9233 An infinite series of complex terms is a function from ℕ to ℂ. (Contributed by Jim Kingdon, 15-Aug-2021.)
𝑍 = (ℤ≥‘𝑀)    &   (𝜑 → 𝑀 ∈ ℤ)    &   ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    ⇒   (𝜑 → seq𝑀( + , 𝐹, ℂ):𝑍⟶ℂ)
 
15-Aug-2021iseqshft2 9232 Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝐾 ∈ ℤ)    &   ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)))    &   (𝜑 → 𝑆 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝐺‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾)))
 
15-Aug-2021iseqfeq 9231 Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.)
(𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = (𝐺‘𝑘))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   (𝜑 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐺, 𝑆))
 
13-Aug-2021absdivapd 9791 Absolute value distributes over division. (Contributed by Jim Kingdon, 13-Aug-2021.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵)))
 
13-Aug-2021absrpclapd 9784 The absolute value of a complex number apart from zero is a positive real. (Contributed by Jim Kingdon, 13-Aug-2021.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    ⇒   (𝜑 → (abs‘𝐴) ∈ ℝ+)
 
13-Aug-2021absdivapzi 9750 Absolute value distributes over division. (Contributed by Jim Kingdon, 13-Aug-2021.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    ⇒   (𝐵 # 0 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵)))
 
13-Aug-2021absgt0ap 9695 The absolute value of a number apart from zero is positive. (Contributed by Jim Kingdon, 13-Aug-2021.)
(𝐴 ∈ ℂ → (𝐴 # 0 ↔ 0 < (abs‘𝐴)))
 
13-Aug-2021recvalap 9693 Reciprocal expressed with a real denominator. (Contributed by Jim Kingdon, 13-Aug-2021.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / 𝐴) = ((∗‘𝐴) / ((abs‘𝐴)↑2)))
 
13-Aug-2021sqap0 9320 A number is apart from zero iff its square is apart from zero. (Contributed by Jim Kingdon, 13-Aug-2021.)
(𝐴 ∈ ℂ → ((𝐴↑2) # 0 ↔ 𝐴 # 0))
 
13-Aug-2021leltapd 7628 '<_' implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐴 ≤ 𝐵)    ⇒   (𝜑 → (𝐴 < 𝐵 ↔ 𝐵 # 𝐴))
 
13-Aug-2021leltap 7615 '<_' implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐵 ↔ 𝐵 # 𝐴))
 
12-Aug-2021abssubap0 9686 If the absolute value of a complex number is less than a real, its difference from the real is apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ (abs‘𝐴) < 𝐵) → (𝐵 − 𝐴) # 0)
 
12-Aug-2021ltabs 9683 A number which is less than its absolute value is negative. (Contributed by Jim Kingdon, 12-Aug-2021.)
((𝐴 ∈ ℝ ∧ 𝐴 < (abs‘𝐴)) → 𝐴 < 0)
 
12-Aug-2021absext 9661 Strong extensionality for absolute value. (Contributed by Jim Kingdon, 12-Aug-2021.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) # (abs‘𝐵) → 𝐴 # 𝐵))
 
12-Aug-2021sq11ap 9414 Analogue to sq11 9326 but for apartness. (Contributed by Jim Kingdon, 12-Aug-2021.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴↑2) # (𝐵↑2) ↔ 𝐴 # 𝐵))
 
12-Aug-2021subap0d 7631 Two numbers apart from each other have difference apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐴 # 𝐵)    ⇒   (𝜑 → (𝐴 − 𝐵) # 0)
 
12-Aug-2021ltapd 7627 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    ⇒   (𝜑 → 𝐴 # 𝐵)
 
12-Aug-2021gtapd 7626 'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    ⇒   (𝜑 → 𝐵 # 𝐴)
 
12-Aug-2021ltapi 7625 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    ⇒   (𝐴 < 𝐵 → 𝐵 # 𝐴)
 
12-Aug-2021ltapii 7624 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐴 < 𝐵    ⇒   𝐴 # 𝐵
 
12-Aug-2021gtapii 7623 'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐴 < 𝐵    ⇒   𝐵 # 𝐴
 
12-Aug-2021ltap 7622 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 # 𝐴)
 
11-Aug-2021absexpzap 9676 Absolute value of integer exponentiation. (Contributed by Jim Kingdon, 11-Aug-2021.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁))
 
11-Aug-2021absdivap 9668 Absolute value distributes over division. (Contributed by Jim Kingdon, 11-Aug-2021.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵)))
 
11-Aug-2021abs00ap 9660 The absolute value of a number is apart from zero iff the number is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
(𝐴 ∈ ℂ → ((abs‘𝐴) # 0 ↔ 𝐴 # 0))
 
11-Aug-2021absrpclap 9659 The absolute value of a number apart from zero is a positive real. (Contributed by Jim Kingdon, 11-Aug-2021.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (abs‘𝐴) ∈ ℝ+)
 
11-Aug-2021sqrt11ap 9636 Analogue to sqrt11 9637 but for apartness. (Contributed by Jim Kingdon, 11-Aug-2021.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((√‘𝐴) # (√‘𝐵) ↔ 𝐴 # 𝐵))
 
11-Aug-2021cjap0d 9548 A number which is apart from zero has a complex conjugate which is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    ⇒   (𝜑 → (∗‘𝐴) # 0)
 
11-Aug-2021ap0gt0d 7630 A nonzero nonnegative number is positive. (Contributed by Jim Kingdon, 11-Aug-2021.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 𝐴 # 0)    ⇒   (𝜑 → 0 < 𝐴)
 
11-Aug-2021ap0gt0 7629 A nonnegative number is apart from zero if and only if it is positive. (Contributed by Jim Kingdon, 11-Aug-2021.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 # 0 ↔ 0 < 𝐴))
 
10-Aug-2021rersqrtthlem 9628 Lemma for resqrtth 9629. (Contributed by Jim Kingdon, 10-Aug-2021.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (((√‘𝐴)↑2) = 𝐴 ∧ 0 ≤ (√‘𝐴)))
 
10-Aug-2021rersqreu 9626 Existence and uniqueness for the real square root function. (Contributed by Jim Kingdon, 10-Aug-2021.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ∃!𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥))
 
10-Aug-2021rsqrmo 9625 Uniqueness for the square root function. (Contributed by Jim Kingdon, 10-Aug-2021.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ∃*𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥))
 
9-Aug-2021resqrexlemoverl 9619 Lemma for resqrex 9624. Every term in the sequence is an overestimate compared with the limit 𝐿. Although this theorem is stated in terms of a particular sequence the proof could be adapted for any decreasing convergent sequence. (Contributed by Jim Kingdon, 9-Aug-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 𝐿 ∈ ℝ)    &   (𝜑 → ∀𝑒 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑒)))    &   (𝜑 → 𝐾 ∈ ℕ)    ⇒   (𝜑 → 𝐿 ≤ (𝐹‘𝐾))
 
8-Aug-2021resqrexlemga 9621 Lemma for resqrex 9624. The sequence formed by squaring each term of 𝐹 converges to 𝐴. (Contributed by Mario Carneiro and Jim Kingdon, 8-Aug-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 𝐿 ∈ ℝ)    &   (𝜑 → ∀𝑒 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑒)))    &   𝐺 = (𝑥 ∈ ℕ ↦ ((𝐹‘𝑥)↑2))    ⇒   (𝜑 → ∀𝑒 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐺‘𝑘) < (𝐴 + 𝑒) ∧ 𝐴 < ((𝐺‘𝑘) + 𝑒)))
 
8-Aug-2021resqrexlemglsq 9620 Lemma for resqrex 9624. The sequence formed by squaring each term of 𝐹 converges to (𝐿↑2). (Contributed by Mario Carneiro and Jim Kingdon, 8-Aug-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 𝐿 ∈ ℝ)    &   (𝜑 → ∀𝑒 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑒)))    &   𝐺 = (𝑥 ∈ ℕ ↦ ((𝐹‘𝑥)↑2))    ⇒   (𝜑 → ∀𝑒 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐺‘𝑘) < ((𝐿↑2) + 𝑒) ∧ (𝐿↑2) < ((𝐺‘𝑘) + 𝑒)))
 
8-Aug-2021recvguniqlem 9592 Lemma for recvguniq 9593. Some of the rearrangements of the expressions. (Contributed by Jim Kingdon, 8-Aug-2021.)
(𝜑 → 𝐹:ℕ⟶ℝ)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐾 ∈ ℕ)    &   (𝜑 → 𝐴 < ((𝐹‘𝐾) + ((𝐴 − 𝐵) / 2)))    &   (𝜑 → (𝐹‘𝐾) < (𝐵 + ((𝐴 − 𝐵) / 2)))    ⇒   (𝜑 → ⊥)
 
8-Aug-2021ifcldcd 3358 Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.)
(𝜑 → 𝐴 ∈ 𝐶)    &   (𝜑 → 𝐵 ∈ 𝐶)    &   (𝜑 → DECID 𝜓)    ⇒   (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)
 
8-Aug-2021ifbothdc 3357 A wff 𝜃 containing a conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 8-Aug-2021.)
(𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃))    &   (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃))    ⇒   ((𝜓 ∧ 𝜒 ∧ DECID 𝜑) → 𝜃)
 
7-Aug-2021resqrexlemsqa 9622 Lemma for resqrex 9624. The square of a limit is 𝐴. (Contributed by Jim Kingdon, 7-Aug-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 𝐿 ∈ ℝ)    &   (𝜑 → ∀𝑒 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑒)))    ⇒   (𝜑 → (𝐿↑2) = 𝐴)
 
7-Aug-2021resqrexlemgt0 9618 Lemma for resqrex 9624. A limit is nonnegative. (Contributed by Jim Kingdon, 7-Aug-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 𝐿 ∈ ℝ)    &   (𝜑 → ∀𝑒 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ≥‘𝑗)((𝐹‘𝑖) < (𝐿 + 𝑒) ∧ 𝐿 < ((𝐹‘𝑖) + 𝑒)))    ⇒   (𝜑 → 0 ≤ 𝐿)
 
7-Aug-2021recvguniq 9593 Limits are unique. (Contributed by Jim Kingdon, 7-Aug-2021.)
(𝜑 → 𝐹:ℕ⟶ℝ)    &   (𝜑 → 𝐿 ∈ ℝ)    &   (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) < (𝐿 + 𝑥) ∧ 𝐿 < ((𝐹‘𝑘) + 𝑥)))    &   (𝜑 → 𝑀 ∈ ℝ)    &   (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) < (𝑀 + 𝑥) ∧ 𝑀 < ((𝐹‘𝑘) + 𝑥)))    ⇒   (𝜑 → 𝐿 = 𝑀)
 
6-Aug-2021resqrexlemcvg 9617 Lemma for resqrex 9624. The sequence has a limit. (Contributed by Jim Kingdon, 6-Aug-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    ⇒   (𝜑 → ∃𝑟 ∈ ℝ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ≥‘𝑗)((𝐹‘𝑖) < (𝑟 + 𝑥) ∧ 𝑟 < ((𝐹‘𝑖) + 𝑥)))
 
6-Aug-2021cvg1nlemcxze 9581 Lemma for cvg1n 9585. Rearranging an expression related to the rate of convergence. (Contributed by Jim Kingdon, 6-Aug-2021.)
(𝜑 → 𝐶 ∈ ℝ+)    &   (𝜑 → 𝑋 ∈ ℝ+)    &   (𝜑 → 𝑍 ∈ ℕ)    &   (𝜑 → 𝐸 ∈ ℕ)    &   (𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → ((((𝐶 · 2) / 𝑋) / 𝑍) + 𝐴) < 𝐸)    ⇒   (𝜑 → (𝐶 / (𝐸 · 𝑍)) < (𝑋 / 2))
 
1-Aug-2021cvg1n 9585 Convergence of real sequences.

This is a version of caucvgre 9580 with a constant multiplier 𝐶 on the rate of convergence. That is, all terms after the nth term must be within 𝐶 / 𝑛 of the nth term.

(Contributed by Jim Kingdon, 1-Aug-2021.)

(𝜑 → 𝐹:ℕ⟶ℝ)    &   (𝜑 → 𝐶 ∈ ℝ+)    &   (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (𝐶 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (𝐶 / 𝑛))))    ⇒   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ≥‘𝑗)((𝐹‘𝑖) < (𝑦 + 𝑥) ∧ 𝑦 < ((𝐹‘𝑖) + 𝑥)))
 
1-Aug-2021cvg1nlemres 9584 Lemma for cvg1n 9585. The original sequence 𝐹 has a limit (turns out it is the same as the limit of the modified sequence 𝐺). (Contributed by Jim Kingdon, 1-Aug-2021.)
(𝜑 → 𝐹:ℕ⟶ℝ)    &   (𝜑 → 𝐶 ∈ ℝ+)    &   (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (𝐶 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (𝐶 / 𝑛))))    &   𝐺 = (𝑗 ∈ ℕ ↦ (𝐹‘(𝑗 · 𝑍)))    &   (𝜑 → 𝑍 ∈ ℕ)    &   (𝜑 → 𝐶 < 𝑍)    ⇒   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ≥‘𝑗)((𝐹‘𝑖) < (𝑦 + 𝑥) ∧ 𝑦 < ((𝐹‘𝑖) + 𝑥)))
 
1-Aug-2021cvg1nlemcau 9583 Lemma for cvg1n 9585. By selecting spaced out terms for the modified sequence 𝐺, the terms are within 1 / 𝑛 (without the constant 𝐶). (Contributed by Jim Kingdon, 1-Aug-2021.)
(𝜑 → 𝐹:ℕ⟶ℝ)    &   (𝜑 → 𝐶 ∈ ℝ+)    &   (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (𝐶 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (𝐶 / 𝑛))))    &   𝐺 = (𝑗 ∈ ℕ ↦ (𝐹‘(𝑗 · 𝑍)))    &   (𝜑 → 𝑍 ∈ ℕ)    &   (𝜑 → 𝐶 < 𝑍)    ⇒   (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)((𝐺‘𝑛) < ((𝐺‘𝑘) + (1 / 𝑛)) ∧ (𝐺‘𝑘) < ((𝐺‘𝑛) + (1 / 𝑛))))
 
1-Aug-2021cvg1nlemf 9582 Lemma for cvg1n 9585. The modified sequence 𝐺 is a sequence. (Contributed by Jim Kingdon, 1-Aug-2021.)
(𝜑 → 𝐹:ℕ⟶ℝ)    &   (𝜑 → 𝐶 ∈ ℝ+)    &   (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (𝐶 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (𝐶 / 𝑛))))    &   𝐺 = (𝑗 ∈ ℕ ↦ (𝐹‘(𝑗 · 𝑍)))    &   (𝜑 → 𝑍 ∈ ℕ)    &   (𝜑 → 𝐶 < 𝑍)    ⇒   (𝜑 → 𝐺:ℕ⟶ℝ)
 
31-Jul-2021resqrexlemnm 9616 Lemma for resqrex 9624. The difference between two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 31-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝑁 ≤ 𝑀)    ⇒   (𝜑 → ((𝐹‘𝑁) − (𝐹‘𝑀)) < ((((𝐹‘1)↑2) · 2) / (2↑(𝑁 − 1))))
 
31-Jul-2021resqrexlemdecn 9610 Lemma for resqrex 9624. The sequence is decreasing. (Contributed by Jim Kingdon, 31-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝑁 < 𝑀)    ⇒   (𝜑 → (𝐹‘𝑀) < (𝐹‘𝑁))
 
30-Jul-2021resqrexlemnmsq 9615 Lemma for resqrex 9624. The difference between the squares of two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 30-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝑁 ≤ 𝑀)    ⇒   (𝜑 → (((𝐹‘𝑁)↑2) − ((𝐹‘𝑀)↑2)) < (((𝐹‘1)↑2) / (4↑(𝑁 − 1))))
 
30-Jul-2021divmuldivapd 7806 Multiplication of two ratios. (Contributed by Jim Kingdon, 30-Jul-2021.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐷 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    &   (𝜑 → 𝐷 # 0)    ⇒   (𝜑 → ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 · 𝐶) / (𝐵 · 𝐷)))
 
29-Jul-2021resqrexlemcalc3 9614 Lemma for resqrex 9624. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    ⇒   ((𝜑 ∧ 𝑁 ∈ ℕ) → (((𝐹‘𝑁)↑2) − 𝐴) ≤ (((𝐹‘1)↑2) / (4↑(𝑁 − 1))))
 
29-Jul-2021resqrexlemcalc2 9613 Lemma for resqrex 9624. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    ⇒   ((𝜑 ∧ 𝑁 ∈ ℕ) → (((𝐹‘(𝑁 + 1))↑2) − 𝐴) ≤ ((((𝐹‘𝑁)↑2) − 𝐴) / 4))
 
29-Jul-2021resqrexlemcalc1 9612 Lemma for resqrex 9624. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    ⇒   ((𝜑 ∧ 𝑁 ∈ ℕ) → (((𝐹‘(𝑁 + 1))↑2) − 𝐴) = (((((𝐹‘𝑁)↑2) − 𝐴)↑2) / (4 · ((𝐹‘𝑁)↑2))))
 
29-Jul-2021resqrexlemlo 9611 Lemma for resqrex 9624. A (variable) lower bound for each term of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    ⇒   ((𝜑 ∧ 𝑁 ∈ ℕ) → (1 / (2↑𝑁)) < (𝐹‘𝑁))
 
29-Jul-2021resqrexlemdec 9609 Lemma for resqrex 9624. The sequence is decreasing. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    ⇒   ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝐹‘(𝑁 + 1)) < (𝐹‘𝑁))
 
28-Jul-2021resqrexlemp1rp 9604 Lemma for resqrex 9624. Applying the recursion rule yields a positive real (expressed in a way that will help apply iseqf 9224 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    ⇒   ((𝜑 ∧ (𝐵 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+)) → (𝐵(𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2))𝐶) ∈ ℝ+)
 
28-Jul-2021resqrexlem1arp 9603 Lemma for resqrex 9624. 1 + 𝐴 is a positive real (expressed in a way that will help apply iseqf 9224 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    ⇒   ((𝜑 ∧ 𝑁 ∈ ℕ) → ((ℕ × {(1 + 𝐴)})‘𝑁) ∈ ℝ+)
 
28-Jul-2021rpap0d 8628 A positive real is apart from zero. (Contributed by Jim Kingdon, 28-Jul-2021.)
(𝜑 → 𝐴 ∈ ℝ+)    ⇒   (𝜑 → 𝐴 # 0)
 
27-Jul-2021resqrexlemex 9623 Lemma for resqrex 9624. Existence of square root given a sequence which converges to the square root. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    ⇒   (𝜑 → ∃𝑥 ∈ ℝ (0 ≤ 𝑥 ∧ (𝑥↑2) = 𝐴))
 
27-Jul-2021resqrexlemover 9608 Lemma for resqrex 9624. Each element of the sequence is an overestimate. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    ⇒   ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝐴 < ((𝐹‘𝑁)↑2))
 
27-Jul-2021resqrexlemfp1 9607 Lemma for resqrex 9624. Recursion rule. This sequence is the ancient method for computing square roots, often known as the babylonian method, although known to many ancient cultures. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    ⇒   ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝐹‘(𝑁 + 1)) = (((𝐹‘𝑁) + (𝐴 / (𝐹‘𝑁))) / 2))
 
27-Jul-2021resqrexlemf1 9606 Lemma for resqrex 9624. Initial value. Although this sequence converges to the square root with any positive initial value, this choice makes various steps in the proof of convergence easier. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    ⇒   (𝜑 → (𝐹‘1) = (1 + 𝐴))
 
27-Jul-2021resqrexlemf 9605 Lemma for resqrex 9624. The sequence is a function. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    ⇒   (𝜑 → 𝐹:ℕ⟶ℝ+)
 
23-Jul-2021iseqf 9224 Range of the recursive sequence builder. (Contributed by Jim Kingdon, 23-Jul-2021.)
𝑍 = (ℤ≥‘𝑀)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑀 ∈ ℤ)    &   ((𝜑 ∧ 𝑥 ∈ 𝑍) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   (𝜑 → seq𝑀( + , 𝐹, 𝑆):𝑍⟶𝑆)
 
22-Jul-2021ialgrlemconst 9882 Lemma for ialgr0 9883. Closure of a constant function, in a form suitable for theorems such as iseq1 9222 or iseqfn 9221. (Contributed by Jim Kingdon, 22-Jul-2021.)
𝑍 = (ℤ≥‘𝑀)    &   (𝜑 → 𝐴 ∈ 𝑆)    ⇒   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆)
 
22-Jul-2021ialgrlem1st 9881 Lemma for ialgr0 9883. Expressing algrflemg 5851 in a form suitable for theorems such as iseq1 9222 or iseqfn 9221. (Contributed by Jim Kingdon, 22-Jul-2021.)
(𝜑 → 𝐹:𝑆⟶𝑆)    ⇒   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆)
 
22-Jul-2021algrflemg 5851 Lemma for algrf and related theorems. (Contributed by Jim Kingdon, 22-Jul-2021.)
((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹‘𝐵))
 
21-Jul-2021zmod1congr 9183 Two arbitrary integers are congruent modulo 1, see example 4 in [ApostolNT] p. 107. (Contributed by AV, 21-Jul-2021.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 mod 1) = (𝐵 mod 1))
 
20-Jul-2021caucvgrelemcau 9579 Lemma for caucvgre 9580. Converting the Cauchy condition. (Contributed by Jim Kingdon, 20-Jul-2021.)
(𝜑 → 𝐹:ℕ⟶ℝ)    &   (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (1 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (1 / 𝑛))))    ⇒   (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ ℕ (𝑛 <ℝ 𝑘 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))
 
20-Jul-2021caucvgrelemrec 9578 Two ways to express a reciprocal. (Contributed by Jim Kingdon, 20-Jul-2021.)
((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (℩𝑟 ∈ ℝ (𝐴 · 𝑟) = 1) = (1 / 𝐴))
 
19-Jul-2021caucvgre 9580 Convergence of real sequences.

A Cauchy sequence (as defined here, which has a rate of convergence built in) of real numbers converges to a real number. Specifically on rate of convergence, all terms after the nth term must be within 1 / 𝑛 of the nth term.

(Contributed by Jim Kingdon, 19-Jul-2021.)

(𝜑 → 𝐹:ℕ⟶ℝ)    &   (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (1 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (1 / 𝑛))))    ⇒   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ≥‘𝑗)((𝐹‘𝑖) < (𝑦 + 𝑥) ∧ 𝑦 < ((𝐹‘𝑖) + 𝑥)))
 
19-Jul-2021ax-caucvg 7004 Completeness. Axiom for real and complex numbers, justified by theorem axcaucvg 6974.

A Cauchy sequence (as defined here, which has a rate convergence built in) of real numbers converges to a real number. Specifically on rate of convergence, all terms after the nth term must be within 1 / 𝑛 of the nth term.

This axiom should not be used directly; instead use caucvgre 9580 (which is the same, but stated in terms of the ℕ and 1 / 𝑛 notations). (Contributed by Jim Kingdon, 19-Jul-2021.) (New usage is discouraged.)

𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝜑 → 𝐹:𝑁⟶ℝ)    &   (𝜑 → ∀𝑛 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑛 <ℝ 𝑘 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))    ⇒   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ (0 <ℝ 𝑥 → ∃𝑗 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑗 <ℝ 𝑘 → ((𝐹‘𝑘) <ℝ (𝑦 + 𝑥) ∧ 𝑦 <ℝ ((𝐹‘𝑘) + 𝑥)))))
 
18-Jul-2021mulnqpr 6675 Multiplication of fractions embedded into positive reals. One can either multiply the fractions as fractions, or embed them into positive reals and multiply them as positive reals, and get the same result. (Contributed by Jim Kingdon, 18-Jul-2021.)
((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))
 
18-Jul-2021mulnqprlemfu 6674 Lemma for mulnqpr 6675. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ⊆ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)))
 
18-Jul-2021mulnqprlemfl 6673 Lemma for mulnqpr 6675. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ⊆ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)))
 
18-Jul-2021mulnqprlemru 6672 Lemma for mulnqpr 6675. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))
 
18-Jul-2021mulnqprlemrl 6671 Lemma for mulnqpr 6675. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))
 
18-Jul-2021lt2mulnq 6503 Ordering property of multiplication for positive fractions. (Contributed by Jim Kingdon, 18-Jul-2021.)
(((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ (𝐶 ∈ Q ∧ 𝐷 ∈ Q)) → ((𝐴 <Q 𝐵 ∧ 𝐶 <Q 𝐷) → (𝐴 ·Q 𝐶) <Q (𝐵 ·Q 𝐷)))
 
17-Jul-2021recidpirqlemcalc 6933 Lemma for recidpirq 6934. Rearranging some of the expressions. (Contributed by Jim Kingdon, 17-Jul-2021.)
(𝜑 → 𝐴 ∈ P)    &   (𝜑 → 𝐵 ∈ P)    &   (𝜑 → (𝐴 ·P 𝐵) = 1P)    ⇒   (𝜑 → ((((𝐴 +P 1P) ·P (𝐵 +P 1P)) +P (1P ·P 1P)) +P 1P) = ((((𝐴 +P 1P) ·P 1P) +P (1P ·P (𝐵 +P 1P))) +P (1P +P 1P)))
 
17-Jul-2021recidpipr 6932 Another way of saying that a number times its reciprocal is one. (Contributed by Jim Kingdon, 17-Jul-2021.)
(𝑁 ∈ N → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) = 1P)
 
15-Jul-2021rereceu 6963 The reciprocal from axprecex 6954 is unique. (Contributed by Jim Kingdon, 15-Jul-2021.)
((𝐴 ∈ ℝ ∧ 0 <ℝ 𝐴) → ∃!𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
 
15-Jul-2021recidpirq 6934 A real number times its reciprocal is one, where reciprocal is expressed with *Q. (Contributed by Jim Kingdon, 15-Jul-2021.)
(𝑁 ∈ N → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = 1)
 
15-Jul-2021recnnre 6927 Embedding the reciprocal of a natural number into ℝ. (Contributed by Jim Kingdon, 15-Jul-2021.)
(𝑁 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ℝ)
 
15-Jul-2021pitore 6926 Embedding from N to ℝ. Similar to pitonn 6924 but separate in the sense that we have not proved nnssre 7918 yet. (Contributed by Jim Kingdon, 15-Jul-2021.)
(𝑁 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ℝ)
 
15-Jul-2021pitoregt0 6925 Embedding from N to ℝ yields a number greater than zero. (Contributed by Jim Kingdon, 15-Jul-2021.)
(𝑁 ∈ N → 0 <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
 
14-Jul-2021adddivflid 9134 The floor of a sum of an integer and a fraction is equal to the integer iff the denominator of the fraction is less than the numerator. (Contributed by AV, 14-Jul-2021.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℕ) → (𝐵 < 𝐶 ↔ (⌊‘(𝐴 + (𝐵 / 𝐶))) = 𝐴))
 
14-Jul-2021nnindnn 6967 Principle of Mathematical Induction (inference schema). This is a counterpart to nnind 7930 designed for real number axioms which involve natural numbers (notably, axcaucvg 6974). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝑧 = 1 → (𝜑 ↔ 𝜓))    &   (𝑧 = 𝑘 → (𝜑 ↔ 𝜒))    &   (𝑧 = (𝑘 + 1) → (𝜑 ↔ 𝜃))    &   (𝑧 = 𝐴 → (𝜑 ↔ 𝜏))    &   𝜓    &   (𝑘 ∈ 𝑁 → (𝜒 → 𝜃))    ⇒   (𝐴 ∈ 𝑁 → 𝜏)
 
14-Jul-2021peano5nnnn 6966 Peano's inductive postulate. This is a counterpart to peano5nni 7917 designed for real number axioms which involve natural numbers (notably, axcaucvg 6974). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}    ⇒   ((1 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝑧 + 1) ∈ 𝐴) → 𝑁 ⊆ 𝐴)
 
14-Jul-2021peano2nnnn 6929 A successor of a positive integer is a positive integer. This is a counterpart to peano2nn 7926 designed for real number axioms which involve to natural numbers (notably, axcaucvg 6974). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}    ⇒   (𝐴 ∈ 𝑁 → (𝐴 + 1) ∈ 𝑁)
 
14-Jul-2021peano1nnnn 6928 One is an element of ℕ. This is a counterpart to 1nn 7925 designed for real number axioms which involve natural numbers (notably, axcaucvg 6974). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}    ⇒   1 ∈ 𝑁
 
14-Jul-2021addvalex 6920 Existence of a sum. This is dependent on how we define + so once we proceed to real number axioms we will replace it with theorems such as addcl 7006. (Contributed by Jim Kingdon, 14-Jul-2021.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 + 𝐵) ∈ V)
 
13-Jul-2021nntopi 6968 Mapping from ℕ to N. (Contributed by Jim Kingdon, 13-Jul-2021.)
𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}    ⇒   (𝐴 ∈ 𝑁 → ∃𝑧 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝐴)
 
13-Jul-2021ltrennb 6930 Ordering of natural numbers with <N or <ℝ. (Contributed by Jim Kingdon, 13-Jul-2021.)
((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <N 𝐾 ↔ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
 
12-Jul-2021ltrenn 6931 Ordering of natural numbers with <N or <ℝ. (Contributed by Jim Kingdon, 12-Jul-2021.)
(𝐽 <N 𝐾 → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
 
11-Jul-2021recriota 6964 Two ways to express the reciprocal of a natural number. (Contributed by Jim Kingdon, 11-Jul-2021.)
(𝑁 ∈ N → (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1) = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
 
11-Jul-2021elrealeu 6906 The real number mapping in elreal 6905 is unique. (Contributed by Jim Kingdon, 11-Jul-2021.)
(𝐴 ∈ ℝ ↔ ∃!𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
 
10-Jul-2021axcaucvglemres 6973 Lemma for axcaucvg 6974. Mapping the limit from N and R. (Contributed by Jim Kingdon, 10-Jul-2021.)
𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝜑 → 𝐹:𝑁⟶ℝ)    &   (𝜑 → ∀𝑛 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑛 <ℝ 𝑘 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))    &   𝐺 = (𝑗 ∈ N ↦ (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))    ⇒   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ (0 <ℝ 𝑥 → ∃𝑗 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑗 <ℝ 𝑘 → ((𝐹‘𝑘) <ℝ (𝑦 + 𝑥) ∧ 𝑦 <ℝ ((𝐹‘𝑘) + 𝑥)))))
 
10-Jul-2021axcaucvglemval 6971 Lemma for axcaucvg 6974. Value of sequence when mapping to N and R. (Contributed by Jim Kingdon, 10-Jul-2021.)
𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝜑 → 𝐹:𝑁⟶ℝ)    &   (𝜑 → ∀𝑛 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑛 <ℝ 𝑘 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))    &   𝐺 = (𝑗 ∈ N ↦ (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))    ⇒   ((𝜑 ∧ 𝐽 ∈ N) → (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺‘𝐽), 0R⟩)
 
10-Jul-2021axcaucvglemcl 6969 Lemma for axcaucvg 6974. Mapping to N and R. (Contributed by Jim Kingdon, 10-Jul-2021.)
𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝜑 → 𝐹:𝑁⟶ℝ)    ⇒   ((𝜑 ∧ 𝐽 ∈ N) → (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩) ∈ R)
 
9-Jul-2021axcaucvglemcau 6972 Lemma for axcaucvg 6974. The result of mapping to N and R satisfies the Cauchy condition. (Contributed by Jim Kingdon, 9-Jul-2021.)
𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝜑 → 𝐹:𝑁⟶ℝ)    &   (𝜑 → ∀𝑛 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑛 <ℝ 𝑘 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))    &   𝐺 = (𝑗 ∈ N ↦ (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))    ⇒   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐺‘𝑛) <R ((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺‘𝑘) <R ((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
 
9-Jul-2021axcaucvglemf 6970 Lemma for axcaucvg 6974. Mapping to N and R yields a sequence. (Contributed by Jim Kingdon, 9-Jul-2021.)
𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝜑 → 𝐹:𝑁⟶ℝ)    &   (𝜑 → ∀𝑛 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑛 <ℝ 𝑘 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))    &   𝐺 = (𝑗 ∈ N ↦ (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))    ⇒   (𝜑 → 𝐺:N⟶R)
 
8-Jul-2021fldiv4p1lem1div2 9147 The floor of an integer equal to 3 or greater than 4, increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.)
((𝑁 = 3 ∨ 𝑁 ∈ (ℤ≥‘5)) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
 
8-Jul-2021divfl0 9138 The floor of a fraction is 0 iff the denominator is less than the numerator. (Contributed by AV, 8-Jul-2021.)
((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (⌊‘(𝐴 / 𝐵)) = 0))
 
8-Jul-2021div4p1lem1div2 8177 An integer greater than 5, divided by 4 and increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.)
((𝑁 ∈ ℝ ∧ 6 ≤ 𝑁) → ((𝑁 / 4) + 1) ≤ ((𝑁 − 1) / 2))
 
8-Jul-2021axcaucvg 6974 Real number completeness axiom. A Cauchy sequence with a modulus of convergence converges. This is basically Corollary 11.2.13 of [HoTT], p. (varies). The HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within 1 / 𝑛 of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis).

Because we are stating this axiom before we have introduced notations for ℕ or division, we use 𝑁 for the natural numbers and express a reciprocal in terms of ℩.

This construction-dependent theorem should not be referenced directly; instead, use ax-caucvg 7004. (Contributed by Jim Kingdon, 8-Jul-2021.) (New usage is discouraged.)

𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}    &   (𝜑 → 𝐹:𝑁⟶ℝ)    &   (𝜑 → ∀𝑛 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑛 <ℝ 𝑘 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))    ⇒   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ (0 <ℝ 𝑥 → ∃𝑗 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑗 <ℝ 𝑘 → ((𝐹‘𝑘) <ℝ (𝑦 + 𝑥) ∧ 𝑦 <ℝ ((𝐹‘𝑘) + 𝑥)))))
 
7-Jul-2021ltadd1sr 6861 Adding one to a signed real yields a larger signed real. (Contributed by Jim Kingdon, 7-Jul-2021.)
(𝐴 ∈ R → 𝐴 <R (𝐴 +R 1R))
 
7-Jul-2021lteupri 6715 The difference from ltexpri 6711 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.)
(𝐴<P 𝐵 → ∃!𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵)
 
4-Jul-2021caucvgsrlemasr 6874 Lemma for caucvgsr 6886. The lower bound is a signed real. (Contributed by Jim Kingdon, 4-Jul-2021.)
(𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚))    ⇒   (𝜑 → 𝐴 ∈ R)
 
3-Jul-2021caucvgsrlemoffres 6884 Lemma for caucvgsr 6886. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
(𝜑 → 𝐹:N⟶R)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚))    &   𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R)))    ⇒   (𝜑 → ∃𝑦 ∈ R ∀𝑥 ∈ R (0R <R 𝑥 → ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → ((𝐹‘𝑘) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹‘𝑘) +R 𝑥)))))
 
3-Jul-2021caucvgsrlemoffgt1 6883 Lemma for caucvgsr 6886. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
(𝜑 → 𝐹:N⟶R)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚))    &   𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R)))    ⇒   (𝜑 → ∀𝑚 ∈ N 1R <R (𝐺‘𝑚))
 
3-Jul-2021caucvgsrlemoffcau 6882 Lemma for caucvgsr 6886. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
(𝜑 → 𝐹:N⟶R)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚))    &   𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R)))    ⇒   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐺‘𝑛) <R ((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺‘𝑘) <R ((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
 
3-Jul-2021caucvgsrlemofff 6881 Lemma for caucvgsr 6886. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
(𝜑 → 𝐹:N⟶R)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚))    &   𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R)))    ⇒   (𝜑 → 𝐺:N⟶R)
 
3-Jul-2021caucvgsrlemoffval 6880 Lemma for caucvgsr 6886. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
(𝜑 → 𝐹:N⟶R)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚))    &   𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R)))    ⇒   ((𝜑 ∧ 𝐽 ∈ N) → ((𝐺‘𝐽) +R 𝐴) = ((𝐹‘𝐽) +R 1R))
 
2-Jul-2021caucvgsrlemcl 6873 Lemma for caucvgsr 6886. Terms of the sequence from caucvgsrlemgt1 6879 can be mapped to positive reals. (Contributed by Jim Kingdon, 2-Jul-2021.)
(𝜑 → 𝐹:N⟶R)    &   (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚))    ⇒   ((𝜑 ∧ 𝐴 ∈ N) → (℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) ∈ P)
 
29-Jun-2021ledivge1le 8652 If a number is less than or equal to another number, the number divided by a positive number greater than or equal to one is less than or equal to the other number. (Contributed by AV, 29-Jun-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ∧ (𝐶 ∈ ℝ+ ∧ 1 ≤ 𝐶)) → (𝐴 ≤ 𝐵 → (𝐴 / 𝐶) ≤ 𝐵))
 
29-Jun-2021divle1le 8651 A real number divided by a positive real number is less than or equal to 1 iff the real number is less than or equal to the positive real number. (Contributed by AV, 29-Jun-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 / 𝐵) ≤ 1 ↔ 𝐴 ≤ 𝐵))
 
29-Jun-2021caucvgsrlemfv 6875 Lemma for caucvgsr 6886. Coercing sequence value from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)
(𝜑 → 𝐹:N⟶R)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚))    &   𝐺 = (𝑥 ∈ N ↦ (℩𝑦 ∈ P (𝐹‘𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))    ⇒   ((𝜑 ∧ 𝐴 ∈ N) → [⟨((𝐺‘𝐴) +P 1P), 1P⟩] ~R = (𝐹‘𝐴))
 
29-Jun-2021prsrriota 6872 Mapping a restricted iota from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)
((𝐴 ∈ R ∧ 0R <R 𝐴) → [⟨((℩𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴) +P 1P), 1P⟩] ~R = 𝐴)
 
29-Jun-2021prsrlt 6871 Mapping from positive real ordering to signed real ordering. (Contributed by Jim Kingdon, 29-Jun-2021.)
((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ [⟨(𝐴 +P 1P), 1P⟩] ~R <R [⟨(𝐵 +P 1P), 1P⟩] ~R ))
 
29-Jun-2021prsradd 6870 Mapping from positive real addition to signed real addition. (Contributed by Jim Kingdon, 29-Jun-2021.)
((𝐴 ∈ P ∧ 𝐵 ∈ P) → [⟨((𝐴 +P 𝐵) +P 1P), 1P⟩] ~R = ([⟨(𝐴 +P 1P), 1P⟩] ~R +R [⟨(𝐵 +P 1P), 1P⟩] ~R ))
 
25-Jun-2021caucvgsrlembound 6878 Lemma for caucvgsr 6886. Defining the boundedness condition in terms of positive reals. (Contributed by Jim Kingdon, 25-Jun-2021.)
(𝜑 → 𝐹:N⟶R)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚))    &   𝐺 = (𝑥 ∈ N ↦ (℩𝑦 ∈ P (𝐹‘𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))    ⇒   (𝜑 → ∀𝑚 ∈ N 1P<P (𝐺‘𝑚))
 
25-Jun-2021prsrpos 6869 Mapping from a positive real to a signed real yields a result greater than zero. (Contributed by Jim Kingdon, 25-Jun-2021.)
(𝐴 ∈ P → 0R <R [⟨(𝐴 +P 1P), 1P⟩] ~R )
 
25-Jun-2021prsrcl 6868 Mapping from a positive real to a signed real. (Contributed by Jim Kingdon, 25-Jun-2021.)
(𝐴 ∈ P → [⟨(𝐴 +P 1P), 1P⟩] ~R ∈ R)
 
25-Jun-2021srpospr 6867 Mapping from a signed real greater than zero to a positive real. (Contributed by Jim Kingdon, 25-Jun-2021.)
((𝐴 ∈ R ∧ 0R <R 𝐴) → ∃!𝑥 ∈ P [⟨(𝑥 +P 1P), 1P⟩] ~R = 𝐴)
 
23-Jun-2021caucvgsrlemcau 6877 Lemma for caucvgsr 6886. Defining the Cauchy condition in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.)
(𝜑 → 𝐹:N⟶R)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚))    &   𝐺 = (𝑥 ∈ N ↦ (℩𝑦 ∈ P (𝐹‘𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))    ⇒   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐺‘𝑛)<P ((𝐺‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐺‘𝑘)<P ((𝐺‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
 
23-Jun-2021caucvgsrlemf 6876 Lemma for caucvgsr 6886. Defining the sequence in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.)
(𝜑 → 𝐹:N⟶R)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚))    &   𝐺 = (𝑥 ∈ N ↦ (℩𝑦 ∈ P (𝐹‘𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))    ⇒   (𝜑 → 𝐺:N⟶P)
 
22-Jun-2021caucvgsrlemgt1 6879 Lemma for caucvgsr 6886. A Cauchy sequence whose terms are greater than one converges. (Contributed by Jim Kingdon, 22-Jun-2021.)
(𝜑 → 𝐹:N⟶R)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚))    ⇒   (𝜑 → ∃𝑦 ∈ R ∀𝑥 ∈ R (0R <R 𝑥 → ∃𝑗 ∈ N ∀𝑖 ∈ N (𝑗 <N 𝑖 → ((𝐹‘𝑖) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹‘𝑖) +R 𝑥)))))
 
20-Jun-2021caucvgsr 6886 A Cauchy sequence of signed reals with a modulus of convergence converges to a signed real. This is basically Corollary 11.2.13 of [HoTT], p. (varies). The HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within 1 / 𝑛 of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis).

This is similar to caucvgprpr 6810 but is for signed reals rather than positive reals.

Here is an outline of how we prove it:

1. Choose a lower bound for the sequence (see caucvgsrlembnd 6885).

2. Offset each element of the sequence so that each element of the resulting sequence is greater than one (greater than zero would not suffice, because the limit as well as the elements of the sequence need to be positive) (see caucvgsrlemofff 6881).

3. Since a signed real (element of R) which is greater than zero can be mapped to a positive real (element of P), perform that mapping on each element of the sequence and invoke caucvgprpr 6810 to get a limit (see caucvgsrlemgt1 6879).

4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 6879).

5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 6884). (Contributed by Jim Kingdon, 20-Jun-2021.)

(𝜑 → 𝐹:N⟶R)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    ⇒   (𝜑 → ∃𝑦 ∈ R ∀𝑥 ∈ R (0R <R 𝑥 → ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → ((𝐹‘𝑘) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹‘𝑘) +R 𝑥)))))
 
19-Jun-20212tnp1ge0ge0 9143 Two times an integer plus one is not negative iff the integer is not negative. (Contributed by AV, 19-Jun-2021.)
(𝑁 ∈ ℤ → (0 ≤ ((2 · 𝑁) + 1) ↔ 0 ≤ 𝑁))
 
19-Jun-2021caucvgsrlembnd 6885 Lemma for caucvgsr 6886. A Cauchy sequence with a lower bound converges. (Contributed by Jim Kingdon, 19-Jun-2021.)
(𝜑 → 𝐹:N⟶R)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚))    ⇒   (𝜑 → ∃𝑦 ∈ R ∀𝑥 ∈ R (0R <R 𝑥 → ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → ((𝐹‘𝑘) <R (𝑦 +R 𝑥) ∧ 𝑦 <R ((𝐹‘𝑘) +R 𝑥)))))
 
19-Jun-2021caucvgprprlemclphr 6803 Lemma for caucvgprpr 6810. The putative limit is a positive real. Like caucvgprprlemcl 6802 but without a distinct variable constraint between 𝜑 and 𝑟. (Contributed by Jim Kingdon, 19-Jun-2021.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩    ⇒   (𝜑 → 𝐿 ∈ P)
 
19-Jun-2021ltnqpr 6691 We can order fractions via <Q or <P. (Contributed by Jim Kingdon, 19-Jun-2021.)
((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))
 
17-Jun-2021caucvgprprlemnbj 6791 Lemma for caucvgprpr 6810. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 17-Jun-2021.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → 𝐵 ∈ N)    &   (𝜑 → 𝐽 ∈ N)    ⇒   (𝜑 → ¬ (((𝐹‘𝐵) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P (𝐹‘𝐽))
 
16-Jun-2021caucvgprprlemexbt 6804 Lemma for caucvgprpr 6810. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 16-Jun-2021.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩    &   (𝜑 → 𝑄 ∈ Q)    &   (𝜑 → 𝑇 ∈ P)    &   (𝜑 → (𝐿 +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)<P 𝑇)    ⇒   (𝜑 → ∃𝑏 ∈ N (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q 𝑄}, {𝑞 ∣ 𝑄 <Q 𝑞}⟩)<P 𝑇)
 
16-Jun-2021prplnqu 6718 Membership in the upper cut of a sum of a positive real and a fraction. (Contributed by Jim Kingdon, 16-Jun-2021.)
(𝜑 → 𝑋 ∈ P)    &   (𝜑 → 𝑄 ∈ Q)    &   (𝜑 → 𝐴 ∈ (2nd ‘(𝑋 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)))    ⇒   (𝜑 → ∃𝑦 ∈ (2nd ‘𝑋)(𝑦 +Q 𝑄) = 𝐴)
 
15-Jun-2021caucvgprprlemexb 6805 Lemma for caucvgprpr 6810. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 15-Jun-2021.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩    &   (𝜑 → 𝑄 ∈ P)    &   (𝜑 → 𝑅 ∈ N)    ⇒   (𝜑 → (((𝐿 +P 𝑄) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹‘𝑅) +P 𝑄) → ∃𝑏 ∈ N (((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P (𝑄 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑅, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑅, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))<P ((𝐹‘𝑅) +P 𝑄)))
 
5-Jun-2021caucvgprprlemaddq 6806 Lemma for caucvgprpr 6810. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 5-Jun-2021.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩    &   (𝜑 → 𝑋 ∈ P)    &   (𝜑 → 𝑄 ∈ P)    &   (𝜑 → ∃𝑟 ∈ N (𝑋 +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹‘𝑟) +P 𝑄))    ⇒   (𝜑 → 𝑋<P (𝐿 +P 𝑄))
 
27-Mar-2021notnotd 560 Deduction associated with notnot 559 and notnoti 574. (Contributed by Jarvin Udandy, 2-Sep-2016.) Avoid biconditional. (Revised by Wolf Lammen, 27-Mar-2021.)
(𝜑 → 𝜓)    ⇒   (𝜑 → ¬ ¬ 𝜓)
 
20-Mar-2021subfzo0 9097 The difference between two elements in a half-open range of nonnegative integers is greater than the negation of the upper bound and less than the upper bound of the range. (Contributed by AV, 20-Mar-2021.)
((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁)) → (-𝑁 < (𝐼 − 𝐽) ∧ (𝐼 − 𝐽) < 𝑁))
 
3-Mar-2021caucvgprprlemval 6786 Lemma for caucvgprpr 6810. Cauchy condition expressed in terms of classes. (Contributed by Jim Kingdon, 3-Mar-2021.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    ⇒   ((𝜑 ∧ 𝐴 <N 𝐵) → ((𝐹‘𝐴)<P ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐴, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ (𝐹‘𝐵)<P ((𝐹‘𝐴) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐴, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)))
 
27-Feb-2021recnnpr 6646 The reciprocal of a positive integer, as a positive real. (Contributed by Jim Kingdon, 27-Feb-2021.)
(𝐴 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
 
12-Feb-2021caucvgprprlemnjltk 6789 Lemma for caucvgprpr 6810. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → 𝐾 ∈ N)    &   (𝜑 → 𝐽 ∈ N)    &   (𝜑 → 𝑆 ∈ Q)    ⇒   ((𝜑 ∧ 𝐽 <N 𝐾) → ¬ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩))
 
12-Feb-2021caucvgprprlemnkeqj 6788 Lemma for caucvgprpr 6810. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → 𝐾 ∈ N)    &   (𝜑 → 𝐽 ∈ N)    &   (𝜑 → 𝑆 ∈ Q)    ⇒   ((𝜑 ∧ 𝐾 = 𝐽) → ¬ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩))
 
12-Feb-2021caucvgprprlemnkltj 6787 Lemma for caucvgprpr 6810. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → 𝐾 ∈ N)    &   (𝜑 → 𝐽 ∈ N)    &   (𝜑 → 𝑆 ∈ Q)    ⇒   ((𝜑 ∧ 𝐾 <N 𝐽) → ¬ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩))
 
12-Feb-2021caucvgprprlemcbv 6785 Lemma for caucvgprpr 6810. Change bound variables in Cauchy condition. (Contributed by Jim Kingdon, 12-Feb-2021.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    ⇒   (𝜑 → ∀𝑎 ∈ N ∀𝑏 ∈ N (𝑎 <N 𝑏 → ((𝐹‘𝑎)<P ((𝐹‘𝑏) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑏)<P ((𝐹‘𝑎) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
 
8-Feb-2021caucvgprprlemloccalc 6782 Lemma for caucvgprpr 6810. Rearranging some expressions for caucvgprprlemloc 6801. (Contributed by Jim Kingdon, 8-Feb-2021.)
(𝜑 → 𝑆 <Q 𝑇)    &   (𝜑 → 𝑌 ∈ Q)    &   (𝜑 → (𝑆 +Q 𝑌) = 𝑇)    &   (𝜑 → 𝑋 ∈ Q)    &   (𝜑 → (𝑋 +Q 𝑋) <Q 𝑌)    &   (𝜑 → 𝑀 ∈ N)    &   (𝜑 → (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑋)    ⇒   (𝜑 → (⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ⟨{𝑙 ∣ 𝑙 <Q 𝑇}, {𝑢 ∣ 𝑇 <Q 𝑢}⟩)
 
5-Feb-2021fvinim0ffz 9096 The function values for the borders of a finite interval of integers, which is the domain of the function, are not in the image of the interior of the interval iff the intersection of the images of the interior and the borders is empty. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 5-Feb-2021.)
((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ ↔ ((𝐹‘0) ∉ (𝐹 “ (1..^𝐾)) ∧ (𝐹‘𝐾) ∉ (𝐹 “ (1..^𝐾)))))
 
28-Jan-2021caucvgprprlemelu 6784 Lemma for caucvgprpr 6810. Membership in the upper cut of the putative limit. (Contributed by Jim Kingdon, 28-Jan-2021.)
𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩    ⇒   (𝑋 ∈ (2nd ‘𝐿) ↔ (𝑋 ∈ Q ∧ ∃𝑏 ∈ N ((𝐹‘𝑏) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑋}, {𝑞 ∣ 𝑋 <Q 𝑞}⟩))
 
21-Jan-2021caucvgprprlemell 6783 Lemma for caucvgprpr 6810. Membership in the lower cut of the putative limit. (Contributed by Jim Kingdon, 21-Jan-2021.)
𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩    ⇒   (𝑋 ∈ (1st ‘𝐿) ↔ (𝑋 ∈ Q ∧ ∃𝑏 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)))
 
20-Jan-2021caucvgprprlemnkj 6790 Lemma for caucvgprpr 6810. Part of disjointness. (Contributed by Jim Kingdon, 20-Jan-2021.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → 𝐾 ∈ N)    &   (𝜑 → 𝐽 ∈ N)    &   (𝜑 → 𝑆 ∈ Q)    ⇒   (𝜑 → ¬ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩))
 
8-Jan-2021ltnqpri 6692 We can order fractions via <Q or <P. (Contributed by Jim Kingdon, 8-Jan-2021.)
(𝐴 <Q 𝐵 → ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)
 
29-Dec-2020caucvgprprlemmu 6793 Lemma for caucvgprpr 6810. The upper cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩    ⇒   (𝜑 → ∃𝑡 ∈ Q 𝑡 ∈ (2nd ‘𝐿))
 
29-Dec-2020caucvgprprlemml 6792 Lemma for caucvgprpr 6810. The lower cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩    ⇒   (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿))
 
21-Dec-2020caucvgprprlemloc 6801 Lemma for caucvgprpr 6810. The putative limit is located. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩    ⇒   (𝜑 → ∀𝑠 ∈ Q ∀𝑡 ∈ Q (𝑠 <Q 𝑡 → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑡 ∈ (2nd ‘𝐿))))
 
21-Dec-2020caucvgprprlemdisj 6800 Lemma for caucvgprpr 6810. The putative limit is disjoint. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩    ⇒   (𝜑 → ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)))
 
21-Dec-2020caucvgprprlemrnd 6799 Lemma for caucvgprpr 6810. The putative limit is rounded. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩    ⇒   (𝜑 → (∀𝑠 ∈ Q (𝑠 ∈ (1st ‘𝐿) ↔ ∃𝑡 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿))) ∧ ∀𝑡 ∈ Q (𝑡 ∈ (2nd ‘𝐿) ↔ ∃𝑠 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)))))
 
21-Dec-2020caucvgprprlemupu 6798 Lemma for caucvgprpr 6810. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩    ⇒   ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)) → 𝑡 ∈ (2nd ‘𝐿))
 
21-Dec-2020caucvgprprlemopu 6797 Lemma for caucvgprpr 6810. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩    ⇒   ((𝜑 ∧ 𝑡 ∈ (2nd ‘𝐿)) → ∃𝑠 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑠 ∈ (2nd ‘𝐿)))
 
21-Dec-2020caucvgprprlemlol 6796 Lemma for caucvgprpr 6810. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩    ⇒   ((𝜑 ∧ 𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)) → 𝑠 ∈ (1st ‘𝐿))
 
21-Dec-2020caucvgprprlemopl 6795 Lemma for caucvgprpr 6810. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩    ⇒   ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑡 ∈ Q (𝑠 <Q 𝑡 ∧ 𝑡 ∈ (1st ‘𝐿)))
 
21-Dec-2020caucvgprprlemm 6794 Lemma for caucvgprpr 6810. The putative limit is inhabited. (Contributed by Jim Kingdon, 21-Dec-2020.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩    ⇒   (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑡 ∈ Q 𝑡 ∈ (2nd ‘𝐿)))
 
29-Nov-2020nqpru 6650 Comparing a fraction to a real can be done by whether it is an element of the upper cut, or by <P. (Contributed by Jim Kingdon, 29-Nov-2020.)
((𝐴 ∈ Q ∧ 𝐵 ∈ P) → (𝐴 ∈ (2nd ‘𝐵) ↔ 𝐵<P ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩))
 
28-Nov-2020caucvgprprlemk 6781 Lemma for caucvgprpr 6810. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 28-Nov-2020.)
(𝜑 → 𝐽 <N 𝐾)    &   (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)    ⇒   (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
 
25-Nov-2020caucvgprprlem2 6808 Lemma for caucvgprpr 6810. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩    &   (𝜑 → 𝑄 ∈ P)    &   (𝜑 → 𝐽 <N 𝐾)    &   (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)    ⇒   (𝜑 → 𝐿<P ((𝐹‘𝐾) +P 𝑄))
 
25-Nov-2020caucvgprprlem1 6807 Lemma for caucvgprpr 6810. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩    &   (𝜑 → 𝑄 ∈ P)    &   (𝜑 → 𝐽 <N 𝐾)    &   (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)    ⇒   (𝜑 → (𝐹‘𝐾)<P (𝐿 +P 𝑄))
 
25-Nov-2020archrecpr 6762 Archimedean principle for positive reals (reciprocal version). (Contributed by Jim Kingdon, 25-Nov-2020.)
(𝐴 ∈ P → ∃𝑗 ∈ N ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝑢}⟩<P 𝐴)
 
21-Nov-2020caucvgprprlemlim 6809 Lemma for caucvgprpr 6810. The putative limit is a limit. (Contributed by Jim Kingdon, 21-Nov-2020.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩    ⇒   (𝜑 → ∀𝑥 ∈ P ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → ((𝐹‘𝑘)<P (𝐿 +P 𝑥) ∧ 𝐿<P ((𝐹‘𝑘) +P 𝑥))))
 
21-Nov-2020caucvgprprlemcl 6802 Lemma for caucvgprpr 6810. The putative limit is a positive real. (Contributed by Jim Kingdon, 21-Nov-2020.)
(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩    ⇒   (𝜑 → 𝐿 ∈ P)
 
17-Nov-2020pm3.2 126 Join antecedents with conjunction. Theorem *3.2 of [WhiteheadRussell] p. 111. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.) (Proof shortened by Jia Ming, 17-Nov-2020.)
(𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
 
14-Nov-2020caucvgprpr 6810 A Cauchy sequence of positive reals with a modulus of convergence converges to a positive real. This is basically Corollary 11.2.13 of [HoTT], p. (varies) (one key difference being that this is for positive reals rather than signed reals). Also, the HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within 1 / 𝑛 of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis). We also specify that every term needs to be larger than a given value 𝐴, to avoid the case where we have positive terms which "converge" to zero (which is not a positive real).

This is similar to caucvgpr 6780 except that values of the sequence are positive reals rather than positive fractions. Reading that proof first (or cauappcvgpr 6760) might help in understanding this one, as they are slightly simpler but similarly structured. (Contributed by Jim Kingdon, 14-Nov-2020.)

(𝜑 → 𝐹:N⟶P)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))    &   (𝜑 → ∀𝑚 ∈ N 𝐴<P (𝐹‘𝑚))    ⇒   (𝜑 → ∃𝑦 ∈ P ∀𝑥 ∈ P ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → ((𝐹‘𝑘)<P (𝑦 +P 𝑥) ∧ 𝑦<P ((𝐹‘𝑘) +P 𝑥))))
 
27-Oct-2020bj-omssonALT 10088 Alternate proof of bj-omsson 10087. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
ω ⊆ On
 
27-Oct-2020bj-omsson 10087 Constructive proof of omsson 4335. See also bj-omssonALT 10088. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.
ω ⊆ On
 
27-Oct-2020bj-nnelon 10084 A natural number is an ordinal. Constructive proof of nnon 4332. Can also be proved from bj-omssonALT 10088. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
(𝐴 ∈ ω → 𝐴 ∈ On)
 
27-Oct-2020bj-nnord 10083 A natural number is an ordinal. Constructive proof of nnord 4334. Can also be proved from bj-nnelon 10084 if the latter is proved from bj-omssonALT 10088. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.)
(𝐴 ∈ ω → Ord 𝐴)
 
27-Oct-2020bj-axempty2 10014 Axiom of the empty set from bounded separation, alternate version to bj-axempty 10013. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3883 instead. (New usage is discouraged.)
∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
 
25-Oct-2020bj-indind 10056 If 𝐴 is inductive and 𝐵 is "inductive in 𝐴", then (𝐴 ∩ 𝐵) is inductive. (Contributed by BJ, 25-Oct-2020.)
((Ind 𝐴 ∧ (∅ ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵))) → Ind (𝐴 ∩ 𝐵))
 
25-Oct-2020bj-axempty 10013 Axiom of the empty set from bounded separation. It is provable from bounded separation since the intuitionistic FOL used in iset.mm assumes a non-empty universe. See axnul 3882. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3883 instead. (New usage is discouraged.)
∃𝑥∀𝑦 ∈ 𝑥 ⊥
 
25-Oct-2020bj-axemptylem 10012 Lemma for bj-axempty 10013 and bj-axempty2 10014. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3883 instead. (New usage is discouraged.)
∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥)
 
23-Oct-2020caucvgprlemnkj 6764 Lemma for caucvgpr 6780. Part of disjointness. (Contributed by Jim Kingdon, 23-Oct-2020.)
(𝜑 → 𝐹:N⟶Q)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → 𝐾 ∈ N)    &   (𝜑 → 𝐽 ∈ N)    &   (𝜑 → 𝑆 ∈ Q)    ⇒   (𝜑 → ¬ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆))
 
20-Oct-2020caucvgprlemupu 6770 Lemma for caucvgpr 6780. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 20-Oct-2020.)
(𝜑 → 𝐹:N⟶Q)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩    ⇒   ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)) → 𝑟 ∈ (2nd ‘𝐿))
 
20-Oct-2020caucvgprlemopu 6769 Lemma for caucvgpr 6780. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.)
(𝜑 → 𝐹:N⟶Q)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩    ⇒   ((𝜑 ∧ 𝑟 ∈ (2nd ‘𝐿)) → ∃𝑠 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)))
 
20-Oct-2020caucvgprlemlol 6768 Lemma for caucvgpr 6780. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 20-Oct-2020.)
(𝜑 → 𝐹:N⟶Q)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩    ⇒   ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) → 𝑠 ∈ (1st ‘𝐿))
 
20-Oct-2020caucvgprlemopl 6767 Lemma for caucvgpr 6780. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.)
(𝜑 → 𝐹:N⟶Q)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩    ⇒   ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)))
 
18-Oct-2020caucvgprlemnbj 6765 Lemma for caucvgpr 6780. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 18-Oct-2020.)
(𝜑 → 𝐹:N⟶Q)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → 𝐵 ∈ N)    &   (𝜑 → 𝐽 ∈ N)    ⇒   (𝜑 → ¬ (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐽))
 
9-Oct-2020caucvgprlemladdfu 6775 Lemma for caucvgpr 6780. Adding 𝑆 after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 9-Oct-2020.)
(𝜑 → 𝐹:N⟶Q)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩    &   (𝜑 → 𝑆 ∈ Q)    ⇒   (𝜑 → (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)) ⊆ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) +Q 𝑆) <Q 𝑢})
 
9-Oct-2020caucvgprlemk 6763 Lemma for caucvgpr 6780. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 9-Oct-2020.)
(𝜑 → 𝐽 <N 𝐾)    &   (𝜑 → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑄)    ⇒   (𝜑 → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑄)
 
8-Oct-2020caucvgprlemladdrl 6776 Lemma for caucvgpr 6780. Adding 𝑆 after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 8-Oct-2020.)
(𝜑 → 𝐹:N⟶Q)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩    &   (𝜑 → 𝑆 ∈ Q)    ⇒   (𝜑 → {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q ((𝐹‘𝑗) +Q 𝑆)} ⊆ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)))
 
3-Oct-2020caucvgprlem2 6778 Lemma for caucvgpr 6780. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.)
(𝜑 → 𝐹:N⟶Q)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩    &   (𝜑 → 𝑄 ∈ Q)    &   (𝜑 → 𝐽 <N 𝐾)    &   (𝜑 → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑄)    ⇒   (𝜑 → 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹‘𝐾) +Q 𝑄) <Q 𝑢}⟩)
 
3-Oct-2020caucvgprlem1 6777 Lemma for caucvgpr 6780. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.)
(𝜑 → 𝐹:N⟶Q)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩    &   (𝜑 → 𝑄 ∈ Q)    &   (𝜑 → 𝐽 <N 𝐾)    &   (𝜑 → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑄)    ⇒   (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝐾)}, {𝑢 ∣ (𝐹‘𝐾) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩))
 
3-Oct-2020ltnnnq 6521 Ordering of positive integers via <N or <Q is equivalent. (Contributed by Jim Kingdon, 3-Oct-2020.)
((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ [⟨𝐴, 1𝑜⟩] ~Q <Q [⟨𝐵, 1𝑜⟩] ~Q ))
 
1-Oct-2020caucvgprlemlim 6779 Lemma for caucvgpr 6780. The putative limit is a limit. (Contributed by Jim Kingdon, 1-Oct-2020.)
(𝜑 → 𝐹:N⟶Q)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩    ⇒   (𝜑 → ∀𝑥 ∈ Q ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩)))
 
27-Sep-2020caucvgprlemloc 6773 Lemma for caucvgpr 6780. The putative limit is located. (Contributed by Jim Kingdon, 27-Sep-2020.)
(𝜑 → 𝐹:N⟶Q)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩    ⇒   (𝜑 → ∀𝑠 ∈ Q ∀𝑟 ∈ Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿))))
 
27-Sep-2020caucvgprlemdisj 6772 Lemma for caucvgpr 6780. The putative limit is disjoint. (Contributed by Jim Kingdon, 27-Sep-2020.)
(𝜑 → 𝐹:N⟶Q)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩    ⇒   (𝜑 → ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)))
 
27-Sep-2020caucvgprlemrnd 6771 Lemma for caucvgpr 6780. The putative limit is rounded. (Contributed by Jim Kingdon, 27-Sep-2020.)
(𝜑 → 𝐹:N⟶Q)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩    ⇒   (𝜑 → (∀𝑠 ∈ Q (𝑠 ∈ (1st ‘𝐿) ↔ ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐿) ↔ ∃𝑠 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)))))
 
27-Sep-2020caucvgprlemm 6766 Lemma for caucvgpr 6780. The putative limit is inhabited. (Contributed by Jim Kingdon, 27-Sep-2020.)
(𝜑 → 𝐹:N⟶Q)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩    ⇒   (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿)))
 
27-Sep-2020archrecnq 6761 Archimedean principle for fractions (reciprocal version). (Contributed by Jim Kingdon, 27-Sep-2020.)
(𝐴 ∈ Q → ∃𝑗 ∈ N (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) <Q 𝐴)
 
26-Sep-2020caucvgprlemcl 6774 Lemma for caucvgpr 6780. The putative limit is a positive real. (Contributed by Jim Kingdon, 26-Sep-2020.)
(𝜑 → 𝐹:N⟶Q)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩    ⇒   (𝜑 → 𝐿 ∈ P)
 
26-Aug-2020sqrt0rlem 9601 Lemma for sqrt0 9602. (Contributed by Jim Kingdon, 26-Aug-2020.)
((𝐴 ∈ ℝ ∧ ((𝐴↑2) = 0 ∧ 0 ≤ 𝐴)) ↔ 𝐴 = 0)
 
23-Aug-2020sqrtrval 9598 Value of square root function. (Contributed by Jim Kingdon, 23-Aug-2020.)
(𝐴 ∈ ℝ → (√‘𝐴) = (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)))
 
23-Aug-2020df-rsqrt 9596 Define a function whose value is the square root of a nonnegative real number.

Defining the square root for complex numbers has one difficult part: choosing between the two roots. The usual way to define a principal square root for all complex numbers relies on excluded middle or something similar. But in the case of a nonnegative real number, we don't have the complications presented for general complex numbers, and we can choose the nonnegative root.

(Contributed by Jim Kingdon, 23-Aug-2020.)

√ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℝ ((𝑦↑2) = 𝑥 ∧ 0 ≤ 𝑦)))
 
19-Aug-2020addnqpr 6659 Addition of fractions embedded into positive reals. One can either add the fractions as fractions, or embed them into positive reals and add them as positive reals, and get the same result. (Contributed by Jim Kingdon, 19-Aug-2020.)
((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))
 
19-Aug-2020addnqprlemfu 6658 Lemma for addnqpr 6659. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ⊆ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)))
 
19-Aug-2020addnqprlemfl 6657 Lemma for addnqpr 6659. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ⊆ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)))
 
19-Aug-2020addnqprlemru 6656 Lemma for addnqpr 6659. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))
 
19-Aug-2020addnqprlemrl 6655 Lemma for addnqpr 6659. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))
 
15-Aug-2020caucvgprlemcanl 6742 Lemma for cauappcvgprlemladdrl 6755. Cancelling a term from both sides. (Contributed by Jim Kingdon, 15-Aug-2020.)
(𝜑 → 𝐿 ∈ P)    &   (𝜑 → 𝑆 ∈ Q)    &   (𝜑 → 𝑅 ∈ Q)    &   (𝜑 → 𝑄 ∈ Q)    ⇒   (𝜑 → ((𝑅 +Q 𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩)) ↔ 𝑅 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))))
 
4-Aug-2020cauappcvgprlemupu 6747 Lemma for cauappcvgpr 6760. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 4-Aug-2020.)
(𝜑 → 𝐹:Q⟶Q)    &   (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩    ⇒   ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)) → 𝑟 ∈ (2nd ‘𝐿))
 
4-Aug-2020cauappcvgprlemopu 6746 Lemma for cauappcvgpr 6760. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.)
(𝜑 → 𝐹:Q⟶Q)    &   (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩    ⇒   ((𝜑 ∧ 𝑟 ∈ (2nd ‘𝐿)) → ∃𝑠 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)))
 
4-Aug-2020cauappcvgprlemlol 6745 Lemma for cauappcvgpr 6760. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 4-Aug-2020.)
(𝜑 → 𝐹:Q⟶Q)    &   (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩    ⇒   ((𝜑 ∧ 𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)) → 𝑠 ∈ (1st ‘𝐿))
 
4-Aug-2020cauappcvgprlemopl 6744 Lemma for cauappcvgpr 6760. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.)
(𝜑 → 𝐹:Q⟶Q)    &   (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩    ⇒   ((𝜑 ∧ 𝑠 ∈ (1st ‘𝐿)) → ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿)))
 
18-Jul-2020cauappcvgprlemloc 6750 Lemma for cauappcvgpr 6760. The putative limit is located. (Contributed by Jim Kingdon, 18-Jul-2020.)
(𝜑 → 𝐹:Q⟶Q)    &   (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩    ⇒   (𝜑 → ∀𝑠 ∈ Q ∀𝑟 ∈ Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿))))
 
18-Jul-2020cauappcvgprlemdisj 6749 Lemma for cauappcvgpr 6760. The putative limit is disjoint. (Contributed by Jim Kingdon, 18-Jul-2020.)
(𝜑 → 𝐹:Q⟶Q)    &   (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩    ⇒   (𝜑 → ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)))
 
18-Jul-2020cauappcvgprlemrnd 6748 Lemma for cauappcvgpr 6760. The putative limit is rounded. (Contributed by Jim Kingdon, 18-Jul-2020.)
(𝜑 → 𝐹:Q⟶Q)    &   (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩    ⇒   (𝜑 → (∀𝑠 ∈ Q (𝑠 ∈ (1st ‘𝐿) ↔ ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐿) ↔ ∃𝑠 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)))))
 
18-Jul-2020cauappcvgprlemm 6743 Lemma for cauappcvgpr 6760. The putative limit is inhabited. (Contributed by Jim Kingdon, 18-Jul-2020.)
(𝜑 → 𝐹:Q⟶Q)    &   (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩    ⇒   (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿)))
 
11-Jul-2020cauappcvgprlemladdrl 6755 Lemma for cauappcvgprlemladd 6756. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
(𝜑 → 𝐹:Q⟶Q)    &   (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩    &   (𝜑 → 𝑆 ∈ Q)    ⇒   (𝜑 → (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩) ⊆ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)))
 
11-Jul-2020cauappcvgprlemladdru 6754 Lemma for cauappcvgprlemladd 6756. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
(𝜑 → 𝐹:Q⟶Q)    &   (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩    &   (𝜑 → 𝑆 ∈ Q)    ⇒   (𝜑 → (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩) ⊆ (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)))
 
11-Jul-2020cauappcvgprlemladdfl 6753 Lemma for cauappcvgprlemladd 6756. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
(𝜑 → 𝐹:Q⟶Q)    &   (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩    &   (𝜑 → 𝑆 ∈ Q)    ⇒   (𝜑 → (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩))
 
11-Jul-2020cauappcvgprlemladdfu 6752 Lemma for cauappcvgprlemladd 6756. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
(𝜑 → 𝐹:Q⟶Q)    &   (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩    &   (𝜑 → 𝑆 ∈ Q)    ⇒   (𝜑 → (2nd ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩))
 
8-Jul-2020nqprl 6649 Comparing a fraction to a real can be done by whether it is an element of the lower cut, or by <P. (Contributed by Jim Kingdon, 8-Jul-2020.)
((𝐴 ∈ Q ∧ 𝐵 ∈ P) → (𝐴 ∈ (1st ‘𝐵) ↔ ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩<P 𝐵))
 
24-Jun-2020nqprlu 6645 The canonical embedding of the rationals into the reals. (Contributed by Jim Kingdon, 24-Jun-2020.)
(𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ∈ P)
 
23-Jun-2020cauappcvgprlem2 6758 Lemma for cauappcvgpr 6760. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.)
(𝜑 → 𝐹:Q⟶Q)    &   (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩    &   (𝜑 → 𝑄 ∈ Q)    &   (𝜑 → 𝑅 ∈ Q)    ⇒   (𝜑 → 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹‘𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩)
 
23-Jun-2020cauappcvgprlem1 6757 Lemma for cauappcvgpr 6760. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.)
(𝜑 → 𝐹:Q⟶Q)    &   (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩    &   (𝜑 → 𝑄 ∈ Q)    &   (𝜑 → 𝑅 ∈ Q)    ⇒   (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑄)}, {𝑢 ∣ (𝐹‘𝑄) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩))
 
23-Jun-2020cauappcvgprlemladd 6756 Lemma for cauappcvgpr 6760. This takes 𝐿 and offsets it by the positive fraction 𝑆. (Contributed by Jim Kingdon, 23-Jun-2020.)
(𝜑 → 𝐹:Q⟶Q)    &   (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩    &   (𝜑 → 𝑆 ∈ Q)    ⇒   (𝜑 → (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q ((𝐹‘𝑞) +Q 𝑆)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q (((𝐹‘𝑞) +Q 𝑞) +Q 𝑆) <Q 𝑢}⟩)
 
20-Jun-2020cauappcvgprlemlim 6759 Lemma for cauappcvgpr 6760. The putative limit is a limit. (Contributed by Jim Kingdon, 20-Jun-2020.)
(𝜑 → 𝐹:Q⟶Q)    &   (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩    ⇒   (𝜑 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩))
 
20-Jun-2020cauappcvgprlemcl 6751 Lemma for cauappcvgpr 6760. The putative limit is a positive real. (Contributed by Jim Kingdon, 20-Jun-2020.)
(𝜑 → 𝐹:Q⟶Q)    &   (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))    &   𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩    ⇒   (𝜑 → 𝐿 ∈ P)
 
19-Jun-2020cauappcvgpr 6760 A Cauchy approximation has a limit. A Cauchy approximation, here 𝐹, is similar to a Cauchy sequence but is indexed by the desired tolerance (that is, how close together terms needs to be) rather than by natural numbers. This is basically Theorem 11.2.12 of [HoTT], p. (varies) with a few differences such as that we are proving the existence of a limit without anything about how fast it converges (that is, mere existence instead of existence, in HoTT terms), and that the codomain of 𝐹 is Q rather than P. We also specify that every term needs to be larger than a fraction 𝐴, to avoid the case where we have positive terms which "converge" to zero (which is not a positive real).

This proof (including its lemmas) is similar to the proofs of caucvgpr 6780 and caucvgprpr 6810 but is somewhat simpler, so reading this one first may help understanding the other two.

(Contributed by Jim Kingdon, 19-Jun-2020.)

(𝜑 → 𝐹:Q⟶Q)    &   (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))    &   (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))    ⇒   (𝜑 → ∃𝑦 ∈ P ∀𝑞 ∈ Q ∀𝑟 ∈ Q (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑞)}, {𝑢 ∣ (𝐹‘𝑞) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹‘𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩))
 
18-Jun-2020caucvgpr 6780 A Cauchy sequence of positive fractions with a modulus of convergence converges to a positive real. This is basically Corollary 11.2.13 of [HoTT], p. (varies) (one key difference being that this is for positive reals rather than signed reals). Also, the HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within 1 / 𝑛 of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis). We also specify that every term needs to be larger than a fraction 𝐴, to avoid the case where we have positive terms which "converge" to zero (which is not a positive real).

This proof (including its lemmas) is similar to the proofs of cauappcvgpr 6760 and caucvgprpr 6810. Reading cauappcvgpr 6760 first (the simplest of the three) might help understanding the other two.

(Contributed by Jim Kingdon, 18-Jun-2020.)

(𝜑 → 𝐹:N⟶Q)    &   (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))    &   (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))    ⇒   (𝜑 → ∃𝑦 ∈ P ∀𝑥 ∈ Q ∃𝑗 ∈ N ∀𝑘 ∈ N (𝑗 <N 𝑘 → (⟨{𝑙 ∣ 𝑙 <Q (𝐹‘𝑘)}, {𝑢 ∣ (𝐹‘𝑘) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑥}, {𝑢 ∣ 𝑥 <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝑘) +Q 𝑥)}, {𝑢 ∣ ((𝐹‘𝑘) +Q 𝑥) <Q 𝑢}⟩)))
 
15-Jun-2020imdivapd 9575 Imaginary part of a division. Related to remul2 9473. (Contributed by Jim Kingdon, 15-Jun-2020.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    ⇒   (𝜑 → (ℑ‘(𝐵 / 𝐴)) = ((ℑ‘𝐵) / 𝐴))
 
15-Jun-2020redivapd 9574 Real part of a division. Related to remul2 9473. (Contributed by Jim Kingdon, 15-Jun-2020.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    ⇒   (𝜑 → (ℜ‘(𝐵 / 𝐴)) = ((ℜ‘𝐵) / 𝐴))
 
15-Jun-2020cjdivapd 9568 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 15-Jun-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → (∗‘(𝐴 / 𝐵)) = ((∗‘𝐴) / (∗‘𝐵)))
 
15-Jun-2020riotaexg 5472 Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.)
(𝐴 ∈ 𝑉 → (℩𝑥 ∈ 𝐴 𝜓) ∈ V)
 
14-Jun-2020cjdivapi 9535 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    ⇒   (𝐵 # 0 → (∗‘(𝐴 / 𝐵)) = ((∗‘𝐴) / (∗‘𝐵)))
 
14-Jun-2020cjdivap 9509 Complex conjugate distributes over division. (Contributed by Jim Kingdon, 14-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (∗‘(𝐴 / 𝐵)) = ((∗‘𝐴) / (∗‘𝐵)))
 
14-Jun-2020cjap0 9507 A number is apart from zero iff its complex conjugate is apart from zero. (Contributed by Jim Kingdon, 14-Jun-2020.)
(𝐴 ∈ ℂ → (𝐴 # 0 ↔ (∗‘𝐴) # 0))
 
14-Jun-2020cjap 9506 Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((∗‘𝐴) # (∗‘𝐵) ↔ 𝐴 # 𝐵))
 
14-Jun-2020imdivap 9481 Imaginary part of a division. Related to immul2 9480. (Contributed by Jim Kingdon, 14-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (ℑ‘(𝐴 / 𝐵)) = ((ℑ‘𝐴) / 𝐵))
 
14-Jun-2020redivap 9474 Real part of a division. Related to remul2 9473. (Contributed by Jim Kingdon, 14-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (ℜ‘(𝐴 / 𝐵)) = ((ℜ‘𝐴) / 𝐵))
 
14-Jun-2020mulreap 9464 A product with a real multiplier apart from zero is real iff the multiplicand is real. (Contributed by Jim Kingdon, 14-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (𝐴 ∈ ℝ ↔ (𝐵 · 𝐴) ∈ ℝ))
 
13-Jun-2020sqgt0apd 9408 The square of a real apart from zero is positive. (Contributed by Jim Kingdon, 13-Jun-2020.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐴 # 0)    ⇒   (𝜑 → 0 < (𝐴↑2))
 
13-Jun-2020reexpclzapd 9405 Closure of exponentiation of reals. (Contributed by Jim Kingdon, 13-Jun-2020.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐴 # 0)    &   (𝜑 → 𝑁 ∈ ℤ)    ⇒   (𝜑 → (𝐴↑𝑁) ∈ ℝ)
 
13-Jun-2020expdivapd 9395 Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 13-Jun-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    &   (𝜑 → 𝑁 ∈ ℕ0)    ⇒   (𝜑 → ((𝐴 / 𝐵)↑𝑁) = ((𝐴↑𝑁) / (𝐵↑𝑁)))
 
13-Jun-2020sqdivapd 9394 Distribution of square over division. (Contributed by Jim Kingdon, 13-Jun-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2)))
 
12-Jun-2020expsubapd 9392 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝑀 ∈ ℤ)    ⇒   (𝜑 → (𝐴↑(𝑀 − 𝑁)) = ((𝐴↑𝑀) / (𝐴↑𝑁)))
 
12-Jun-2020expm1apd 9391 Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    &   (𝜑 → 𝑁 ∈ ℤ)    ⇒   (𝜑 → (𝐴↑(𝑁 − 1)) = ((𝐴↑𝑁) / 𝐴))
 
12-Jun-2020expp1zapd 9390 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    &   (𝜑 → 𝑁 ∈ ℤ)    ⇒   (𝜑 → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴))
 
12-Jun-2020exprecapd 9389 Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    &   (𝜑 → 𝑁 ∈ ℤ)    ⇒   (𝜑 → ((1 / 𝐴)↑𝑁) = (1 / (𝐴↑𝑁)))
 
12-Jun-2020expnegapd 9388 Value of a complex number raised to a negative power. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    &   (𝜑 → 𝑁 ∈ ℤ)    ⇒   (𝜑 → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))
 
12-Jun-2020expap0d 9387 Nonnegative integer exponentiation is nonzero if its mantissa is nonzero. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    &   (𝜑 → 𝑁 ∈ ℤ)    ⇒   (𝜑 → (𝐴↑𝑁) # 0)
 
12-Jun-2020expclzapd 9386 Closure law for integer exponentiation. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    &   (𝜑 → 𝑁 ∈ ℤ)    ⇒   (𝜑 → (𝐴↑𝑁) ∈ ℂ)
 
12-Jun-2020sqrecapd 9385 Square of reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    ⇒   (𝜑 → ((1 / 𝐴)↑2) = (1 / (𝐴↑2)))
 
12-Jun-2020sqgt0api 9339 The square of a nonzero real is positive. (Contributed by Jim Kingdon, 12-Jun-2020.)
𝐴 ∈ ℝ    ⇒   (𝐴 # 0 → 0 < (𝐴↑2))
 
12-Jun-2020sqdivapi 9337 Distribution of square over division. (Contributed by Jim Kingdon, 12-Jun-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0    ⇒   ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2))
 
11-Jun-2020sqgt0ap 9322 The square of a nonzero real is positive. (Contributed by Jim Kingdon, 11-Jun-2020.)
((𝐴 ∈ ℝ ∧ 𝐴 # 0) → 0 < (𝐴↑2))
 
11-Jun-2020sqdivap 9318 Distribution of square over division. (Contributed by Jim Kingdon, 11-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2)))
 
11-Jun-2020expdivap 9305 Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 11-Jun-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝑁 ∈ ℕ0) → ((𝐴 / 𝐵)↑𝑁) = ((𝐴↑𝑁) / (𝐵↑𝑁)))
 
11-Jun-2020expm1ap 9304 Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑(𝑁 − 1)) = ((𝐴↑𝑁) / 𝐴))
 
11-Jun-2020expp1zap 9303 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴))
 
11-Jun-2020expsubap 9302 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝐴↑(𝑀 − 𝑁)) = ((𝐴↑𝑀) / (𝐴↑𝑁)))
 
11-Jun-2020expmulzap 9301 Product of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝐴↑(𝑀 · 𝑁)) = ((𝐴↑𝑀)↑𝑁))
 
10-Jun-2020expaddzap 9299 Sum of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 10-Jun-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝐴↑(𝑀 + 𝑁)) = ((𝐴↑𝑀) · (𝐴↑𝑁)))
 
10-Jun-2020expaddzaplem 9298 Lemma for expaddzap 9299. (Contributed by Jim Kingdon, 10-Jun-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑀 ∈ ℝ ∧ -𝑀 ∈ ℕ) ∧ 𝑁 ∈ ℕ0) → (𝐴↑(𝑀 + 𝑁)) = ((𝐴↑𝑀) · (𝐴↑𝑁)))
 
10-Jun-2020exprecap 9296 Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 10-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → ((1 / 𝐴)↑𝑁) = (1 / (𝐴↑𝑁)))
 
10-Jun-2020mulexpzap 9295 Integer exponentiation of a product. (Contributed by Jim Kingdon, 10-Jun-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝑁 ∈ ℤ) → ((𝐴 · 𝐵)↑𝑁) = ((𝐴↑𝑁) · (𝐵↑𝑁)))
 
10-Jun-2020expap0i 9287 Integer exponentiation is apart from zero if its mantissa is apart from zero. (Contributed by Jim Kingdon, 10-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) # 0)
 
10-Jun-2020expap0 9285 Positive integer exponentiation is apart from zero iff its mantissa is apart from zero. That it is easier to prove this first, and then prove expeq0 9286 in terms of it, rather than the other way around, is perhaps an illustration of the maxim "In constructive analysis, the apartness is more basic [ than ] equality." ([Geuvers], p. 1). (Contributed by Jim Kingdon, 10-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) # 0 ↔ 𝐴 # 0))
 
10-Jun-2020mvllmulapd 7809 Move LHS left multiplication to RHS. (Contributed by Jim Kingdon, 10-Jun-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    &   (𝜑 → (𝐴 · 𝐵) = 𝐶)    ⇒   (𝜑 → 𝐵 = (𝐶 / 𝐴))
 
9-Jun-2020expclzap 9280 Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℂ)
 
9-Jun-2020expclzaplem 9279 Closure law for integer exponentiation. Lemma for expclzap 9280 and expap0i 9287. (Contributed by Jim Kingdon, 9-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0})
 
9-Jun-2020reexpclzap 9275 Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.)
((𝐴 ∈ ℝ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ)
 
9-Jun-2020neg1ap0 8026 -1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
-1 # 0
 
8-Jun-2020expcl2lemap 9267 Lemma for proving integer exponentiation closure laws. (Contributed by Jim Kingdon, 8-Jun-2020.)
𝐹 ⊆ ℂ    &   ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 · 𝑦) ∈ 𝐹)    &   1 ∈ 𝐹    &   ((𝑥 ∈ 𝐹 ∧ 𝑥 # 0) → (1 / 𝑥) ∈ 𝐹)    ⇒   ((𝐴 ∈ 𝐹 ∧ 𝐴 # 0 ∧ 𝐵 ∈ ℤ) → (𝐴↑𝐵) ∈ 𝐹)
 
8-Jun-2020expn1ap0 9265 A number to the negative one power is the reciprocal. (Contributed by Jim Kingdon, 8-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴↑-1) = (1 / 𝐴))
 
8-Jun-2020expineg2 9264 Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑁 ∈ ℂ ∧ -𝑁 ∈ ℕ0)) → (𝐴↑𝑁) = (1 / (𝐴↑-𝑁)))
 
8-Jun-2020expnegap0 9263 Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℕ0) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))
 
8-Jun-2020expinnval 9258 Value of exponentiation to positive integer powers. (Contributed by Jim Kingdon, 8-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁))
 
7-Jun-2020expival 9257 Value of exponentiation to integer powers. (Contributed by Jim Kingdon, 7-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (𝐴↑𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))))
 
7-Jun-2020expivallem 9256 Lemma for expival 9257. If we take a complex number apart from zero and raise it to a positive integer power, the result is apart from zero. (Contributed by Jim Kingdon, 7-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℕ) → (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁) # 0)
 
7-Jun-2020df-iexp 9255 Define exponentiation to nonnegative integer powers. This definition is not meant to be used directly; instead, exp0 9259 and expp1 9262 provide the standard recursive definition. The up-arrow notation is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976) and is convenient for us since we don't have superscripts. 10-Jun-2005: The definition was extended to include zero exponents, so that 0↑0 = 1 per the convention of Definition 10-4.1 of [Gleason] p. 134. 4-Jun-2014: The definition was extended to include negative integer exponents. The case 𝑥 = 0, 𝑦 < 0 gives the value (1 / 0), so we will avoid this case in our theorems. (Contributed by Jim Kingdon, 7-Jun-2020.)
↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}), ℂ)‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}), ℂ)‘-𝑦)))))
 
7-Jun-2020halfge0 8141 One-half is not negative. (Contributed by AV, 7-Jun-2020.)
0 ≤ (1 / 2)
 
4-Jun-2020iseqfveq 9230 Equality of sequences. (Contributed by Jim Kingdon, 4-Jun-2020.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘))    &   (𝜑 → 𝑆 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝑀( + , 𝐺, 𝑆)‘𝑁))
 
3-Jun-2020iseqfeq2 9229 Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
(𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))    &   (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (𝐺‘𝐾))    &   (𝜑 → 𝑆 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → (𝐺‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝐾 + 1))) → (𝐹‘𝑘) = (𝐺‘𝑘))    ⇒   (𝜑 → (seq𝑀( + , 𝐹, 𝑆) ↾ (ℤ≥‘𝐾)) = seq𝐾( + , 𝐺, 𝑆))
 
3-Jun-2020iseqfveq2 9228 Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
(𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))    &   (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (𝐺‘𝐾))    &   (𝜑 → 𝑆 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → (𝐺‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾))    &   ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘))    ⇒   (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁))
 
1-Jun-2020iseqcl 9223 Closure properties of the recursive sequence builder. (Contributed by Jim Kingdon, 1-Jun-2020.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝑆 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) ∈ 𝑆)
 
1-Jun-2020fzdcel 8904 Decidability of membership in a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝐾 ∈ (𝑀...𝑁))
 
1-Jun-2020fztri3or 8903 Trichotomy in terms of a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 < 𝑀 ∨ 𝐾 ∈ (𝑀...𝑁) ∨ 𝑁 < 𝐾))
 
1-Jun-2020zdclt 8318 Integer < is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 < 𝐵)
 
31-May-2020iseqp1 9225 Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 31-May-2020.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝑆 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁) + (𝐹‘(𝑁 + 1))))
 
31-May-2020iseq1 9222 Value of the sequence builder function at its initial value. (Contributed by Jim Kingdon, 31-May-2020.)
(𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹‘𝑀))
 
31-May-2020iseqovex 9219 Closure of a function used in proving sequence builder theorems. This can be thought of as a lemma for the small number of sequence builder theorems which need it. (Contributed by Jim Kingdon, 31-May-2020.)
((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) ∈ 𝑆)
 
31-May-2020frecuzrdgcl 9199 Closure law for the recursive definition generator on upper integers. See comment in frec2uz0d 9185 for the description of 𝐺 as the mapping from ω to (ℤ≥‘𝐶). (Contributed by Jim Kingdon, 31-May-2020.)
(𝜑 → 𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝐴 ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑 → 𝑇 = ran 𝑅)    ⇒   ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑇‘𝐵) ∈ 𝑆)
 
30-May-2020iseqfn 9221 The sequence builder function is a function. (Contributed by Jim Kingdon, 30-May-2020.)
(𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   (𝜑 → seq𝑀( + , 𝐹, 𝑆) Fn (ℤ≥‘𝑀))
 
30-May-2020iseqval 9220 Value of the sequence builder function. (Contributed by Jim Kingdon, 30-May-2020.)
𝑅 = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩)    &   ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   (𝜑 → seq𝑀( + , 𝐹, 𝑆) = ran 𝑅)
 
30-May-2020nfiseq 9218 Hypothesis builder for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
Ⅎ𝑥𝑀    &   Ⅎ𝑥 +    &   Ⅎ𝑥𝐹    &   Ⅎ𝑥𝑆    ⇒   Ⅎ𝑥seq𝑀( + , 𝐹, 𝑆)
 
30-May-2020iseqeq4 9217 Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
(𝑆 = 𝑇 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐹, 𝑇))
 
30-May-2020iseqeq3 9216 Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
(𝐹 = 𝐺 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐺, 𝑆))
 
30-May-2020iseqeq2 9215 Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
( + = 𝑄 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀(𝑄, 𝐹, 𝑆))
 
30-May-2020iseqeq1 9214 Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
(𝑀 = 𝑁 → seq𝑀( + , 𝐹, 𝑆) = seq𝑁( + , 𝐹, 𝑆))
 
30-May-2020nffrec 5982 Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
Ⅎ𝑥𝐹    &   Ⅎ𝑥𝐴    ⇒   Ⅎ𝑥frec(𝐹, 𝐴)
 
30-May-2020freceq2 5980 Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
(𝐴 = 𝐵 → frec(𝐹, 𝐴) = frec(𝐹, 𝐵))
 
30-May-2020freceq1 5979 Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
(𝐹 = 𝐺 → frec(𝐹, 𝐴) = frec(𝐺, 𝐴))
 
29-May-2020df-iseq 9212 Define a general-purpose operation that builds a recursive sequence (i.e. a function on the positive integers ℕ or some other upper integer set) whose value at an index is a function of its previous value and the value of an input sequence at that index. This definition is complicated, but fortunately it is not intended to be used directly. Instead, the only purpose of this definition is to provide us with an object that has the properties expressed by iseq1 9222 and iseqp1 9225. Typically, those are the main theorems that would be used in practice.

The first operand in the parentheses is the operation that is applied to the previous value and the value of the input sequence (second operand). The operand to the left of the parenthesis is the integer to start from. For example, for the operation +, an input sequence 𝐹 with values 1, 1/2, 1/4, 1/8,... would be transformed into the output sequence seq1( + , 𝐹, ℚ) with values 1, 3/2, 7/4, 15/8,.., so that (seq1( + , 𝐹, ℚ)‘1) = 1, (seq1( + , 𝐹, ℚ)‘2) = 3/2, etc. In other words, seq𝑀( + , 𝐹, ℚ) transforms a sequence 𝐹 into an infinite series.

Internally, the frec function generates as its values a set of ordered pairs starting at ⟨𝑀, (𝐹‘𝑀)⟩, with the first member of each pair incremented by one in each successive value. So, the range of frec is exactly the sequence we want, and we just extract the range and throw away the domain.

(Contributed by Jim Kingdon, 29-May-2020.)

seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
 
28-May-2020frecuzrdgsuc 9201 Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 9185 for the description of 𝐺 as the mapping from ω to (ℤ≥‘𝐶). (Contributed by Jim Kingdon, 28-May-2020.)
(𝜑 → 𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝐴 ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑 → 𝑇 = ran 𝑅)    ⇒   ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑇‘(𝐵 + 1)) = (𝐵𝐹(𝑇‘𝐵)))
 
27-May-2020frecuzrdg0 9200 Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 9185 for the description of 𝐺 as the mapping from ω to (ℤ≥‘𝐶). (Contributed by Jim Kingdon, 27-May-2020.)
(𝜑 → 𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝐴 ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑 → 𝑇 = ran 𝑅)    ⇒   (𝜑 → (𝑇‘𝐶) = 𝐴)
 
27-May-2020frecuzrdgrrn 9194 The function 𝑅 (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of 𝑆. (Contributed by Jim Kingdon, 27-May-2020.)
(𝜑 → 𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝐴 ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    ⇒   ((𝜑 ∧ 𝐷 ∈ ω) → (𝑅‘𝐷) ∈ ((ℤ≥‘𝐶) × 𝑆))
 
27-May-2020dffun5r 4914 A way of proving a relation is a function, analogous to mo2r 1952. (Contributed by Jim Kingdon, 27-May-2020.)
((Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧)) → Fun 𝐴)
 
26-May-2020frecuzrdgfn 9198 The recursive definition generator on upper integers is a function. See comment in frec2uz0d 9185 for the description of 𝐺 as the mapping from ω to (ℤ≥‘𝐶). (Contributed by Jim Kingdon, 26-May-2020.)
(𝜑 → 𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝐴 ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑 → 𝑇 = ran 𝑅)    ⇒   (𝜑 → 𝑇 Fn (ℤ≥‘𝐶))
 
26-May-2020frecuzrdglem 9197 A helper lemma for the value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 26-May-2020.)
(𝜑 → 𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝐴 ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑 → 𝐵 ∈ (ℤ≥‘𝐶))    ⇒   (𝜑 → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ ran 𝑅)
 
26-May-2020frecuzrdgrom 9196 The function 𝑅 (used in the definition of the recursive definition generator on upper integers) is a function defined for all natural numbers. (Contributed by Jim Kingdon, 26-May-2020.)
(𝜑 → 𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝐴 ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    ⇒   (𝜑 → 𝑅 Fn ω)
 
25-May-2020divlt1lt 8650 A real number divided by a positive real number is less than 1 iff the real number is less than the positive real number. (Contributed by AV, 25-May-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 / 𝐵) < 1 ↔ 𝐴 < 𝐵))
 
25-May-2020freccl 5993 Closure for finite recursion. (Contributed by Jim Kingdon, 25-May-2020.)
(𝜑 → ∀𝑧(𝐹‘𝑧) ∈ V)    &   (𝜑 → 𝐴 ∈ 𝑆)    &   ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐹‘𝑧) ∈ 𝑆)    &   (𝜑 → 𝐵 ∈ ω)    ⇒   (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)
 
24-May-2020frec2uzrdg 9195 A helper lemma for the value of a recursive definition generator on upper integers (typically either ℕ or ℕ0) with characteristic function 𝐹(𝑥, 𝑦) and initial value 𝐴. This lemma shows that evaluating 𝑅 at an element of ω gives an ordered pair whose first element is the index (translated from ω to (ℤ≥‘𝐶)). See comment in frec2uz0d 9185 which describes 𝐺 and the index translation. (Contributed by Jim Kingdon, 24-May-2020.)
(𝜑 → 𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝐴 ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    &   𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)    &   (𝜑 → 𝐵 ∈ ω)    ⇒   (𝜑 → (𝑅‘𝐵) = ⟨(𝐺‘𝐵), (2nd ‘(𝑅‘𝐵))⟩)
 
21-May-2020fzofig 9208 Half-open integer sets are finite. (Contributed by Jim Kingdon, 21-May-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀..^𝑁) ∈ Fin)
 
21-May-2020fzfigd 9207 Deduction form of fzfig 9206. (Contributed by Jim Kingdon, 21-May-2020.)
(𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    ⇒   (𝜑 → (𝑀...𝑁) ∈ Fin)
 
19-May-2020fzfig 9206 A finite interval of integers is finite. (Contributed by Jim Kingdon, 19-May-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) ∈ Fin)
 
19-May-2020frechashgf1o 9205 𝐺 maps ω one-to-one onto ℕ0. (Contributed by Jim Kingdon, 19-May-2020.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    ⇒   𝐺:ω–1-1-onto→ℕ0
 
19-May-2020ssfiexmid 6336 If any subset of a finite set is finite, excluded middle follows. One direction of Theorem 2.1 of [Bauer], p. 485. (Contributed by Jim Kingdon, 19-May-2020.)
∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin)    ⇒   (𝜑 ∨ ¬ 𝜑)
 
19-May-2020enm 6294 A set equinumerous to an inhabited set is inhabited. (Contributed by Jim Kingdon, 19-May-2020.)
((𝐴 ≈ 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∃𝑦 𝑦 ∈ 𝐵)
 
18-May-2020frecfzen2 9204 The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Jim Kingdon, 18-May-2020.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    ⇒   (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) ≈ (◡𝐺‘((𝑁 + 1) − 𝑀)))
 
18-May-2020frecfzennn 9203 The cardinality of a finite set of sequential integers. (See frec2uz0d 9185 for a description of the hypothesis.) (Contributed by Jim Kingdon, 18-May-2020.)
𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)    ⇒   (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (◡𝐺‘𝑁))
 
17-May-2020frec2uzisod 9193 𝐺 (see frec2uz0d 9185) is an isomorphism from natural ordinals to upper integers. (Contributed by Jim Kingdon, 17-May-2020.)
(𝜑 → 𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    ⇒   (𝜑 → 𝐺 Isom E , < (ω, (ℤ≥‘𝐶)))
 
17-May-2020frec2uzf1od 9192 𝐺 (see frec2uz0d 9185) is a one-to-one onto mapping. (Contributed by Jim Kingdon, 17-May-2020.)
(𝜑 → 𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    ⇒   (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶))
 
17-May-2020frec2uzrand 9191 Range of 𝐺 (see frec2uz0d 9185). (Contributed by Jim Kingdon, 17-May-2020.)
(𝜑 → 𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    ⇒   (𝜑 → ran 𝐺 = (ℤ≥‘𝐶))
 
16-May-2020frec2uzlt2d 9190 The mapping 𝐺 (see frec2uz0d 9185) preserves order. (Contributed by Jim Kingdon, 16-May-2020.)
(𝜑 → 𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑 → 𝐴 ∈ ω)    &   (𝜑 → 𝐵 ∈ ω)    ⇒   (𝜑 → (𝐴 ∈ 𝐵 ↔ (𝐺‘𝐴) < (𝐺‘𝐵)))
 
16-May-2020frec2uzltd 9189 Less-than relation for 𝐺 (see frec2uz0d 9185). (Contributed by Jim Kingdon, 16-May-2020.)
(𝜑 → 𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑 → 𝐴 ∈ ω)    &   (𝜑 → 𝐵 ∈ ω)    ⇒   (𝜑 → (𝐴 ∈ 𝐵 → (𝐺‘𝐴) < (𝐺‘𝐵)))
 
16-May-2020frec2uzuzd 9188 The value 𝐺 (see frec2uz0d 9185) at an ordinal natural number is in the upper integers. (Contributed by Jim Kingdon, 16-May-2020.)
(𝜑 → 𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑 → 𝐴 ∈ ω)    ⇒   (𝜑 → (𝐺‘𝐴) ∈ (ℤ≥‘𝐶))
 
16-May-2020frec2uzsucd 9187 The value of 𝐺 (see frec2uz0d 9185) at a successor. (Contributed by Jim Kingdon, 16-May-2020.)
(𝜑 → 𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑 → 𝐴 ∈ ω)    ⇒   (𝜑 → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) + 1))
 
16-May-2020frec2uzzd 9186 The value of 𝐺 (see frec2uz0d 9185) is an integer. (Contributed by Jim Kingdon, 16-May-2020.)
(𝜑 → 𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    &   (𝜑 → 𝐴 ∈ ω)    ⇒   (𝜑 → (𝐺‘𝐴) ∈ ℤ)
 
16-May-2020frec2uz0d 9185 The mapping 𝐺 is a one-to-one mapping from ω onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number 𝐶 (normally 0 for the upper integers ℕ0 or 1 for the upper integers ℕ), 1 maps to 𝐶 + 1, etc. This theorem shows the value of 𝐺 at ordinal natural number zero. (Contributed by Jim Kingdon, 16-May-2020.)
(𝜑 → 𝐶 ∈ ℤ)    &   𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)    ⇒   (𝜑 → (𝐺‘∅) = 𝐶)
 
15-May-2020nntri3 6075 A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-May-2020.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 ↔ (¬ 𝐴 ∈ 𝐵 ∧ ¬ 𝐵 ∈ 𝐴)))
 
14-May-2020rdgifnon2 5967 The recursive definition generator is a function on ordinal numbers. (Contributed by Jim Kingdon, 14-May-2020.)
((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → rec(𝐹, 𝐴) Fn On)
 
14-May-2020rdgtfr 5961 The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 14-May-2020.)
((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → (Fun (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) ∧ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘𝑓) ∈ V))
 
13-May-2020frecfnom 5986 The function generated by finite recursive definition generation is a function on omega. (Contributed by Jim Kingdon, 13-May-2020.)
((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → frec(𝐹, 𝐴) Fn ω)
 
13-May-2020frecabex 5984 The class abstraction from df-frec 5978 exists. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 13-May-2020.)
(𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → ∀𝑦(𝐹‘𝑦) ∈ V)    &   (𝜑 → 𝐴 ∈ 𝑊)    ⇒   (𝜑 → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑆 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑆‘𝑚))) ∨ (dom 𝑆 = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ V)
 
8-May-2020tfr0 5937 Transfinite recursion at the empty set. (Contributed by Jim Kingdon, 8-May-2020.)
𝐹 = recs(𝐺)    ⇒   ((𝐺‘∅) ∈ 𝑉 → (𝐹‘∅) = (𝐺‘∅))
 
7-May-2020frec0g 5983 The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.)
(𝐴 ∈ 𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
 
3-May-2020dcned 2212 Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)
(𝜑 → DECID 𝐴 = 𝐵)    ⇒   (𝜑 → DECID 𝐴 ≠ 𝐵)
 
2-May-2020ax-arch 7003 Archimedean axiom. Definition 3.1(2) of [Geuvers], p. 9. Axiom for real and complex numbers, justified by theorem axarch 6965.

This axiom should not be used directly; instead use arch 8178 (which is the same, but stated in terms of ℕ and <). (Contributed by Jim Kingdon, 2-May-2020.) (New usage is discouraged.)

(𝐴 ∈ ℝ → ∃𝑛 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 <ℝ 𝑛)
 
30-Apr-2020ltexnqi 6507 Ordering on positive fractions in terms of existence of sum. (Contributed by Jim Kingdon, 30-Apr-2020.)
(𝐴 <Q 𝐵 → ∃𝑥 ∈ Q (𝐴 +Q 𝑥) = 𝐵)
 
26-Apr-2020pitonnlem1p1 6922 Lemma for pitonn 6924. Simplifying an expression involving signed reals. (Contributed by Jim Kingdon, 26-Apr-2020.)
(𝐴 ∈ P → [⟨(𝐴 +P (1P +P 1P)), (1P +P 1P)⟩] ~R = [⟨(𝐴 +P 1P), 1P⟩] ~R )
 
26-Apr-2020addnqpr1 6660 Addition of one to a fraction embedded into a positive real. One can either add the fraction one to the fraction, or the positive real one to the positive real, and get the same result. Special case of addnqpr 6659. (Contributed by Jim Kingdon, 26-Apr-2020.)
(𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 1Q)}, {𝑢 ∣ (𝐴 +Q 1Q) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P 1P))
 
26-Apr-2020addpinq1 6562 Addition of one to the numerator of a fraction whose denominator is one. (Contributed by Jim Kingdon, 26-Apr-2020.)
(𝐴 ∈ N → [⟨(𝐴 +N 1𝑜), 1𝑜⟩] ~Q = ([⟨𝐴, 1𝑜⟩] ~Q +Q 1Q))
 
26-Apr-2020nnnq 6520 The canonical embedding of positive integers into positive fractions. (Contributed by Jim Kingdon, 26-Apr-2020.)
(𝐴 ∈ N → [⟨𝐴, 1𝑜⟩] ~Q ∈ Q)
 
24-Apr-2020pitonnlem2 6923 Lemma for pitonn 6924. Two ways to add one to a number. (Contributed by Jim Kingdon, 24-Apr-2020.)
(𝐾 ∈ N → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
 
24-Apr-2020pitonnlem1 6921 Lemma for pitonn 6924. Two ways to write the number one. (Contributed by Jim Kingdon, 24-Apr-2020.)
⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1
 
23-Apr-2020archsr 6866 For any signed real, there is an integer that is greater than it. This is also known as the "archimedean property". The expression [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R is the embedding of the positive integer 𝑥 into the signed reals. (Contributed by Jim Kingdon, 23-Apr-2020.)
(𝐴 ∈ R → ∃𝑥 ∈ N 𝐴 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
 
23-Apr-2020nnprlu 6651 The canonical embedding of positive integers into the positive reals. (Contributed by Jim Kingdon, 23-Apr-2020.)
(𝐴 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝐴, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐴, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P)
 
22-Apr-2020axarch 6965 Archimedean axiom. The Archimedean property is more naturally stated once we have defined ℕ. Unless we find another way to state it, we'll just use the right hand side of dfnn2 7916 in stating what we mean by "natural number" in the context of this axiom.

This construction-dependent theorem should not be referenced directly; instead, use ax-arch 7003. (Contributed by Jim Kingdon, 22-Apr-2020.) (New usage is discouraged.)

(𝐴 ∈ ℝ → ∃𝑛 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 <ℝ 𝑛)
 
22-Apr-2020pitonn 6924 Mapping from N to ℕ. (Contributed by Jim Kingdon, 22-Apr-2020.)
(𝑁 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)})
 
22-Apr-2020archpr 6741 For any positive real, there is an integer that is greater than it. This is also known as the "archimedean property". The integer 𝑥 is embedded into the reals as described at nnprlu 6651. (Contributed by Jim Kingdon, 22-Apr-2020.)
(𝐴 ∈ P → ∃𝑥 ∈ N 𝐴<P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩)
 
20-Apr-2020fzo0m 9047 A half-open integer range based at 0 is inhabited precisely if the upper bound is a positive integer. (Contributed by Jim Kingdon, 20-Apr-2020.)
(∃𝑥 𝑥 ∈ (0..^𝐴) ↔ 𝐴 ∈ ℕ)
 
20-Apr-2020fzom 9020 A half-open integer interval is inhabited iff it contains its left endpoint. (Contributed by Jim Kingdon, 20-Apr-2020.)
(∃𝑥 𝑥 ∈ (𝑀..^𝑁) ↔ 𝑀 ∈ (𝑀..^𝑁))
 
18-Apr-2020eluzdc 8547 Membership of an integer in an upper set of integers is decidable. (Contributed by Jim Kingdon, 18-Apr-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑁 ∈ (ℤ≥‘𝑀))
 
17-Apr-2020zlelttric 8290 Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴))
 
16-Apr-2020fznlem 8905 A finite set of sequential integers is empty if the bounds are reversed. (Contributed by Jim Kingdon, 16-Apr-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 → (𝑀...𝑁) = ∅))
 
15-Apr-2020fzm 8902 Properties of a finite interval of integers which is inhabited. (Contributed by Jim Kingdon, 15-Apr-2020.)
(∃𝑥 𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (ℤ≥‘𝑀))
 
15-Apr-2020xpdom3m 6308 A set is dominated by its Cartesian product with an inhabited set. Exercise 6 of [Suppes] p. 98. (Contributed by Jim Kingdon, 15-Apr-2020.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ ∃𝑥 𝑥 ∈ 𝐵) → 𝐴 ≼ (𝐴 × 𝐵))
 
13-Apr-2020snfig 6291 A singleton is finite. (Contributed by Jim Kingdon, 13-Apr-2020.)
(𝐴 ∈ 𝑉 → {𝐴} ∈ Fin)
 
13-Apr-2020en1bg 6280 A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Jim Kingdon, 13-Apr-2020.)
(𝐴 ∈ 𝑉 → (𝐴 ≈ 1𝑜 ↔ 𝐴 = {∪ 𝐴}))
 
10-Apr-2020negm 8550 The image under negation of an inhabited set of reals is inhabited. (Contributed by Jim Kingdon, 10-Apr-2020.)
((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∃𝑦 𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴})
 
8-Apr-2020zleloe 8292 Integer 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by Jim Kingdon, 8-Apr-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)))
 
7-Apr-2020zdcle 8317 Integer ≤ is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 ≤ 𝐵)
 
5-Apr-2020divge1 8649 The ratio of a number over a smaller positive number is larger than 1. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 1 ≤ (𝐵 / 𝐴))
 
4-Apr-2020ioorebasg 8844 Open intervals are elements of the set of all open intervals. (Contributed by Jim Kingdon, 4-Apr-2020.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) ∈ ran (,))
 
30-Mar-2020icc0r 8795 An empty closed interval of extended reals. (Contributed by Jim Kingdon, 30-Mar-2020.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐵 < 𝐴 → (𝐴[,]𝐵) = ∅))
 
30-Mar-2020ubioog 8783 An open interval does not contain its right endpoint. (Contributed by Jim Kingdon, 30-Mar-2020.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ¬ 𝐵 ∈ (𝐴(,)𝐵))
 
30-Mar-2020lbioog 8782 An open interval does not contain its left endpoint. (Contributed by Jim Kingdon, 30-Mar-2020.)
((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ¬ 𝐴 ∈ (𝐴(,)𝐵))
 
29-Mar-2020iooidg 8778 An open interval with identical lower and upper bounds is empty. (Contributed by Jim Kingdon, 29-Mar-2020.)
(𝐴 ∈ ℝ* → (𝐴(,)𝐴) = ∅)
 
27-Mar-2020zletric 8289 Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))
 
26-Mar-20204z 8275 4 is an integer. (Contributed by BJ, 26-Mar-2020.)
4 ∈ ℤ
 
25-Mar-2020elfzmlbm 8988 Subtracting the lower bound of a finite set of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
(𝐾 ∈ (𝑀...𝑁) → (𝐾 − 𝑀) ∈ (0...(𝑁 − 𝑀)))
 
25-Mar-2020elfz0add 8979 An element of a finite set of sequential nonnegative integers is an element of an extended finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 28-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝑁 ∈ (0...𝐴) → 𝑁 ∈ (0...(𝐴 + 𝐵))))
 
25-Mar-20202eluzge0 8517 2 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
2 ∈ (ℤ≥‘0)
 
23-Mar-2020rpnegap 8615 Either a real apart from zero or its negation is a positive real, but not both. (Contributed by Jim Kingdon, 23-Mar-2020.)
((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (𝐴 ∈ ℝ+ ⊻ -𝐴 ∈ ℝ+))
 
23-Mar-2020reapltxor 7580 Real apartness in terms of less than (exclusive-or version). (Contributed by Jim Kingdon, 23-Mar-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵 ⊻ 𝐵 < 𝐴)))
 
22-Mar-2020rpcnap0 8603 A positive real is a complex number apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
(𝐴 ∈ ℝ+ → (𝐴 ∈ ℂ ∧ 𝐴 # 0))
 
22-Mar-2020rpreap0 8601 A positive real is a real number apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
(𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 𝐴 # 0))
 
22-Mar-2020rpap0 8599 A positive real is apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
(𝐴 ∈ ℝ+ → 𝐴 # 0)
 
20-Mar-2020qapne 8574 Apartness is equivalent to not equal for rationals. (Contributed by Jim Kingdon, 20-Mar-2020.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 # 𝐵 ↔ 𝐴 ≠ 𝐵))
 
20-Mar-2020divap1d 7776 If two complex numbers are apart, their quotient is apart from one. (Contributed by Jim Kingdon, 20-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    &   (𝜑 → 𝐴 # 𝐵)    ⇒   (𝜑 → (𝐴 / 𝐵) # 1)
 
20-Mar-2020apmul1 7764 Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 # 𝐵 ↔ (𝐴 · 𝐶) # (𝐵 · 𝐶)))
 
19-Mar-2020divfnzn 8556 Division restricted to ℤ × ℕ is a function. Given excluded middle, it would be easy to prove this for ℂ × (ℂ ∖ {0}). The key difference is that an element of ℕ is apart from zero, whereas being an element of ℂ ∖ {0} implies being not equal to zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
( / ↾ (ℤ × ℕ)) Fn (ℤ × ℕ)
 
19-Mar-2020div2negapd 7780 Quotient of two negatives. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → (-𝐴 / -𝐵) = (𝐴 / 𝐵))
 
19-Mar-2020divneg2apd 7779 Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → -(𝐴 / 𝐵) = (𝐴 / -𝐵))
 
19-Mar-2020divnegapd 7778 Move negative sign inside of a division. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → -(𝐴 / 𝐵) = (-𝐴 / 𝐵))
 
19-Mar-2020divap0bd 7777 A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → (𝐴 # 0 ↔ (𝐴 / 𝐵) # 0))
 
19-Mar-2020diveqap0ad 7775 A fraction of complex numbers is zero iff its numerator is. Deduction form of diveqap0 7661. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → ((𝐴 / 𝐵) = 0 ↔ 𝐴 = 0))
 
19-Mar-2020diveqap1ad 7774 The quotient of two complex numbers is one iff they are equal. Deduction form of diveqap1 7682. Generalization of diveqap1d 7773. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → ((𝐴 / 𝐵) = 1 ↔ 𝐴 = 𝐵))
 
19-Mar-2020diveqap1d 7773 Equality in terms of unit ratio. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    &   (𝜑 → (𝐴 / 𝐵) = 1)    ⇒   (𝜑 → 𝐴 = 𝐵)
 
19-Mar-2020diveqap0d 7772 If a ratio is zero, the numerator is zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    &   (𝜑 → (𝐴 / 𝐵) = 0)    ⇒   (𝜑 → 𝐴 = 0)
 
15-Mar-2020nneoor 8340 A positive integer is even or odd. (Contributed by Jim Kingdon, 15-Mar-2020.)
(𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 + 1) / 2) ∈ ℕ))
 
14-Mar-2020zltlen 8319 Integer 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 7621 which is a similar result for real numbers. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴)))
 
14-Mar-2020zdceq 8316 Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 = 𝐵)
 
14-Mar-2020zapne 8315 Apartness is equivalent to not equal for integers. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 # 𝑁 ↔ 𝑀 ≠ 𝑁))
 
14-Mar-2020zltnle 8291 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
 
14-Mar-2020ztri3or 8288 Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ∨ 𝑀 = 𝑁 ∨ 𝑁 < 𝑀))
 
14-Mar-2020ztri3or0 8287 Integer trichotomy (with zero). (Contributed by Jim Kingdon, 14-Mar-2020.)
(𝑁 ∈ ℤ → (𝑁 < 0 ∨ 𝑁 = 0 ∨ 0 < 𝑁))
 
14-Mar-2020zaddcllemneg 8284 Lemma for zaddcl 8285. Special case in which -𝑁 is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ)
 
14-Mar-2020zaddcllempos 8282 Lemma for zaddcl 8285. Special case in which 𝑁 is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ)
 
14-Mar-2020dcne 2216 Decidable equality expressed in terms of ≠. Basically the same as df-dc 743. (Contributed by Jim Kingdon, 14-Mar-2020.)
(DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵))
 
9-Mar-20202muliap0 8149 2 · i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
(2 · i) # 0
 
9-Mar-2020iap0 8148 The imaginary unit i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
i # 0
 
9-Mar-20202ap0 8009 The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
2 # 0
 
9-Mar-20201ne0 7983 1 ≠ 0. See aso 1ap0 7581. (Contributed by Jim Kingdon, 9-Mar-2020.)
1 ≠ 0
 
9-Mar-2020redivclapi 7755 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐵 # 0    ⇒   (𝐴 / 𝐵) ∈ ℝ
 
9-Mar-2020redivclapzi 7754 Closure law for division of reals. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    ⇒   (𝐵 # 0 → (𝐴 / 𝐵) ∈ ℝ)
 
9-Mar-2020rerecclapi 7753 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℝ    &   𝐴 # 0    ⇒   (1 / 𝐴) ∈ ℝ
 
9-Mar-2020rerecclapzi 7752 Closure law for reciprocal. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℝ    ⇒   (𝐴 # 0 → (1 / 𝐴) ∈ ℝ)
 
9-Mar-2020divdivdivapi 7751 Division of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ    &   𝐵 # 0    &   𝐷 # 0    &   𝐶 # 0    ⇒   ((𝐴 / 𝐵) / (𝐶 / 𝐷)) = ((𝐴 · 𝐷) / (𝐵 · 𝐶))
 
9-Mar-2020divadddivapi 7750 Addition of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ    &   𝐵 # 0    &   𝐷 # 0    ⇒   ((𝐴 / 𝐵) + (𝐶 / 𝐷)) = (((𝐴 · 𝐷) + (𝐶 · 𝐵)) / (𝐵 · 𝐷))
 
9-Mar-2020divmul13api 7749 Swap denominators of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ    &   𝐵 # 0    &   𝐷 # 0    ⇒   ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐶 / 𝐵) · (𝐴 / 𝐷))
 
9-Mar-2020divmuldivapi 7748 Multiplication of two ratios. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ    &   𝐵 # 0    &   𝐷 # 0    ⇒   ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 · 𝐶) / (𝐵 · 𝐷))
 
9-Mar-2020div11api 7747 One-to-one relationship for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐶 # 0    ⇒   ((𝐴 / 𝐶) = (𝐵 / 𝐶) ↔ 𝐴 = 𝐵)
 
9-Mar-2020div23api 7746 A commutative/associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐶 # 0    ⇒   ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵)
 
9-Mar-2020divdirapi 7745 Distribution of division over addition. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐶 # 0    ⇒   ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶))
 
9-Mar-2020divassapi 7744 An associative law for division. (Contributed by Jim Kingdon, 9-Mar-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐶 # 0    ⇒   ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶))
 
8-Mar-2020nnap0 7943 A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝐴 ∈ ℕ → 𝐴 # 0)
 
8-Mar-2020divdivap2d 7797 Division by a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    &   (𝜑 → 𝐶 # 0)    ⇒   (𝜑 → (𝐴 / (𝐵 / 𝐶)) = ((𝐴 · 𝐶) / 𝐵))
 
8-Mar-2020divdivap1d 7796 Division into a fraction. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    &   (𝜑 → 𝐶 # 0)    ⇒   (𝜑 → ((𝐴 / 𝐵) / 𝐶) = (𝐴 / (𝐵 · 𝐶)))
 
8-Mar-2020dmdcanap2d 7795 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    &   (𝜑 → 𝐶 # 0)    ⇒   (𝜑 → ((𝐴 / 𝐵) · (𝐵 / 𝐶)) = (𝐴 / 𝐶))
 
8-Mar-2020dmdcanapd 7794 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    &   (𝜑 → 𝐶 # 0)    ⇒   (𝜑 → ((𝐵 / 𝐶) · (𝐴 / 𝐵)) = (𝐴 / 𝐶))
 
8-Mar-2020divcanap7d 7793 Cancel equal divisors in a division. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    &   (𝜑 → 𝐶 # 0)    ⇒   (𝜑 → ((𝐴 / 𝐶) / (𝐵 / 𝐶)) = (𝐴 / 𝐵))
 
8-Mar-2020divcanap5rd 7792 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    &   (𝜑 → 𝐶 # 0)    ⇒   (𝜑 → ((𝐴 · 𝐶) / (𝐵 · 𝐶)) = (𝐴 / 𝐵))
 
8-Mar-2020divcanap5d 7791 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    &   (𝜑 → 𝐶 # 0)    ⇒   (𝜑 → ((𝐶 · 𝐴) / (𝐶 · 𝐵)) = (𝐴 / 𝐵))
 
8-Mar-2020divdiv32apd 7790 Swap denominators in a division. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    &   (𝜑 → 𝐶 # 0)    ⇒   (𝜑 → ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵))
 
8-Mar-2020div13apd 7789 A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → ((𝐴 / 𝐵) · 𝐶) = ((𝐶 / 𝐵) · 𝐴))
 
8-Mar-2020div32apd 7788 A commutative/associative law for division. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → ((𝐴 / 𝐵) · 𝐶) = (𝐴 · (𝐶 / 𝐵)))
 
8-Mar-2020divmulapd 7787 Relationship between division and multiplication. (Contributed by Jim Kingdon, 8-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → ((𝐴 / 𝐵) = 𝐶 ↔ (𝐵 · 𝐶) = 𝐴))
 
7-Mar-2020nn1gt1 7947 A positive integer is either one or greater than one. This is for ℕ; 0elnn 4340 is a similar theorem for ω (the natural numbers as ordinals). (Contributed by Jim Kingdon, 7-Mar-2020.)
(𝐴 ∈ ℕ → (𝐴 = 1 ∨ 1 < 𝐴))
 
5-Mar-2020crap0 7910 The real representation of complex numbers is apart from zero iff one of its terms is apart from zero. (Contributed by Jim Kingdon, 5-Mar-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 # 0 ∨ 𝐵 # 0) ↔ (𝐴 + (i · 𝐵)) # 0))
 
3-Mar-2020rec11apd 7786 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    &   (𝜑 → 𝐵 # 0)    &   (𝜑 → (1 / 𝐴) = (1 / 𝐵))    ⇒   (𝜑 → 𝐴 = 𝐵)
 
3-Mar-2020ddcanapd 7785 Cancellation in a double division. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → (𝐴 / (𝐴 / 𝐵)) = 𝐵)
 
3-Mar-2020divcanap6d 7784 Cancellation of inverted fractions. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → ((𝐴 / 𝐵) · (𝐵 / 𝐴)) = 1)
 
3-Mar-2020recdivap2d 7783 Division into a reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → ((1 / 𝐴) / 𝐵) = (1 / (𝐴 · 𝐵)))
 
3-Mar-2020recdivapd 7782 The reciprocal of a ratio. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → (1 / (𝐴 / 𝐵)) = (𝐵 / 𝐴))
 
3-Mar-2020divap0d 7781 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → (𝐴 / 𝐵) # 0)
 
3-Mar-2020div0apd 7763 Division into zero is zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    ⇒   (𝜑 → (0 / 𝐴) = 0)
 
3-Mar-2020dividapd 7762 A number divided by itself is one. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    ⇒   (𝜑 → (𝐴 / 𝐴) = 1)
 
3-Mar-2020recrecapd 7761 A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    ⇒   (𝜑 → (1 / (1 / 𝐴)) = 𝐴)
 
3-Mar-2020recidap2d 7760 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    ⇒   (𝜑 → ((1 / 𝐴) · 𝐴) = 1)
 
3-Mar-2020recidapd 7759 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    ⇒   (𝜑 → (𝐴 · (1 / 𝐴)) = 1)
 
3-Mar-2020recap0d 7758 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    ⇒   (𝜑 → (1 / 𝐴) # 0)
 
3-Mar-2020recclapd 7757 Closure law for reciprocal. (Contributed by Jim Kingdon, 3-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    ⇒   (𝜑 → (1 / 𝐴) ∈ ℂ)
 
2-Mar-2020div11apd 7805 One-to-one relationship for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐶 # 0)    &   (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐶))    ⇒   (𝜑 → 𝐴 = 𝐵)
 
2-Mar-2020divsubdirapd 7804 Distribution of division over subtraction. (Contributed by Jim Kingdon, 2-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐶 # 0)    ⇒   (𝜑 → ((𝐴 − 𝐵) / 𝐶) = ((𝐴 / 𝐶) − (𝐵 / 𝐶)))
 
2-Mar-2020divdirapd 7803 Distribution of division over addition. (Contributed by Jim Kingdon, 2-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐶 # 0)    ⇒   (𝜑 → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶)))
 
2-Mar-2020div23apd 7802 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐶 # 0)    ⇒   (𝜑 → ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵))
 
2-Mar-2020div12apd 7801 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐶 # 0)    ⇒   (𝜑 → (𝐴 · (𝐵 / 𝐶)) = (𝐵 · (𝐴 / 𝐶)))
 
2-Mar-2020divassapd 7800 An associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐶 # 0)    ⇒   (𝜑 → ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶)))
 
2-Mar-2020divmulap3d 7799 Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐶 # 0)    ⇒   (𝜑 → ((𝐴 / 𝐶) = 𝐵 ↔ 𝐴 = (𝐵 · 𝐶)))
 
2-Mar-2020divmulap2d 7798 Relationship between division and multiplication. (Contributed by Jim Kingdon, 2-Mar-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐶 # 0)    ⇒   (𝜑 → ((𝐴 / 𝐶) = 𝐵 ↔ 𝐴 = (𝐶 · 𝐵)))
 
29-Feb-2020prodgt0gt0 7817 Infer that a multiplicand is positive from a positive multiplier and positive product. See prodgt0 7818 for the same theorem with 0 < 𝐴 replaced by the weaker condition 0 ≤ 𝐴. (Contributed by Jim Kingdon, 29-Feb-2020.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 0 < (𝐴 · 𝐵))) → 0 < 𝐵)
 
29-Feb-2020redivclapd 7808 Closure law for division of reals. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → (𝐴 / 𝐵) ∈ ℝ)
 
29-Feb-2020rerecclapd 7807 Closure law for reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐴 # 0)    ⇒   (𝜑 → (1 / 𝐴) ∈ ℝ)
 
29-Feb-2020divcanap4d 7771 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → ((𝐴 · 𝐵) / 𝐵) = 𝐴)
 
29-Feb-2020divcanap3d 7770 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → ((𝐵 · 𝐴) / 𝐵) = 𝐴)
 
29-Feb-2020divrecap2d 7769 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴))
 
29-Feb-2020divrecapd 7768 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵)))
 
29-Feb-2020divcanap2d 7767 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → (𝐵 · (𝐴 / 𝐵)) = 𝐴)
 
29-Feb-2020divcanap1d 7766 A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → ((𝐴 / 𝐵) · 𝐵) = 𝐴)
 
29-Feb-2020divclapd 7765 Closure law for division. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → (𝐴 / 𝐵) ∈ ℂ)
 
29-Feb-2020divdiv32api 7743 Swap denominators in a division. (Contributed by Jim Kingdon, 29-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐵 # 0    &   𝐶 # 0    ⇒   ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵)
 
29-Feb-2020divmulapi 7742 Relationship between division and multiplication. (Contributed by Jim Kingdon, 29-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐵 # 0    ⇒   ((𝐴 / 𝐵) = 𝐶 ↔ (𝐵 · 𝐶) = 𝐴)
 
28-Feb-2020divdiv23apzi 7741 Swap denominators in a division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    ⇒   ((𝐵 # 0 ∧ 𝐶 # 0) → ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵))
 
28-Feb-2020divdirapzi 7740 Distribution of division over addition. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    ⇒   (𝐶 # 0 → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶)))
 
28-Feb-2020divmulapzi 7739 Relationship between division and multiplication. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    ⇒   (𝐵 # 0 → ((𝐴 / 𝐵) = 𝐶 ↔ (𝐵 · 𝐶) = 𝐴))
 
28-Feb-2020divassapzi 7738 An associative law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    ⇒   (𝐶 # 0 → ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶)))
 
28-Feb-2020rec11apii 7737 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐴 # 0    &   𝐵 # 0    ⇒   ((1 / 𝐴) = (1 / 𝐵) ↔ 𝐴 = 𝐵)
 
28-Feb-2020divap0i 7736 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐴 # 0    &   𝐵 # 0    ⇒   (𝐴 / 𝐵) # 0
 
28-Feb-2020divcanap4i 7735 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0    ⇒   ((𝐴 · 𝐵) / 𝐵) = 𝐴
 
28-Feb-2020divcanap3i 7734 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0    ⇒   ((𝐵 · 𝐴) / 𝐵) = 𝐴
 
28-Feb-2020divrecapi 7733 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0    ⇒   (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))
 
28-Feb-2020divcanap1i 7732 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0    ⇒   ((𝐴 / 𝐵) · 𝐵) = 𝐴
 
28-Feb-2020divcanap2i 7731 A cancellation law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0    ⇒   (𝐵 · (𝐴 / 𝐵)) = 𝐴
 
28-Feb-2020divclapi 7730 Closure law for division. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐵 # 0    ⇒   (𝐴 / 𝐵) ∈ ℂ
 
28-Feb-2020rec11api 7729 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 28-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    ⇒   ((𝐴 # 0 ∧ 𝐵 # 0) → ((1 / 𝐴) = (1 / 𝐵) ↔ 𝐴 = 𝐵))
 
28-Feb-2020ltleap 7621 Less than in terms of non-strict order and apartness. (Contributed by Jim Kingdon, 28-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 # 𝐵)))
 
27-Feb-2020divcanap4zi 7728 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    ⇒   (𝐵 # 0 → ((𝐴 · 𝐵) / 𝐵) = 𝐴)
 
27-Feb-2020divcanap3zi 7727 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    ⇒   (𝐵 # 0 → ((𝐵 · 𝐴) / 𝐵) = 𝐴)
 
27-Feb-2020divrecapzi 7726 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    ⇒   (𝐵 # 0 → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵)))
 
27-Feb-2020divcanap2zi 7725 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    ⇒   (𝐵 # 0 → (𝐵 · (𝐴 / 𝐵)) = 𝐴)
 
27-Feb-2020divcanap1zi 7724 A cancellation law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    ⇒   (𝐵 # 0 → ((𝐴 / 𝐵) · 𝐵) = 𝐴)
 
27-Feb-2020divclapzi 7723 Closure law for division. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    ⇒   (𝐵 # 0 → (𝐴 / 𝐵) ∈ ℂ)
 
27-Feb-2020recidapzi 7715 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    ⇒   (𝐴 # 0 → (𝐴 · (1 / 𝐴)) = 1)
 
27-Feb-2020recap0apzi 7714 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    ⇒   (𝐴 # 0 → (1 / 𝐴) # 0)
 
27-Feb-2020recclapzi 7713 Closure law for reciprocal. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℂ    ⇒   (𝐴 # 0 → (1 / 𝐴) ∈ ℂ)
 
27-Feb-2020divneg2ap 7712 Move negative sign inside of a division. (Contributed by Jim Kingdon, 27-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → -(𝐴 / 𝐵) = (𝐴 / -𝐵))
 
27-Feb-2020div2negap 7711 Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (-𝐴 / -𝐵) = (𝐴 / 𝐵))
 
27-Feb-2020negap0 7620 A number is apart from zero iff its negative is apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
(𝐴 ∈ ℂ → (𝐴 # 0 ↔ -𝐴 # 0))
 
27-Feb-2020gt0ap0d 7619 Positive implies apart from zero. Because of the way we define #, 𝐴 must be an element of ℝ, not just ℝ*. (Contributed by Jim Kingdon, 27-Feb-2020.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)    ⇒   (𝜑 → 𝐴 # 0)
 
27-Feb-2020gt0ap0ii 7618 Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℝ    &   0 < 𝐴    ⇒   𝐴 # 0
 
27-Feb-2020gt0ap0i 7617 Positive means apart from zero (useful for ordering theorems involving division). (Contributed by Jim Kingdon, 27-Feb-2020.)
𝐴 ∈ ℝ    ⇒   (0 < 𝐴 → 𝐴 # 0)
 
27-Feb-2020gt0ap0 7616 Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 # 0)
 
26-Feb-20202times 8038 Two times a number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.) (Proof shortened by AV, 26-Feb-2020.)
(𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴))
 
26-Feb-2020redivclap 7707 Closure law for division of reals. (Contributed by Jim Kingdon, 26-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 # 0) → (𝐴 / 𝐵) ∈ ℝ)
 
26-Feb-2020rerecclap 7706 Closure law for reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℝ)
 
26-Feb-2020conjmulap 7705 Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy this relationship. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝑃 ∈ ℂ ∧ 𝑃 # 0) ∧ (𝑄 ∈ ℂ ∧ 𝑄 # 0)) → (((1 / 𝑃) + (1 / 𝑄)) = 1 ↔ ((𝑃 − 1) · (𝑄 − 1)) = 1))
 
26-Feb-2020divsubdivap 7704 Subtraction of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐶) − (𝐵 / 𝐷)) = (((𝐴 · 𝐷) − (𝐵 · 𝐶)) / (𝐶 · 𝐷)))
 
26-Feb-2020divadddivap 7703 Addition of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐶) + (𝐵 / 𝐷)) = (((𝐴 · 𝐷) + (𝐵 · 𝐶)) / (𝐶 · 𝐷)))
 
26-Feb-2020ddcanap 7702 Cancellation in a double division. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (𝐴 / (𝐴 / 𝐵)) = 𝐵)
 
26-Feb-2020recdivap2 7701 Division into a reciprocal. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((1 / 𝐴) / 𝐵) = (1 / (𝐴 · 𝐵)))
 
26-Feb-2020divdivap2 7700 Division by a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 / (𝐵 / 𝐶)) = ((𝐴 · 𝐶) / 𝐵))
 
26-Feb-2020divdivap1 7699 Division into a fraction. (Contributed by Jim Kingdon, 26-Feb-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐵) / 𝐶) = (𝐴 / (𝐵 · 𝐶)))
 
26-Feb-2020dmdcanap 7698 Cancellation law for division and multiplication. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵) · (𝐶 / 𝐴)) = (𝐶 / 𝐵))
 
26-Feb-2020divcanap7 7697 Cancel equal divisors in a division. (Contributed by Jim Kingdon, 26-Feb-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) / (𝐵 / 𝐶)) = (𝐴 / 𝐵))
 
26-Feb-2020divdiv32ap 7696 Swap denominators in a division. (Contributed by Jim Kingdon, 26-Feb-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐵) / 𝐶) = ((𝐴 / 𝐶) / 𝐵))
 
26-Feb-2020divcanap6 7695 Cancellation of inverted fractions. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((𝐴 / 𝐵) · (𝐵 / 𝐴)) = 1)
 
26-Feb-2020recdivap 7694 The reciprocal of a ratio. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (1 / (𝐴 / 𝐵)) = (𝐵 / 𝐴))
 
26-Feb-2020divmuleqap 7693 Cross-multiply in an equality of ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐶) = (𝐵 / 𝐷) ↔ (𝐴 · 𝐷) = (𝐵 · 𝐶)))
 
26-Feb-2020divmul24ap 7692 Swap the numerators in the product of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐶) · (𝐵 / 𝐷)) = ((𝐴 / 𝐷) · (𝐵 / 𝐶)))
 
26-Feb-2020divmul13ap 7691 Swap the denominators in the product of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐶) · (𝐵 / 𝐷)) = ((𝐵 / 𝐶) · (𝐴 / 𝐷)))
 
25-Feb-2020divcanap5 7690 Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐶 · 𝐴) / (𝐶 · 𝐵)) = (𝐴 / 𝐵))
 
25-Feb-2020divdivdivap 7689 Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by Jim Kingdon, 25-Feb-2020.)
(((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐵) / (𝐶 / 𝐷)) = ((𝐴 · 𝐷) / (𝐵 · 𝐶)))
 
25-Feb-2020divmuldivap 7688 Multiplication of two ratios. (Contributed by Jim Kingdon, 25-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐷 ∈ ℂ ∧ 𝐷 # 0))) → ((𝐴 / 𝐶) · (𝐵 / 𝐷)) = ((𝐴 · 𝐵) / (𝐶 · 𝐷)))
 
25-Feb-2020rec11rap 7687 Mutual reciprocals. (Contributed by Jim Kingdon, 25-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((1 / 𝐴) = 𝐵 ↔ (1 / 𝐵) = 𝐴))
 
25-Feb-2020rec11ap 7686 Reciprocal is one-to-one. (Contributed by Jim Kingdon, 25-Feb-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((1 / 𝐴) = (1 / 𝐵) ↔ 𝐴 = 𝐵))
 
25-Feb-2020recrecap 7685 A number is equal to the reciprocal of its reciprocal. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / (1 / 𝐴)) = 𝐴)
 
25-Feb-2020divnegap 7683 Move negative sign inside of a division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → -(𝐴 / 𝐵) = (-𝐴 / 𝐵))
 
25-Feb-2020diveqap1 7682 Equality in terms of unit ratio. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵) = 1 ↔ 𝐴 = 𝐵))
 
25-Feb-2020div0ap 7679 Division into zero is zero. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (0 / 𝐴) = 0)
 
25-Feb-2020dividap 7678 A number divided by itself is one. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 / 𝐴) = 1)
 
25-Feb-2020div11ap 7677 One-to-one relationship for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = (𝐵 / 𝐶) ↔ 𝐴 = 𝐵))
 
25-Feb-2020divcanap4 7676 A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 · 𝐵) / 𝐵) = 𝐴)
 
25-Feb-2020divcanap3 7675 A cancellation law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐵 · 𝐴) / 𝐵) = 𝐴)
 
25-Feb-2020divdirap 7674 Distribution of division over addition. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶)))
 
25-Feb-2020div12ap 7673 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 · (𝐵 / 𝐶)) = (𝐵 · (𝐴 / 𝐶)))
 
25-Feb-2020div13ap 7672 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵) · 𝐶) = ((𝐶 / 𝐵) · 𝐴))
 
25-Feb-2020div32ap 7671 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝐶 ∈ ℂ) → ((𝐴 / 𝐵) · 𝐶) = (𝐴 · (𝐶 / 𝐵)))
 
25-Feb-2020div23ap 7670 A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵))
 
25-Feb-2020divassap 7669 An associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶)))
 
25-Feb-2020divrecap2 7668 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 25-Feb-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴))
 
24-Feb-2020conventions 9892 Unless there is a reason to diverge, we follow the conventions of the Metamath Proof Explorer (aka "set.mm"). This list of conventions is intended to be read in conjunction with the corresponding conventions in the Metamath Proof Explorer, and only the differences are described below.

  • Minimizing axioms and the axiom of choice. We prefer proofs that depend on fewer and/or weaker axioms, even if the proofs are longer. In particular, our choice of IZF (Intuitionistic Zermelo-Fraenkel) over CZF (Constructive Zermelo-Fraenkel, a weaker system) was just an expedient choice because IZF is easier to formalize in metamath. You can find some development using CZF in BJ's mathbox starting at ax-bd0 9933 (and the section header just above it). As for the axiom of choice, the full axiom of choice implies excluded middle as seen at acexmid 5511, although some authors will use countable choice or dependent choice. For example, countable choice or excluded middle is needed to show that the Cauchy reals coincide with the Dedekind reals - Corollary 11.4.3 of [HoTT], p. (varies).
  • Junk/undefined results. Much of the discussion of this topic in the Metamath Proof Explorer applies except that certain techniques are not available to us. For example, the Metamath Proof Explorer will often say "if a function is evaluated within its domain, a certain result follows; if the function is evaluated outside its domain, the same result follows. Since the function must be evaluated within its domain or outside it, the result follows unconditionally" (the use of excluded middle in this argument is perhaps obvious when stated this way). For this reason, we generally need to prove we are evaluating functions within their domains and avoid the reverse closure theorems of the Metamath Proof Explorer.
  • Bibliography references. The bibliography for the Intuitionistic Logic Explorer is separate from the one for the Metamath Proof Explorer but feel free to copy-paste a citation in either direction in order to cite it.

Label naming conventions

Here are a few of the label naming conventions:

  • Suffixes. We follow the conventions of the Metamath Proof Explorer with a few additions. A biconditional in set.mm which is an implication in iset.mm should have a "r" (for the reverse direction), or "i"/"im" (for the forward direction) appended. A theorem in set.mm which has a decidability condition added should add "dc" to the theorem name. A theorem in set.mm where "nonempty class" is changed to "inhabited class" should add "m" (for member) to the theorem name.

The following table shows some commonly-used abbreviations in labels which are not found in the Metamath Proof Explorer, in alphabetical order. For each abbreviation we provide a mnenomic to help you remember it, the source theorem/assumption defining it, an expression showing what it looks like, whether or not it is a "syntax fragment" (an abbreviation that indicates a particular kind of syntax), and hyperlinks to label examples that use the abbreviation. The abbreviation is bolded if there is a df-NAME definition but the label fragment is not NAME.

AbbreviationMnenomicSource ExpressionSyntax?Example(s)
apapart df-ap 7573 Yes apadd1 7599, apne 7614

  • Community. The Metamath mailing list also covers the Intuitionistic Logic Explorer and is at: https://groups.google.com/forum/#!forum/metamath.

    • (Contributed by Jim Kingdon, 24-Feb-2020.)

    𝜑    ⇒   𝜑
     
    24-Feb-2020divrecap 7667 Relationship between division and reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵)))
     
    24-Feb-2020recidap2 7666 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ((1 / 𝐴) · 𝐴) = 1)
     
    24-Feb-2020recidap 7665 Multiplication of a number and its reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 · (1 / 𝐴)) = 1)
     
    24-Feb-2020recap0 7664 The reciprocal of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / 𝐴) # 0)
     
    24-Feb-2020mulap0bbd 7641 A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 7639 and consequence of mulap0bd 7638. (Contributed by Jim Kingdon, 24-Feb-2020.)
    (𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 · 𝐵) # 0)    ⇒   (𝜑 → 𝐵 # 0)
     
    24-Feb-2020mulap0bad 7640 A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 7639 and consequence of mulap0bd 7638. (Contributed by Jim Kingdon, 24-Feb-2020.)
    (𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 · 𝐵) # 0)    ⇒   (𝜑 → 𝐴 # 0)
     
    24-Feb-2020mulap0bd 7638 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
    (𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    ⇒   (𝜑 → ((𝐴 # 0 ∧ 𝐵 # 0) ↔ (𝐴 · 𝐵) # 0))
     
    24-Feb-2020mulap0b 7636 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 # 0 ∧ 𝐵 # 0) ↔ (𝐴 · 𝐵) # 0))
     
    24-Feb-2020mulap0r 7606 A product apart from zero. Lemma 2.13 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 24-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐴 · 𝐵) # 0) → (𝐴 # 0 ∧ 𝐵 # 0))
     
    24-Feb-20201ap0 7581 One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
    1 # 0
     
    23-Feb-2020mulap0d 7639 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 23-Feb-2020.)
    (𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐴 # 0)    &   (𝜑 → 𝐵 # 0)    ⇒   (𝜑 → (𝐴 · 𝐵) # 0)
     
    23-Feb-2020mulap0i 7637 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 23-Feb-2020.)
    𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐴 # 0    &   𝐵 # 0    ⇒   (𝐴 · 𝐵) # 0
     
    23-Feb-2020mulext 7605 Strong extensionality for multiplication. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5521. For us, it is proved a different way. (Contributed by Jim Kingdon, 23-Feb-2020.)
    (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 · 𝐵) # (𝐶 · 𝐷) → (𝐴 # 𝐶 ∨ 𝐵 # 𝐷)))
     
    23-Feb-2020mulreim 7595 Complex multiplication in terms of real and imaginary parts. (Contributed by Jim Kingdon, 23-Feb-2020.)
    (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 + (i · 𝐵)) · (𝐶 + (i · 𝐷))) = (((𝐴 · 𝐶) + -(𝐵 · 𝐷)) + (i · ((𝐶 · 𝐵) + (𝐷 · 𝐴)))))
     
    22-Feb-2020divap0 7663 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 22-Feb-2020.)
    (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (𝐴 / 𝐵) # 0)
     
    22-Feb-2020divap0b 7662 The ratio of numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 22-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 # 0 ↔ (𝐴 / 𝐵) # 0))
     
    22-Feb-2020diveqap0 7661 A ratio is zero iff the numerator is zero. (Contributed by Jim Kingdon, 22-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵) = 0 ↔ 𝐴 = 0))
     
    22-Feb-2020divcanap1 7660 A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵) · 𝐵) = 𝐴)
     
    22-Feb-2020divcanap2 7659 A cancellation law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐵 · (𝐴 / 𝐵)) = 𝐴)
     
    22-Feb-2020recclap 7658 Closure law for reciprocal. (Contributed by Jim Kingdon, 22-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / 𝐴) ∈ ℂ)
     
    22-Feb-2020divclap 7657 Closure law for division. (Contributed by Jim Kingdon, 22-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) ∈ ℂ)
     
    22-Feb-2020divmulap3 7656 Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ 𝐴 = (𝐵 · 𝐶)))
     
    22-Feb-2020divmulap2 7655 Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ 𝐴 = (𝐶 · 𝐵)))
     
    22-Feb-2020divmulap 7654 Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ (𝐶 · 𝐵) = 𝐴))
     
    22-Feb-2020mulap0 7635 The product of two numbers apart from zero is apart from zero. Lemma 2.15 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 22-Feb-2020.)
    (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (𝐴 · 𝐵) # 0)
     
    22-Feb-2020mulext2 7604 Right extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐶 · 𝐴) # (𝐶 · 𝐵) → 𝐴 # 𝐵))
     
    22-Feb-2020mulext1 7603 Left extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐶) # (𝐵 · 𝐶) → 𝐴 # 𝐵))
     
    22-Feb-2020remulext2 7591 Right extensionality for real multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
    ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 · 𝐴) # (𝐶 · 𝐵) → 𝐴 # 𝐵))
     
    21-Feb-2020divvalap 7653 Value of division: the (unique) element 𝑥 such that (𝐵 · 𝑥) = 𝐴. This is meaningful only when 𝐵 is apart from zero. (Contributed by Jim Kingdon, 21-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) = (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))
     
    21-Feb-2020receuap 7650 Existential uniqueness of reciprocals. (Contributed by Jim Kingdon, 21-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ∃!𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴)
     
    21-Feb-2020mulcanapi 7648 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
    𝐴 ∈ ℂ    &   𝐵 ∈ ℂ    &   𝐶 ∈ ℂ    &   𝐶 # 0    ⇒   ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵)
     
    21-Feb-2020mulcanap2 7647 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵))
     
    21-Feb-2020mulcanap 7646 Cancellation law for multiplication (full theorem form). (Contributed by Jim Kingdon, 21-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵))
     
    21-Feb-2020mulcanap2ad 7645 Cancellation of a nonzero factor on the right in an equation. One-way deduction form of mulcanap2d 7643. (Contributed by Jim Kingdon, 21-Feb-2020.)
    (𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐶 # 0)    &   (𝜑 → (𝐴 · 𝐶) = (𝐵 · 𝐶))    ⇒   (𝜑 → 𝐴 = 𝐵)
     
    21-Feb-2020mulcanapad 7644 Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcanapd 7642. (Contributed by Jim Kingdon, 21-Feb-2020.)
    (𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐶 # 0)    &   (𝜑 → (𝐶 · 𝐴) = (𝐶 · 𝐵))    ⇒   (𝜑 → 𝐴 = 𝐵)
     
    21-Feb-2020mulcanap2d 7643 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
    (𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐶 # 0)    ⇒   (𝜑 → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵))
     
    21-Feb-2020mulcanapd 7642 Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
    (𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐶 # 0)    ⇒   (𝜑 → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵))
     
    21-Feb-2020apne 7614 Apartness implies negated equality. We cannot in general prove the converse, which is the whole point of having separate notations for apartness and negated equality. (Contributed by Jim Kingdon, 21-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 # 𝐵 → 𝐴 ≠ 𝐵))
     
    21-Feb-2020apti 7613 Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 = 𝐵 ↔ ¬ 𝐴 # 𝐵))
     
    20-Feb-2020recexap 7634 Existence of reciprocal of nonzero complex number. (Contributed by Jim Kingdon, 20-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ∃𝑥 ∈ ℂ (𝐴 · 𝑥) = 1)
     
    20-Feb-2020recexaplem2 7633 Lemma for recexap 7634. (Contributed by Jim Kingdon, 20-Feb-2020.)
    ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 + (i · 𝐵)) # 0) → ((𝐴 · 𝐴) + (𝐵 · 𝐵)) # 0)
     
    19-Feb-2020remulext1 7590 Left extensionality for multiplication. (Contributed by Jim Kingdon, 19-Feb-2020.)
    ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) # (𝐵 · 𝐶) → 𝐴 # 𝐵))
     
    18-Feb-2020ax-pre-mulext 7002 Strong extensionality of multiplication (expressed in terms of <ℝ). Axiom for real and complex numbers, justified by theorem axpre-mulext 6962

    (Contributed by Jim Kingdon, 18-Feb-2020.)

    ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) <ℝ (𝐵 · 𝐶) → (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)))
     
    18-Feb-2020axpre-mulext 6962 Strong extensionality of multiplication (expressed in terms of <ℝ). Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulext 7002.

    (Contributed by Jim Kingdon, 18-Feb-2020.) (New usage is discouraged.)

    ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) <ℝ (𝐵 · 𝐶) → (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)))
     
    18-Feb-2020mulextsr1 6865 Strong extensionality of multiplication of signed reals. (Contributed by Jim Kingdon, 18-Feb-2020.)
    ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → ((𝐴 ·R 𝐶) <R (𝐵 ·R 𝐶) → (𝐴 <R 𝐵 ∨ 𝐵 <R 𝐴)))
     
    18-Feb-2020ltmprr 6740 Ordering property of multiplication. (Contributed by Jim Kingdon, 18-Feb-2020.)
    ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵) → 𝐴<P 𝐵))
     
    17-Feb-2020mulextsr1lem 6864 Lemma for mulextsr1 6865. (Contributed by Jim Kingdon, 17-Feb-2020.)
    (((𝑋 ∈ P ∧ 𝑌 ∈ P) ∧ (𝑍 ∈ P ∧ 𝑊 ∈ P) ∧ (𝑈 ∈ P ∧ 𝑉 ∈ P)) → ((((𝑋 ·P 𝑈) +P (𝑌 ·P 𝑉)) +P ((𝑍 ·P 𝑉) +P (𝑊 ·P 𝑈)))<P (((𝑋 ·P 𝑉) +P (𝑌 ·P 𝑈)) +P ((𝑍 ·P 𝑈) +P (𝑊 ·P 𝑉))) → ((𝑋 +P 𝑊)<P (𝑌 +P 𝑍) ∨ (𝑍 +P 𝑌)<P (𝑊 +P 𝑋))))
     
    17-Feb-2020addextpr 6719 Strong extensionality of addition (ordering version). This is similar to addext 7601 but for positive reals and based on less-than rather than apartness. (Contributed by Jim Kingdon, 17-Feb-2020.)
    (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ((𝐴 +P 𝐵)<P (𝐶 +P 𝐷) → (𝐴<P 𝐶 ∨ 𝐵<P 𝐷)))
     
    16-Feb-2020apadd2 7600 Addition respects apartness. (Contributed by Jim Kingdon, 16-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 ↔ (𝐶 + 𝐴) # (𝐶 + 𝐵)))
     
    16-Feb-2020apcotr 7598 Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 → (𝐴 # 𝐶 ∨ 𝐵 # 𝐶)))
     
    16-Feb-2020apsym 7597 Apartness is symmetric. This theorem for real numbers is part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 # 𝐵 ↔ 𝐵 # 𝐴))
     
    16-Feb-2020apirr 7596 Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.)
    (𝐴 ∈ ℂ → ¬ 𝐴 # 𝐴)
     
    16-Feb-2020reapcotr 7589 Real apartness is cotransitive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.)
    ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 # 𝐵 → (𝐴 # 𝐶 ∨ 𝐵 # 𝐶)))
     
    15-Feb-2020addext 7601 Strong extensionality for addition. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5521. For us, it is proved a different way. (Contributed by Jim Kingdon, 15-Feb-2020.)
    (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) # (𝐶 + 𝐷) → (𝐴 # 𝐶 ∨ 𝐵 # 𝐷)))
     
    14-Feb-2020apneg 7602 Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 # 𝐵 ↔ -𝐴 # -𝐵))
     
    13-Feb-2020apadd1 7599 Addition respects apartness. Analogue of addcan 7191 for apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 # 𝐵 ↔ (𝐴 + 𝐶) # (𝐵 + 𝐶)))
     
    13-Feb-2020reapneg 7588 Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
    ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ -𝐴 # -𝐵))
     
    13-Feb-2020reapadd1 7587 Real addition respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
    ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 + 𝐶) # (𝐵 + 𝐶)))
     
    12-Feb-2020apreim 7594 Complex apartness in terms of real and imaginary parts. (Contributed by Jim Kingdon, 12-Feb-2020.)
    (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 + (i · 𝐵)) # (𝐶 + (i · 𝐷)) ↔ (𝐴 # 𝐶 ∨ 𝐵 # 𝐷)))
     
    8-Feb-2020reapmul1 7586 Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 7764. (Contributed by Jim Kingdon, 8-Feb-2020.)
    ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 𝐶 # 0)) → (𝐴 # 𝐵 ↔ (𝐴 · 𝐶) # (𝐵 · 𝐶)))
     
    8-Feb-2020reapmul1lem 7585 Lemma for reapmul1 7586. (Contributed by Jim Kingdon, 8-Feb-2020.)
    ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 # 𝐵 ↔ (𝐴 · 𝐶) # (𝐵 · 𝐶)))
     
    7-Feb-2020apsqgt0 7592 The square of a real number apart from zero is positive. (Contributed by Jim Kingdon, 7-Feb-2020.)
    ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → 0 < (𝐴 · 𝐴))
     
    6-Feb-2020recexgt0 7571 Existence of reciprocal of positive real number. (Contributed by Jim Kingdon, 6-Feb-2020.)
    ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1))
     
    6-Feb-2020ax-precex 6994 Existence of reciprocal of positive real number. Axiom for real and complex numbers, justified by theorem axprecex 6954. (Contributed by Jim Kingdon, 6-Feb-2020.)
    ((𝐴 ∈ ℝ ∧ 0 <ℝ 𝐴) → ∃𝑥 ∈ ℝ (0 <ℝ 𝑥 ∧ (𝐴 · 𝑥) = 1))
     
    6-Feb-2020axprecex 6954 Existence of positive reciprocal of positive real number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-precex 6994.

    In treatments which assume excluded middle, the 0 <ℝ 𝐴 condition is generally replaced by 𝐴 ≠ 0, and it may not be necessary to state that the reciproacal is positive. (Contributed by Jim Kingdon, 6-Feb-2020.) (New usage is discouraged.)

    ((𝐴 ∈ ℝ ∧ 0 <ℝ 𝐴) → ∃𝑥 ∈ ℝ (0 <ℝ 𝑥 ∧ (𝐴 · 𝑥) = 1))
     
    6-Feb-2020recexgt0sr 6858 The reciprocal of a positive signed real exists and is positive. (Contributed by Jim Kingdon, 6-Feb-2020.)
    (0R <R 𝐴 → ∃𝑥 ∈ R (0R <R 𝑥 ∧ (𝐴 ·R 𝑥) = 1R))
     
    1-Feb-2020reaplt 7579 Real apartness in terms of less than. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 1-Feb-2020.)
    ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
     
    31-Jan-2020apreap 7578 Complex apartness and real apartness agree on the real numbers. (Contributed by Jim Kingdon, 31-Jan-2020.)
    ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ 𝐴 #ℝ 𝐵))
     
    30-Jan-2020rereim 7577 Decomposition of a real number into real part (itself) and imaginary part (zero). (Contributed by Jim Kingdon, 30-Jan-2020.)
    (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐴 = (𝐵 + (i · 𝐶)))) → (𝐵 = 𝐴 ∧ 𝐶 = 0))
     
    30-Jan-2020reapti 7570 Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 7613. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.)
    ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ ¬ 𝐴 #ℝ 𝐵))
     
    29-Jan-2020recexre 7569 Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.)
    ((𝐴 ∈ ℝ ∧ 𝐴 #ℝ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
     
    29-Jan-2020reapval 7567 Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 7579 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.)
    ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 #ℝ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
     
    29-Jan-2020axapti 7090 Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 6999 with ordering on the extended reals.) (Contributed by Jim Kingdon, 29-Jan-2020.)
    ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)) → 𝐴 = 𝐵)
     
    29-Jan-2020ax-pre-apti 6999 Apartness of reals is tight. Axiom for real and complex numbers, justified by theorem axpre-apti 6959. (Contributed by Jim Kingdon, 29-Jan-2020.)
    ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)) → 𝐴 = 𝐵)
     
    29-Jan-2020axpre-apti 6959 Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-apti 6999.

    (Contributed by Jim Kingdon, 29-Jan-2020.) (New usage is discouraged.)

    ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)) → 𝐴 = 𝐵)
     
    29-Jan-2020aptisr 6863 Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.)
    ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ ¬ (𝐴 <R 𝐵 ∨ 𝐵 <R 𝐴)) → 𝐴 = 𝐵)
     
    28-Jan-2020aptipr 6739 Apartness of positive reals is tight. (Contributed by Jim Kingdon, 28-Jan-2020.)
    ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → 𝐴 = 𝐵)
     
    28-Jan-2020aptiprlemu 6738 Lemma for aptipr 6739. (Contributed by Jim Kingdon, 28-Jan-2020.)
    ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (2nd ‘𝐵) ⊆ (2nd ‘𝐴))
     
    28-Jan-2020aptiprleml 6737 Lemma for aptipr 6739. (Contributed by Jim Kingdon, 28-Jan-2020.)
    ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (1st ‘𝐴) ⊆ (1st ‘𝐵))
     
    27-Jan-2020bj-dcbi 10048 Equivalence property for DECID. TODO: solve conflict with dcbi 844; minimize dcbii 747 and dcbid 748 with it, as well as theorems using those. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
    ((𝜑 ↔ 𝜓) → (DECID 𝜑 ↔ DECID 𝜓))
     
    27-Jan-2020bj-notbid 10047 Deduction form of bj-notbi 10045. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
    (𝜑 → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))
     
    27-Jan-2020bj-notbii 10046 Inference associated with bj-notbi 10045. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
    (𝜑 ↔ 𝜓)    ⇒   (¬ 𝜑 ↔ ¬ 𝜓)
     
    27-Jan-2020bj-notbi 10045 Equivalence property for negation. TODO: minimize all theorems using notbid 592 and notbii 594. (Contributed by BJ, 27-Jan-2020.) (Proof modification is discouraged.)
    ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
     
    26-Jan-2020df-ap 7573 Define complex apartness. Definition 6.1 of [Geuvers], p. 17.

    Two numbers are considered apart if it is possible to separate them. One common usage is that we can divide by a number if it is apart from zero (see for example recclap 7658 which says that a number apart from zero has a reciprocal).

    The defining characteristics of an apartness are irreflexivity (apirr 7596), symmetry (apsym 7597), and cotransitivity (apcotr 7598). Apartness implies negated equality, as seen at apne 7614, and the converse would also follow if we assumed excluded middle.

    In addition, apartness of complex numbers is tight, which means that two numbers which are not apart are equal (apti 7613).

    (Contributed by Jim Kingdon, 26-Jan-2020.)

    # = {⟨𝑥, 𝑦⟩ ∣ ∃𝑟 ∈ ℝ ∃𝑠 ∈ ℝ ∃𝑡 ∈ ℝ ∃𝑢 ∈ ℝ ((𝑥 = (𝑟 + (i · 𝑠)) ∧ 𝑦 = (𝑡 + (i · 𝑢))) ∧ (𝑟 #ℝ 𝑡 ∨ 𝑠 #ℝ 𝑢))}
     
    26-Jan-2020reapirr 7568 Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 7596 instead. (Contributed by Jim Kingdon, 26-Jan-2020.)
    (𝐴 ∈ ℝ → ¬ 𝐴 #ℝ 𝐴)
     
    26-Jan-2020df-reap 7566 Define real apartness. Definition in Section 11.2.1 of [HoTT], p. (varies). Although #ℝ is an apartness relation on the reals (see df-ap 7573 for more discussion of apartness relations), for our purposes it is just a stepping stone to defining # which is an apartness relation on complex numbers. On the reals, #ℝ and # agree (apreap 7578). (Contributed by Jim Kingdon, 26-Jan-2020.)
    #ℝ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑥 < 𝑦 ∨ 𝑦 < 𝑥))}
     
    26-Jan-2020gt0add 7564 A positive sum must have a positive addend. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 26-Jan-2020.)
    ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < (𝐴 + 𝐵)) → (0 < 𝐴 ∨ 0 < 𝐵))
     
    17-Jan-2020addcom 7150 Addition commutes. (Contributed by Jim Kingdon, 17-Jan-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
     
    17-Jan-2020ax-addcom 6984 Addition commutes. Axiom for real and complex numbers, justified by theorem axaddcom 6944. Proofs should normally use addcom 7150 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 17-Jan-2020.)
    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
     
    17-Jan-2020axaddcom 6944 Addition commutes. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addcom 6984 be used later. Instead, use addcom 7150.

    In the Metamath Proof Explorer this is not a complex number axiom but is instead proved from other axioms. That proof relies on real number trichotomy and it is not known whether it is possible to prove this from the other axioms without it. (Contributed by Jim Kingdon, 17-Jan-2020.) (New usage is discouraged.)

    ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
     
    16-Jan-2020addid1 7151 0 is an additive identity. (Contributed by Jim Kingdon, 16-Jan-2020.)
    (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
     
    16-Jan-2020ax-0id 6992 0 is an identity element for real addition. Axiom for real and complex numbers, justified by theorem ax0id 6952.

    Proofs should normally use addid1 7151 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 16-Jan-2020.)

    (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
     
    16-Jan-2020ax0id 6952 0 is an identity element for real addition. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-0id 6992.

    In the Metamath Proof Explorer this is not a complex number axiom but is instead proved from other axioms. That proof relies on excluded middle and it is not known whether it is possible to prove this from the other axioms without excluded middle. (Contributed by Jim Kingdon, 16-Jan-2020.) (New usage is discouraged.)

    (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
     
    15-Jan-2020axltwlin 7087 Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 6997 with ordering on the extended reals. (Contributed by Jim Kingdon, 15-Jan-2020.)
    ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐴 < 𝐶 ∨ 𝐶 < 𝐵)))
     
    15-Jan-2020axltirr 7086 Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 6996 with ordering on the extended reals. New proofs should use ltnr 7095 instead for naming consistency. (New usage is discouraged.) (Contributed by Jim Kingdon, 15-Jan-2020.)
    (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
     
    14-Jan-20202pwuninelg 5898 The power set of the power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by Jim Kingdon, 14-Jan-2020.)
    (𝐴 ∈ 𝑉 → ¬ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴)
     
    13-Jan-20201re 7026 1 is a real number. (Contributed by Jim Kingdon, 13-Jan-2020.)
    1 ∈ ℝ
     
    13-Jan-2020ax-1re 6978 1 is a real number. Axiom for real and complex numbers, justified by theorem ax1re 6938. Proofs should use 1re 7026 instead. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.)
    1 ∈ ℝ
     
    13-Jan-2020ax1re 6938 1 is a real number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1re 6978.

    In the Metamath Proof Explorer, this is not a complex number axiom but is proved from ax-1cn 6977 and the other axioms. It is not known whether we can do so here, but the Metamath Proof Explorer proof (accessed 13-Jan-2020) uses excluded middle. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.)

    1 ∈ ℝ
     
    12-Jan-2020ax-pre-ltwlin 6997 Real number less-than is weakly linear. Axiom for real and complex numbers, justified by theorem axpre-ltwlin 6957. (Contributed by Jim Kingdon, 12-Jan-2020.)
    ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵)))
     
    12-Jan-2020ax-pre-ltirr 6996 Real number less-than is irreflexive. Axiom for real and complex numbers, justified by theorem ax-pre-ltirr 6996. (Contributed by Jim Kingdon, 12-Jan-2020.)
    (𝐴 ∈ ℝ → ¬ 𝐴 <ℝ 𝐴)
     
    12-Jan-2020ax-0lt1 6990 0 is less than 1. Axiom for real and complex numbers, justified by theorem ax0lt1 6950. Proofs should normally use 0lt1 7141 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 12-Jan-2020.)
    0 <ℝ 1
     
    12-Jan-2020axpre-ltwlin 6957 Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltwlin 6997. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.)
    ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵)))
     
    12-Jan-2020axpre-ltirr 6956 Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltirr 6996. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.)
    (𝐴 ∈ ℝ → ¬ 𝐴 <ℝ 𝐴)
     
    12-Jan-2020ax0lt1 6950 0 is less than 1. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-0lt1 6990.

    The version of this axiom in the Metamath Proof Explorer reads 1 ≠ 0; here we change it to 0 <ℝ 1. The proof of 0 <ℝ 1 from 1 ≠ 0 in the Metamath Proof Explorer (accessed 12-Jan-2020) relies on real number trichotomy. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.)

    0 <ℝ 1
     
    8-Jan-2020ecidg 6170 A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by Jim Kingdon, 8-Jan-2020.)
    (𝐴 ∈ 𝑉 → [𝐴]◡ E = 𝐴)
     
    5-Jan-2020elfzom1elp1fzo 9058 Membership of an integer incremented by one in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Proof shortened by AV, 5-Jan-2020.)
    ((𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(𝑁 − 1))) → (𝐼 + 1) ∈ (0..^𝑁))
     
    5-Jan-20201idsr 6853 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.)
    (𝐴 ∈ R → (𝐴 ·R 1R) = 𝐴)
     
    4-Jan-2020nn0ge2m1nn 8242 If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 4-Jan-2020.)
    ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ)
     
    4-Jan-2020distrsrg 6844 Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.)
    ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶)))
     
    3-Jan-2020mulasssrg 6843 Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)
    ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → ((𝐴 ·R 𝐵) ·R 𝐶) = (𝐴 ·R (𝐵 ·R 𝐶)))
     
    3-Jan-2020mulcomsrg 6842 Multiplication of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.)
    ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) = (𝐵 ·R 𝐴))
     
    3-Jan-2020addasssrg 6841 Addition of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)
    ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → ((𝐴 +R 𝐵) +R 𝐶) = (𝐴 +R (𝐵 +R 𝐶)))
     
    3-Jan-2020addcomsrg 6840 Addition of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.)
    ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) = (𝐵 +R 𝐴))
     
    3-Jan-2020caovlem2d 5693 Rearrangement of expression involving multiplication (𝐺) and addition (𝐹). (Contributed by Jim Kingdon, 3-Jan-2020.)
    ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐹(𝑦𝐺𝑧)))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) ∈ 𝑆)    &   (𝜑 → 𝐴 ∈ 𝑆)    &   (𝜑 → 𝐵 ∈ 𝑆)    &   (𝜑 → 𝐶 ∈ 𝑆)    &   (𝜑 → 𝐷 ∈ 𝑆)    &   (𝜑 → 𝐻 ∈ 𝑆)    &   (𝜑 → 𝑅 ∈ 𝑆)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)    ⇒   (𝜑 → ((((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻)𝐹(((𝐴𝐺𝐷)𝐹(𝐵𝐺𝐶))𝐺𝑅)) = ((𝐴𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅)))𝐹(𝐵𝐺((𝐶𝐺𝑅)𝐹(𝐷𝐺𝐻)))))
     
    2-Jan-2020bj-d0clsepcl 10049 Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.)
    DECID 𝜑
     
    2-Jan-2020ax-bj-d0cl 10044 Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.)
    BOUNDED 𝜑    ⇒   DECID 𝜑
     
    1-Jan-2020mulcmpblnrlemg 6825 Lemma used in lemma showing compatibility of multiplication. (Contributed by Jim Kingdon, 1-Jan-2020.)
    ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) ∧ ((𝐹 ∈ P ∧ 𝐺 ∈ P) ∧ (𝑅 ∈ P ∧ 𝑆 ∈ P))) → (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐵 ·P 𝐺)) +P ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐵 ·P 𝐹)) +P ((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆))))))
     
    31-Dec-2019eceq1d 6142 Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.)
    (𝜑 → 𝐴 = 𝐵)    ⇒   (𝜑 → [𝐴]𝐶 = [𝐵]𝐶)
     
    30-Dec-2019mulsrmo 6829 There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
    ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))
     
    30-Dec-2019addsrmo 6828 There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
    ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ))
     
    30-Dec-2019prsrlem1 6827 Decomposing signed reals into positive reals. Lemma for addsrpr 6830 and mulsrpr 6831. (Contributed by Jim Kingdon, 30-Dec-2019.)
    (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → ((((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ (𝑠 ∈ P ∧ 𝑓 ∈ P)) ∧ ((𝑢 ∈ P ∧ 𝑡 ∈ P) ∧ (𝑔 ∈ P ∧ ℎ ∈ P))) ∧ ((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ℎ) = (𝑡 +P 𝑔))))
     
    29-Dec-2019bj-omelon 10086 The set ω is an ordinal. Constructive proof of omelon 4331. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
    ω ∈ On
     
    29-Dec-2019bj-omord 10085 The set ω is an ordinal. Constructive proof of ordom 4329. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
    Ord ω
     
    29-Dec-2019bj-omtrans2 10082 The set ω is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)
    Tr ω
     
    29-Dec-2019bj-omtrans 10081 The set ω is transitive. A natural number is included in ω. Constructive proof of elnn 4328.

    The idea is to use bounded induction with the formula 𝑥 ⊆ ω. This formula, in a logic with terms, is bounded. So in our logic without terms, we need to temporarily replace it with 𝑥 ⊆ 𝑎 and then deduce the original claim. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.)

    (𝐴 ∈ ω → 𝐴 ⊆ ω)
     
    29-Dec-2019ltrnqg 6518 Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 6519. (Contributed by Jim Kingdon, 29-Dec-2019.)
    ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ (*Q‘𝐵) <Q (*Q‘𝐴)))
     
    29-Dec-2019rec1nq 6493 Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.)
    (*Q‘1Q) = 1Q
     
    28-Dec-2019recexprlemupu 6726 The upper cut of 𝐵 is upper. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 28-Dec-2019.)
    𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   ((𝐴 ∈ P ∧ 𝑟 ∈ Q) → (∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵)) → 𝑟 ∈ (2nd ‘𝐵)))
     
    28-Dec-2019recexprlemopu 6725 The upper cut of 𝐵 is open. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 28-Dec-2019.)
    𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   ((𝐴 ∈ P ∧ 𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐵)) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵)))
     
    28-Dec-2019recexprlemlol 6724 The lower cut of 𝐵 is lower. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 28-Dec-2019.)
    𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   ((𝐴 ∈ P ∧ 𝑞 ∈ Q) → (∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵)) → 𝑞 ∈ (1st ‘𝐵)))
     
    28-Dec-2019recexprlemopl 6723 The lower cut of 𝐵 is open. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 28-Dec-2019.)
    𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   ((𝐴 ∈ P ∧ 𝑞 ∈ Q ∧ 𝑞 ∈ (1st ‘𝐵)) → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵)))
     
    28-Dec-2019prmuloc2 6665 Positive reals are multiplicatively located. This is a variation of prmuloc 6664 which only constructs one (named) point and is therefore often easier to work with. It states that given a ratio 𝐵, there are elements of the lower and upper cut which have exactly that ratio between them. (Contributed by Jim Kingdon, 28-Dec-2019.)
    ((⟨𝐿, 𝑈⟩ ∈ P ∧ 1Q <Q 𝐵) → ∃𝑥 ∈ 𝐿 (𝑥 ·Q 𝐵) ∈ 𝑈)
     
    28-Dec-20191pru 6654 The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
    (2nd ‘1P) = {𝑥 ∣ 1Q <Q 𝑥}
     
    28-Dec-20191prl 6653 The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
    (1st ‘1P) = {𝑥 ∣ 𝑥 <Q 1Q}
     
    27-Dec-2019recexprlemex 6735 𝐵 is the reciprocal of 𝐴. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
    𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   (𝐴 ∈ P → (𝐴 ·P 𝐵) = 1P)
     
    27-Dec-2019recexprlemss1u 6734 The upper cut of 𝐴 ·P 𝐵 is a subset of the upper cut of one. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
    𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   (𝐴 ∈ P → (2nd ‘(𝐴 ·P 𝐵)) ⊆ (2nd ‘1P))
     
    27-Dec-2019recexprlemss1l 6733 The lower cut of 𝐴 ·P 𝐵 is a subset of the lower cut of one. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
    𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   (𝐴 ∈ P → (1st ‘(𝐴 ·P 𝐵)) ⊆ (1st ‘1P))
     
    27-Dec-2019recexprlem1ssu 6732 The upper cut of one is a subset of the upper cut of 𝐴 ·P 𝐵. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
    𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   (𝐴 ∈ P → (2nd ‘1P) ⊆ (2nd ‘(𝐴 ·P 𝐵)))
     
    27-Dec-2019recexprlem1ssl 6731 The lower cut of one is a subset of the lower cut of 𝐴 ·P 𝐵. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
    𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   (𝐴 ∈ P → (1st ‘1P) ⊆ (1st ‘(𝐴 ·P 𝐵)))
     
    27-Dec-2019recexprlempr 6730 𝐵 is a positive real. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
    𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   (𝐴 ∈ P → 𝐵 ∈ P)
     
    27-Dec-2019recexprlemloc 6729 𝐵 is located. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
    𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   (𝐴 ∈ P → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵))))
     
    27-Dec-2019recexprlemdisj 6728 𝐵 is disjoint. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
    𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   (𝐴 ∈ P → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐵) ∧ 𝑞 ∈ (2nd ‘𝐵)))
     
    27-Dec-2019recexprlemrnd 6727 𝐵 is rounded. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
    𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   (𝐴 ∈ P → (∀𝑞 ∈ Q (𝑞 ∈ (1st ‘𝐵) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐵))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵)))))
     
    27-Dec-2019recexprlemm 6722 𝐵 is inhabited. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
    𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   (𝐴 ∈ P → (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐵) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐵)))
     
    27-Dec-2019recexprlemelu 6721 Membership in the upper cut of 𝐵. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
    𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   (𝐶 ∈ (2nd ‘𝐵) ↔ ∃𝑦(𝑦 <Q 𝐶 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)))
     
    27-Dec-2019recexprlemell 6720 Membership in the lower cut of 𝐵. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)
    𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩    ⇒   (𝐶 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)))
     
    26-Dec-2019ltaprg 6717 Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by Jim Kingdon, 26-Dec-2019.)
    ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
     
    26-Dec-2019prarloc2 6602 A Dedekind cut is arithmetically located. This is a variation of prarloc 6601 which only constructs one (named) point and is therefore often easier to work with. It states that given a tolerance 𝑃, there are elements of the lower and upper cut which are exactly that tolerance from each other. (Contributed by Jim Kingdon, 26-Dec-2019.)
    ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝑃 ∈ Q) → ∃𝑎 ∈ 𝐿 (𝑎 +Q 𝑃) ∈ 𝑈)
     
    25-Dec-2019addcanprlemu 6713 Lemma for addcanprg 6714. (Contributed by Jim Kingdon, 25-Dec-2019.)
    (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd ‘𝐵) ⊆ (2nd ‘𝐶))
     
    25-Dec-2019addcanprleml 6712 Lemma for addcanprg 6714. (Contributed by Jim Kingdon, 25-Dec-2019.)
    (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st ‘𝐵) ⊆ (1st ‘𝐶))
     
    24-Dec-2019addcanprg 6714 Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123. (Contributed by Jim Kingdon, 24-Dec-2019.)
    ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶))
     
    23-Dec-2019ltprordil 6687 If a positive real is less than a second positive real, its lower cut is a subset of the second's lower cut. (Contributed by Jim Kingdon, 23-Dec-2019.)
    (𝐴<P 𝐵 → (1st ‘𝐴) ⊆ (1st ‘𝐵))
     
    22-Dec-2019bj-findis 10104 Principle of induction, using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See bj-bdfindis 10072 for a bounded version not requiring ax-setind 4262. See finds 4323 for a proof in IZF. From this version, it is easy to prove of finds 4323, finds2 4324, finds1 4325. (Contributed by BJ, 22-Dec-2019.) (Proof modification is discouraged.)
    Ⅎ𝑥𝜓    &   Ⅎ𝑥𝜒    &   Ⅎ𝑥𝜃    &   (𝑥 = ∅ → (𝜓 → 𝜑))    &   (𝑥 = 𝑦 → (𝜑 → 𝜒))    &   (𝑥 = suc 𝑦 → (𝜃 → 𝜑))    ⇒   ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑)
     
    21-Dec-2019ltexprlemupu 6702 The upper cut of our constructed difference is upper. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 21-Dec-2019.)
    𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ((𝐴<P 𝐵 ∧ 𝑟 ∈ Q) → (∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶)) → 𝑟 ∈ (2nd ‘𝐶)))
     
    21-Dec-2019ltexprlemopu 6701 The upper cut of our constructed difference is open. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 21-Dec-2019.)
    𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ((𝐴<P 𝐵 ∧ 𝑟 ∈ Q ∧ 𝑟 ∈ (2nd ‘𝐶)) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶)))
     
    21-Dec-2019ltexprlemlol 6700 The lower cut of our constructed difference is lower. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 21-Dec-2019.)
    𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ((𝐴<P 𝐵 ∧ 𝑞 ∈ Q) → (∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐶)) → 𝑞 ∈ (1st ‘𝐶)))
     
    21-Dec-2019ltexprlemopl 6699 The lower cut of our constructed difference is open. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 21-Dec-2019.)
    𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   ((𝐴<P 𝐵 ∧ 𝑞 ∈ Q ∧ 𝑞 ∈ (1st ‘𝐶)) → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐶)))
     
    21-Dec-2019ltexprlemelu 6697 Element in upper cut of the constructed difference. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 21-Dec-2019.)
    𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   (𝑟 ∈ (2nd ‘𝐶) ↔ (𝑟 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd ‘𝐵))))
     
    21-Dec-2019ltexprlemell 6696 Element in lower cut of the constructed difference. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 21-Dec-2019.)
    𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   (𝑞 ∈ (1st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st ‘𝐵))))
     
    17-Dec-2019ltexprlemru 6710 Lemma for ltexpri 6711. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
    𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   (𝐴<P 𝐵 → (2nd ‘𝐵) ⊆ (2nd ‘(𝐴 +P 𝐶)))
     
    17-Dec-2019ltexprlemfu 6709 Lemma for ltexpri 6711. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
    𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   (𝐴<P 𝐵 → (2nd ‘(𝐴 +P 𝐶)) ⊆ (2nd ‘𝐵))
     
    17-Dec-2019ltexprlemrl 6708 Lemma for ltexpri 6711. Reverse directon of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
    𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   (𝐴<P 𝐵 → (1st ‘𝐵) ⊆ (1st ‘(𝐴 +P 𝐶)))
     
    17-Dec-2019ltexprlemfl 6707 Lemma for ltexpri 6711. One directon of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
    𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   (𝐴<P 𝐵 → (1st ‘(𝐴 +P 𝐶)) ⊆ (1st ‘𝐵))
     
    17-Dec-2019ltexprlempr 6706 Our constructed difference is a positive real. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 17-Dec-2019.)
    𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   (𝐴<P 𝐵 → 𝐶 ∈ P)
     
    17-Dec-2019ltexprlemloc 6705 Our constructed difference is located. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 17-Dec-2019.)
    𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   (𝐴<P 𝐵 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐶) ∨ 𝑟 ∈ (2nd ‘𝐶))))
     
    17-Dec-2019ltexprlemdisj 6704 Our constructed difference is disjoint. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 17-Dec-2019.)
    𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   (𝐴<P 𝐵 → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1st ‘𝐶) ∧ 𝑞 ∈ (2nd ‘𝐶)))
     
    17-Dec-2019ltexprlemrnd 6703 Our constructed difference is rounded. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 17-Dec-2019.)
    𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   (𝐴<P 𝐵 → (∀𝑞 ∈ Q (𝑞 ∈ (1st ‘𝐶) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐶))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐶) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐶)))))
     
    17-Dec-2019ltexprlemm 6698 Our constructed difference is inhabited. Lemma for ltexpri 6711. (Contributed by Jim Kingdon, 17-Dec-2019.)
    𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2nd ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1st ‘𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd ‘𝐵))}⟩    ⇒   (𝐴<P 𝐵 → (∃𝑞 ∈ Q 𝑞 ∈ (1st ‘𝐶) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐶)))
     
    16-Dec-2019bj-sbime 9913 A strengthening of sbie 1674 (same proof). (Contributed by BJ, 16-Dec-2019.)
    Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ([𝑦 / 𝑥]𝜑 → 𝜓)
     
    16-Dec-2019bj-sbimeh 9912 A strengthening of sbieh 1673 (same proof). (Contributed by BJ, 16-Dec-2019.)
    (𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ([𝑦 / 𝑥]𝜑 → 𝜓)
     
    16-Dec-2019bj-sbimedh 9911 A strengthening of sbiedh 1670 (same proof). (Contributed by BJ, 16-Dec-2019.)
    (𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜒 → ∀𝑥𝜒))    &   (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))    ⇒   (𝜑 → ([𝑦 / 𝑥]𝜓 → 𝜒))
     
    16-Dec-2019ltsopr 6694 Positive real 'less than' is a weak linear order (in the sense of df-iso 4034). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.)
    <P Or P
     
    15-Dec-2019ltpopr 6693 Positive real 'less than' is a partial ordering. Remark ("< is transitive and irreflexive") preceding Proposition 11.2.3 of [HoTT], p. (varies). Lemma for ltsopr 6694. (Contributed by Jim Kingdon, 15-Dec-2019.)
    <P Po P
     
    15-Dec-2019ltdfpr 6604 More convenient form of df-iltp 6568. (Contributed by Jim Kingdon, 15-Dec-2019.)
    ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (1st ‘𝐵))))
     
    15-Dec-2019prdisj 6590 A Dedekind cut is disjoint. (Contributed by Jim Kingdon, 15-Dec-2019.)
    ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 ∈ Q) → ¬ (𝐴 ∈ 𝐿 ∧ 𝐴 ∈ 𝑈))
     
    13-Dec-20191idpru 6689 Lemma for 1idpr 6690. (Contributed by Jim Kingdon, 13-Dec-2019.)
    (𝐴 ∈ P → (2nd ‘(𝐴 ·P 1P)) = (2nd ‘𝐴))
     
    13-Dec-20191idprl 6688 Lemma for 1idpr 6690. (Contributed by Jim Kingdon, 13-Dec-2019.)
    (𝐴 ∈ P → (1st ‘(𝐴 ·P 1P)) = (1st ‘𝐴))
     
    13-Dec-2019gtnqex 6648 The class of rationals greater than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
    {𝑥 ∣ 𝐴 <Q 𝑥} ∈ V
     
    13-Dec-2019ltnqex 6647 The class of rationals less than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
    {𝑥 ∣ 𝑥 <Q 𝐴} ∈ V
     
    13-Dec-2019sotritrieq 4062 A trichotomy relationship, given a trichotomous order. (Contributed by Jim Kingdon, 13-Dec-2019.)
    𝑅 Or 𝐴    &   ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))    ⇒   ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
     
    12-Dec-2019distrprg 6686 Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 12-Dec-2019.)
    ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P (𝐵 +P 𝐶)) = ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))
     
    12-Dec-2019distrlem5pru 6685 Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
    ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
     
    12-Dec-2019distrlem5prl 6684 Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
    ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))
     
    12-Dec-2019distrlem4pru 6683 Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
    (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
     
    12-Dec-2019distrlem4prl 6682 Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
    (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1st ‘𝐴) ∧ 𝑦 ∈ (1st ‘𝐵)) ∧ (𝑓 ∈ (1st ‘𝐴) ∧ 𝑧 ∈ (1st ‘𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))
     
    12-Dec-2019distrlem1pru 6681 Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
    ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ⊆ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
     
    12-Dec-2019distrlem1prl 6680 Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
    ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) ⊆ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
     
    12-Dec-2019ltdcnq 6495 Less-than for positive fractions is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
    ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → DECID 𝐴 <Q 𝐵)
     
    12-Dec-2019ltdcpi 6421 Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
    ((𝐴 ∈ N ∧ 𝐵 ∈ N) → DECID 𝐴 <N 𝐵)
     
    11-Dec-2019sublt0d 7561 When a subtraction gives a negative result. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
    (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    ⇒   (𝜑 → ((𝐴 − 𝐵) < 0 ↔ 𝐴 < 𝐵))
     
    11-Dec-2019mulassprg 6679 Multiplication of positive reals is associative. Proposition 9-3.7(i) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.)
    ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴 ·P 𝐵) ·P 𝐶) = (𝐴 ·P (𝐵 ·P 𝐶)))
     
    11-Dec-2019mulcomprg 6678 Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.)
    ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴))
     
    11-Dec-2019addassprg 6677 Addition of positive reals is associative. Proposition 9-3.5(i) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.)
    ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴 +P 𝐵) +P 𝐶) = (𝐴 +P (𝐵 +P 𝐶)))
     
    11-Dec-2019addcomprg 6676 Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.)
    ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) = (𝐵 +P 𝐴))
     
    11-Dec-2019genpassg 6624 Associativity of an operation on reals. (Contributed by Jim Kingdon, 11-Dec-2019.)
    𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    &   dom 𝐹 = (P × P)    &   ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓𝐹𝑔) ∈ P)    &   ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓𝐺𝑔)𝐺ℎ) = (𝑓𝐺(𝑔𝐺ℎ)))    ⇒   ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
     
    11-Dec-2019genpassu 6623 Associativity of upper cuts. Lemma for genpassg 6624. (Contributed by Jim Kingdon, 11-Dec-2019.)
    𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    &   dom 𝐹 = (P × P)    &   ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓𝐹𝑔) ∈ P)    &   ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓𝐺𝑔)𝐺ℎ) = (𝑓𝐺(𝑔𝐺ℎ)))    ⇒   ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (2nd ‘((𝐴𝐹𝐵)𝐹𝐶)) = (2nd ‘(𝐴𝐹(𝐵𝐹𝐶))))
     
    11-Dec-2019genpassl 6622 Associativity of lower cuts. Lemma for genpassg 6624. (Contributed by Jim Kingdon, 11-Dec-2019.)
    𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    &   dom 𝐹 = (P × P)    &   ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓𝐹𝑔) ∈ P)    &   ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓𝐺𝑔)𝐺ℎ) = (𝑓𝐺(𝑔𝐺ℎ)))    ⇒   ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (1st ‘((𝐴𝐹𝐵)𝐹𝐶)) = (1st ‘(𝐴𝐹(𝐵𝐹𝐶))))
     
    11-Dec-2019preqlu 6570 Two reals are equal if and only if their lower and upper cuts are. (Contributed by Jim Kingdon, 11-Dec-2019.)
    ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 = 𝐵 ↔ ((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵))))
     
    11-Dec-2019fssd 5055 Expanding the codomain of a mapping, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
    (𝜑 → 𝐹:𝐴⟶𝐵)    &   (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   (𝜑 → 𝐹:𝐴⟶𝐶)
     
    11-Dec-2019iffalsed 3341 Value of the conditional operator when its first argument is false. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
    (𝜑 → ¬ 𝜒)    ⇒   (𝜑 → if(𝜒, 𝐴, 𝐵) = 𝐵)
     
    11-Dec-2019iftrued 3338 Value of the conditional operator when its first argument is true. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
    (𝜑 → 𝜒)    ⇒   (𝜑 → if(𝜒, 𝐴, 𝐵) = 𝐴)
     
    11-Dec-2019neneq 2227 From inequality to non equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
    (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵)
     
    11-Dec-2019neqne 2214 From non equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
    (¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵)
     
    10-Dec-2019mullocprlem 6668 Calculations for mullocpr 6669. (Contributed by Jim Kingdon, 10-Dec-2019.)
    (𝜑 → (𝐴 ∈ P ∧ 𝐵 ∈ P))    &   (𝜑 → (𝑈 ·Q 𝑄) <Q (𝐸 ·Q (𝐷 ·Q 𝑈)))    &   (𝜑 → (𝐸 ·Q (𝐷 ·Q 𝑈)) <Q (𝑇 ·Q (𝐷 ·Q 𝑈)))    &   (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅))    &   (𝜑 → (𝑄 ∈ Q ∧ 𝑅 ∈ Q))    &   (𝜑 → (𝐷 ∈ Q ∧ 𝑈 ∈ Q))    &   (𝜑 → (𝐷 ∈ (1st ‘𝐴) ∧ 𝑈 ∈ (2nd ‘𝐴)))    &   (𝜑 → (𝐸 ∈ Q ∧ 𝑇 ∈ Q))    ⇒   (𝜑 → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
     
    10-Dec-2019mulnqpru 6667 Lemma to prove upward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)
    ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 → 𝑋 ∈ (2nd ‘(𝐴 ·P 𝐵))))
     
    10-Dec-2019mulnqprl 6666 Lemma to prove downward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)
    ((((𝐴 ∈ P ∧ 𝐺 ∈ (1st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 <Q (𝐺 ·Q 𝐻) → 𝑋 ∈ (1st ‘(𝐴 ·P 𝐵))))
     
    9-Dec-2019prmuloclemcalc 6663 Calculations for prmuloc 6664. (Contributed by Jim Kingdon, 9-Dec-2019.)
    (𝜑 → 𝑅 <Q 𝑈)    &   (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))    &   (𝜑 → (𝐴 +Q 𝑋) = 𝐵)    &   (𝜑 → (𝑃 ·Q 𝐵) <Q (𝑅 ·Q 𝑋))    &   (𝜑 → 𝐴 ∈ Q)    &   (𝜑 → 𝐵 ∈ Q)    &   (𝜑 → 𝐷 ∈ Q)    &   (𝜑 → 𝑃 ∈ Q)    &   (𝜑 → 𝑋 ∈ Q)    ⇒   (𝜑 → (𝑈 ·Q 𝐴) <Q (𝐷 ·Q 𝐵))
     
    9-Dec-2019appdiv0nq 6662 Approximate division for positive rationals. This can be thought of as a variation of appdivnq 6661 in which 𝐴 is zero, although it can be stated and proved in terms of positive rationals alone, without zero as such. (Contributed by Jim Kingdon, 9-Dec-2019.)
    ((𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ∃𝑚 ∈ Q (𝑚 ·Q 𝐶) <Q 𝐵)
     
    9-Dec-2019ltmnqi 6501 Ordering property of multiplication for positive fractions. One direction of ltmnqg 6499. (Contributed by Jim Kingdon, 9-Dec-2019.)
    ((𝐴 <Q 𝐵 ∧ 𝐶 ∈ Q) → (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵))
     
    9-Dec-2019ltanqi 6500 Ordering property of addition for positive fractions. One direction of ltanqg 6498. (Contributed by Jim Kingdon, 9-Dec-2019.)
    ((𝐴 <Q 𝐵 ∧ 𝐶 ∈ Q) → (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵))
     
    8-Dec-2019bj-nn0sucALT 10103 Alternate proof of bj-nn0suc 10089, also constructive but from ax-inf2 10101, hence requiring ax-bdsetind 10093. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
    (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
     
    8-Dec-2019bj-omex2 10102 Using bounded set induction and the strong axiom of infinity, ω is a set, that is, we recover ax-infvn 10066 (see bj-2inf 10062 for the equivalence of the latter with bj-omex 10067). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
    ω ∈ V
     
    8-Dec-2019bj-inf2vn2 10100 A sufficient condition for ω to be a set; unbounded version of bj-inf2vn 10099. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
    (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → 𝐴 = ω))
     
    8-Dec-2019bj-inf2vn 10099 A sufficient condition for ω to be a set. See bj-inf2vn2 10100 for the unbounded version from full set induction. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
    BOUNDED 𝐴    ⇒   (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → 𝐴 = ω))
     
    8-Dec-2019bj-inf2vnlem4 10098 Lemma for bj-inf2vn2 10100. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
    (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → 𝐴 ⊆ 𝑍))
     
    8-Dec-2019bj-inf2vnlem3 10097 Lemma for bj-inf2vn 10099. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
    BOUNDED 𝐴    &   BOUNDED 𝑍    ⇒   (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → 𝐴 ⊆ 𝑍))
     
    8-Dec-2019bj-inf2vnlem2 10096 Lemma for bj-inf2vnlem3 10097 and bj-inf2vnlem4 10098. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
    (∀𝑥 ∈ 𝐴 (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦) → (Ind 𝑍 → ∀𝑢(∀𝑡 ∈ 𝑢 (𝑡 ∈ 𝐴 → 𝑡 ∈ 𝑍) → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑍))))
     
    8-Dec-2019bj-inf2vnlem1 10095 Lemma for bj-inf2vn 10099. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
    (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 = ∅ ∨ ∃𝑦 ∈ 𝐴 𝑥 = suc 𝑦)) → Ind 𝐴)
     
    8-Dec-2019bj-bdcel 9957 Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.)
    BOUNDED 𝑦 = 𝐴    ⇒   BOUNDED 𝐴 ∈ 𝑥
     
    8-Dec-2019bj-ex 9902 Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1489 and 19.9ht 1532 or 19.23ht 1386). (Proof modification is discouraged.)
    (∃𝑥𝜑 → 𝜑)
     
    8-Dec-2019mullocpr 6669 Locatedness of multiplication on positive reals. Lemma 12.9 in [BauerTaylor], p. 56 (but where both 𝐴 and 𝐵 are positive, not just 𝐴). (Contributed by Jim Kingdon, 8-Dec-2019.)
    ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 ·P 𝐵)))))
     
    8-Dec-2019prmuloc 6664 Positive reals are multiplicatively located. Lemma 12.8 of [BauerTaylor], p. 56. (Contributed by Jim Kingdon, 8-Dec-2019.)
    ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐴 <Q 𝐵) → ∃𝑑 ∈ Q ∃𝑢 ∈ Q (𝑑 ∈ 𝐿 ∧ 𝑢 ∈ 𝑈 ∧ (𝑢 ·Q 𝐴) <Q (𝑑 ·Q 𝐵)))
     
    8-Dec-2019appdivnq 6661 Approximate division for positive rationals. Proposition 12.7 of [BauerTaylor], p. 55 (a special case where 𝐴 and 𝐵 are positive, as well as 𝐶). Our proof is simpler than the one in BauerTaylor because we have reciprocals. (Contributed by Jim Kingdon, 8-Dec-2019.)
    ((𝐴 <Q 𝐵 ∧ 𝐶 ∈ Q) → ∃𝑚 ∈ Q (𝐴 <Q (𝑚 ·Q 𝐶) ∧ (𝑚 ·Q 𝐶) <Q 𝐵))
     
    8-Dec-2019nqprxx 6644 The canonical embedding of the rationals into the reals, expressed with the same variable for the lower and upper cuts. (Contributed by Jim Kingdon, 8-Dec-2019.)
    (𝐴 ∈ Q → ⟨{𝑥 ∣ 𝑥 <Q 𝐴}, {𝑥 ∣ 𝐴 <Q 𝑥}⟩ ∈ P)
     
    8-Dec-2019nqprloc 6643 A cut produced from a rational is located. Lemma for nqprlu 6645. (Contributed by Jim Kingdon, 8-Dec-2019.)
    (𝐴 ∈ Q → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ∨ 𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥})))
     
    8-Dec-2019nqprdisj 6642 A cut produced from a rational is disjoint. Lemma for nqprlu 6645. (Contributed by Jim Kingdon, 8-Dec-2019.)
    (𝐴 ∈ Q → ∀𝑞 ∈ Q ¬ (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ∧ 𝑞 ∈ {𝑥 ∣ 𝐴 <Q 𝑥}))
     
    8-Dec-2019nqprrnd 6641 A cut produced from a rational is rounded. Lemma for nqprlu 6645. (Contributed by Jim Kingdon, 8-Dec-2019.)
    (𝐴 ∈ Q → (∀𝑞 ∈ Q (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ {𝑥 ∣ 𝑥 <Q 𝐴})) ∧ ∀𝑟 ∈ Q (𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥} ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ {𝑥 ∣ 𝐴 <Q 𝑥}))))
     
    8-Dec-2019nqprm 6640 A cut produced from a rational is inhabited. Lemma for nqprlu 6645. (Contributed by Jim Kingdon, 8-Dec-2019.)
    (𝐴 ∈ Q → (∃𝑞 ∈ Q 𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ∧ ∃𝑟 ∈ Q 𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥}))
     
    8-Dec-2019mpvlu 6637 Value of multiplication on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.)
    ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝐵)𝑥 = (𝑦 ·Q 𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐵)𝑥 = (𝑦 ·Q 𝑧)}⟩)
     
    8-Dec-2019plpvlu 6636 Value of addition on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.)
    ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) = ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ (1st ‘𝐴)∃𝑧 ∈ (1st ‘𝐵)𝑥 = (𝑦 +Q 𝑧)}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐵)𝑥 = (𝑦 +Q 𝑧)}⟩)
     
    7-Dec-2019addlocprlemeqgt 6630 Lemma for addlocpr 6634. This is a step used in both the 𝑄 = (𝐷 +Q 𝐸) and (𝐷 +Q 𝐸) <Q 𝑄 cases. (Contributed by Jim Kingdon, 7-Dec-2019.)
    (𝜑 → 𝐴 ∈ P)    &   (𝜑 → 𝐵 ∈ P)    &   (𝜑 → 𝑄 <Q 𝑅)    &   (𝜑 → 𝑃 ∈ Q)    &   (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)    &   (𝜑 → 𝐷 ∈ (1st ‘𝐴))    &   (𝜑 → 𝑈 ∈ (2nd ‘𝐴))    &   (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))    &   (𝜑 → 𝐸 ∈ (1st ‘𝐵))    &   (𝜑 → 𝑇 ∈ (2nd ‘𝐵))    &   (𝜑 → 𝑇 <Q (𝐸 +Q 𝑃))    ⇒   (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
     
    7-Dec-2019addnqprulem 6626 Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.)
    (((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐺 ∈ 𝑈) ∧ 𝑋 ∈ Q) → (𝑆 <Q 𝑋 → ((𝑋 ·Q (*Q‘𝑆)) ·Q 𝐺) ∈ 𝑈))
     
    7-Dec-2019addnqprllem 6625 Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.)
    (((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐺 ∈ 𝐿) ∧ 𝑋 ∈ Q) → (𝑋 <Q 𝑆 → ((𝑋 ·Q (*Q‘𝑆)) ·Q 𝐺) ∈ 𝐿))
     
    7-Dec-2019lt2addnq 6502 Ordering property of addition for positive fractions. (Contributed by Jim Kingdon, 7-Dec-2019.)
    (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ (𝐶 ∈ Q ∧ 𝐷 ∈ Q)) → ((𝐴 <Q 𝐵 ∧ 𝐶 <Q 𝐷) → (𝐴 +Q 𝐶) <Q (𝐵 +Q 𝐷)))
     
    6-Dec-2019addlocprlem 6633 Lemma for addlocpr 6634. The result, in deduction form. (Contributed by Jim Kingdon, 6-Dec-2019.)
    (𝜑 → 𝐴 ∈ P)    &   (𝜑 → 𝐵 ∈ P)    &   (𝜑 → 𝑄 <Q 𝑅)    &   (𝜑 → 𝑃 ∈ Q)    &   (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)    &   (𝜑 → 𝐷 ∈ (1st ‘𝐴))    &   (𝜑 → 𝑈 ∈ (2nd ‘𝐴))    &   (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))    &   (𝜑 → 𝐸 ∈ (1st ‘𝐵))    &   (𝜑 → 𝑇 ∈ (2nd ‘𝐵))    &   (𝜑 → 𝑇 <Q (𝐸 +Q 𝑃))    ⇒   (𝜑 → (𝑄 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
     
    6-Dec-2019addlocprlemgt 6632 Lemma for addlocpr 6634. The (𝐷 +Q 𝐸) <Q 𝑄 case. (Contributed by Jim Kingdon, 6-Dec-2019.)
    (𝜑 → 𝐴 ∈ P)    &   (𝜑 → 𝐵 ∈ P)    &   (𝜑 → 𝑄 <Q 𝑅)    &   (𝜑 → 𝑃 ∈ Q)    &   (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)    &   (𝜑 → 𝐷 ∈ (1st ‘𝐴))    &   (𝜑 → 𝑈 ∈ (2nd ‘𝐴))    &   (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))    &   (𝜑 → 𝐸 ∈ (1st ‘𝐵))    &   (𝜑 → 𝑇 ∈ (2nd ‘𝐵))    &   (𝜑 → 𝑇 <Q (𝐸 +Q 𝑃))    ⇒   (𝜑 → ((𝐷 +Q 𝐸) <Q 𝑄 → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
     
    6-Dec-2019addlocprlemeq 6631 Lemma for addlocpr 6634. The 𝑄 = (𝐷 +Q 𝐸) case. (Contributed by Jim Kingdon, 6-Dec-2019.)
    (𝜑 → 𝐴 ∈ P)    &   (𝜑 → 𝐵 ∈ P)    &   (𝜑 → 𝑄 <Q 𝑅)    &   (𝜑 → 𝑃 ∈ Q)    &   (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)    &   (𝜑 → 𝐷 ∈ (1st ‘𝐴))    &   (𝜑 → 𝑈 ∈ (2nd ‘𝐴))    &   (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))    &   (𝜑 → 𝐸 ∈ (1st ‘𝐵))    &   (𝜑 → 𝑇 ∈ (2nd ‘𝐵))    &   (𝜑 → 𝑇 <Q (𝐸 +Q 𝑃))    ⇒   (𝜑 → (𝑄 = (𝐷 +Q 𝐸) → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
     
    6-Dec-2019addlocprlemlt 6629 Lemma for addlocpr 6634. The 𝑄 <Q (𝐷 +Q 𝐸) case. (Contributed by Jim Kingdon, 6-Dec-2019.)
    (𝜑 → 𝐴 ∈ P)    &   (𝜑 → 𝐵 ∈ P)    &   (𝜑 → 𝑄 <Q 𝑅)    &   (𝜑 → 𝑃 ∈ Q)    &   (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)    &   (𝜑 → 𝐷 ∈ (1st ‘𝐴))    &   (𝜑 → 𝑈 ∈ (2nd ‘𝐴))    &   (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))    &   (𝜑 → 𝐸 ∈ (1st ‘𝐵))    &   (𝜑 → 𝑇 ∈ (2nd ‘𝐵))    &   (𝜑 → 𝑇 <Q (𝐸 +Q 𝑃))    ⇒   (𝜑 → (𝑄 <Q (𝐷 +Q 𝐸) → 𝑄 ∈ (1st ‘(𝐴 +P 𝐵))))
     
    5-Dec-2019addlocpr 6634 Locatedness of addition on positive reals. Lemma 11.16 in [BauerTaylor], p. 53. The proof in BauerTaylor relies on signed rationals, so we replace it with another proof which applies prarloc 6601 to both 𝐴 and 𝐵, and uses nqtri3or 6494 rather than prloc 6589 to decide whether 𝑞 is too big to be in the lower cut of 𝐴 +P 𝐵 (and deduce that if it is, then 𝑟 must be in the upper cut). What the two proofs have in common is that they take the difference between 𝑞 and 𝑟 to determine how tight a range they need around the real numbers. (Contributed by Jim Kingdon, 5-Dec-2019.)
    ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑟 ∈ (2nd ‘(𝐴 +P 𝐵)))))
     
    5-Dec-2019addnqpru 6628 Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.)
    ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → 𝑋 ∈ (2nd ‘(𝐴 +P 𝐵))))
     
    5-Dec-2019addnqprl 6627 Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.)
    ((((𝐴 ∈ P ∧ 𝐺 ∈ (1st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 <Q (𝐺 +Q 𝐻) → 𝑋 ∈ (1st ‘(𝐴 +P 𝐵))))
     
    5-Dec-2019genpmu 6616 The upper cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Dec-2019.)
    𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    &   ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)    ⇒   ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))
     
    5-Dec-2019genpelxp 6609 Set containing the result of adding or multiplying positive reals. (Contributed by Jim Kingdon, 5-Dec-2019.)
    𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)    ⇒   ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ∈ (𝒫 Q × 𝒫 Q))
     
    3-Dec-2019addassnq0lemcl 6559 A natural number closure law. Lemma for addassnq0 6560. (Contributed by Jim Kingdon, 3-Dec-2019.)
    (((𝐼 ∈ ω ∧ 𝐽 ∈ N) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ N)) → (((𝐼 ·𝑜 𝐿) +𝑜 (𝐽 ·𝑜 𝐾)) ∈ ω ∧ (𝐽 ·𝑜 𝐿) ∈ N))
     
    3-Dec-2019nndir 6069 Distributive law for natural numbers (right-distributivity). (Contributed by Jim Kingdon, 3-Dec-2019.)
    ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +𝑜 𝐵) ·𝑜 𝐶) = ((𝐴 ·𝑜 𝐶) +𝑜 (𝐵 ·𝑜 𝐶)))
     
    1-Dec-2019nnanq0 6556 Addition of non-negative fractions with a common denominator. You can add two fractions with the same denominator by adding their numerators and keeping the same denominator. (Contributed by Jim Kingdon, 1-Dec-2019.)
    ((𝑁 ∈ ω ∧ 𝑀 ∈ ω ∧ 𝐴 ∈ N) → [⟨(𝑁 +𝑜 𝑀), 𝐴⟩] ~Q0 = ([⟨𝑁, 𝐴⟩] ~Q0 +Q0 [⟨𝑀, 𝐴⟩] ~Q0 ))
     
    1-Dec-2019archnqq 6515 For any fraction, there is an integer that is greater than it. This is also known as the "archimedean property". (Contributed by Jim Kingdon, 1-Dec-2019.)
    (𝐴 ∈ Q → ∃𝑥 ∈ N 𝐴 <Q [⟨𝑥, 1𝑜⟩] ~Q )
     
    30-Nov-2019bj-2inf 10062 Two formulations of the axiom of infinity (see ax-infvn 10066 and bj-omex 10067) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
    (ω ∈ V ↔ ∃𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦 → 𝑥 ⊆ 𝑦)))
     
    30-Nov-2019bj-om 10061 A set is equal to ω if and only if it is the smallest inductive set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
    (𝐴 ∈ 𝑉 → (𝐴 = ω ↔ (Ind 𝐴 ∧ ∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥))))
     
    30-Nov-2019bj-ssom 10060 A characterization of subclasses of ω. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
    (∀𝑥(Ind 𝑥 → 𝐴 ⊆ 𝑥) ↔ 𝐴 ⊆ ω)
     
    30-Nov-2019bj-omssind 10059 ω is included in all the inductive sets (but for the moment, we cannot prove that it is included in all the inductive classes). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
    (𝐴 ∈ 𝑉 → (Ind 𝐴 → ω ⊆ 𝐴))
     
    30-Nov-2019bj-omind 10058 ω is an inductive class. (Contributed by BJ, 30-Nov-2019.)
    Ind ω
     
    30-Nov-2019bj-dfom 10057 Alternate definition of ω, as the intersection of all the inductive sets. Proposal: make this the definition. (Contributed by BJ, 30-Nov-2019.)
    ω = ∩ {𝑥 ∣ Ind 𝑥}
     
    30-Nov-2019bj-indint 10055 The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.)
    Ind ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}
     
    30-Nov-2019bj-bdind 10054 Boundedness of the formula "the setvar 𝑥 is an inductive class". (Contributed by BJ, 30-Nov-2019.)
    BOUNDED Ind 𝑥
     
    30-Nov-2019bj-indeq 10053 Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)
    (𝐴 = 𝐵 → (Ind 𝐴 ↔ Ind 𝐵))
     
    30-Nov-2019bj-indsuc 10052 A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.)
    (Ind 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴))
     
    30-Nov-2019df-bj-ind 10051 Define the property of being an inductive class. (Contributed by BJ, 30-Nov-2019.)
    (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))
     
    30-Nov-2019bj-bdsucel 10002 Boundedness of the formula "the successor of the setvar 𝑥 belongs to the setvar 𝑦". (Contributed by BJ, 30-Nov-2019.)
    BOUNDED suc 𝑥 ∈ 𝑦
     
    30-Nov-2019bj-bd0el 9988 Boundedness of the formula "the empty set belongs to the setvar 𝑥". (Contributed by BJ, 30-Nov-2019.)
    BOUNDED ∅ ∈ 𝑥
     
    30-Nov-2019bj-sseq 9931 If two converse inclusions are characterized each by a formula, then equality is characterized by the conjunction of these formulas. (Contributed by BJ, 30-Nov-2019.)
    (𝜑 → (𝜓 ↔ 𝐴 ⊆ 𝐵))    &   (𝜑 → (𝜒 ↔ 𝐵 ⊆ 𝐴))    ⇒   (𝜑 → ((𝜓 ∧ 𝜒) ↔ 𝐴 = 𝐵))
     
    30-Nov-2019nqpnq0nq 6551 A positive fraction plus a non-negative fraction is a positive fraction. (Contributed by Jim Kingdon, 30-Nov-2019.)
    ((𝐴 ∈ Q ∧ 𝐵 ∈ Q0) → (𝐴 +Q0 𝐵) ∈ Q)
     
    30-Nov-2019mulclnq0 6550 Closure of multiplication on non-negative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.)
    ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0) → (𝐴 ·Q0 𝐵) ∈ Q0)
     
    30-Nov-2019prarloclemarch 6516 A version of the Archimedean property. This variation is "stronger" than archnqq 6515 in the sense that we provide an integer which is larger than a given rational 𝐴 even after being multiplied by a second rational 𝐵. (Contributed by Jim Kingdon, 30-Nov-2019.)
    ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ∃𝑥 ∈ N 𝐴 <Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐵))
     
    29-Nov-2019bj-elssuniab 9930 Version of elssuni 3608 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)
    Ⅎ𝑥𝐴    ⇒   (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 → 𝐴 ⊆ ∪ {𝑥 ∣ 𝜑}))
     
    29-Nov-2019bj-intabssel1 9929 Version of intss1 3630 using a class abstraction and implicit substitution. Closed form of intmin3 3642. (Contributed by BJ, 29-Nov-2019.)
    Ⅎ𝑥𝐴    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   (𝐴 ∈ 𝑉 → (𝜓 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴))
     
    29-Nov-2019bj-intabssel 9928 Version of intss1 3630 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.)
    Ⅎ𝑥𝐴    ⇒   (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 → ∩ {𝑥 ∣ 𝜑} ⊆ 𝐴))
     
    29-Nov-2019nq02m 6563 Multiply a non-negative fraction by two. (Contributed by Jim Kingdon, 29-Nov-2019.)
    (𝐴 ∈ Q0 → ([⟨2𝑜, 1𝑜⟩] ~Q0 ·Q0 𝐴) = (𝐴 +Q0 𝐴))
     
    29-Nov-2019distnq0r 6561 Multiplication of non-negative fractions is distributive. Version of distrnq0 6557 with the multiplications commuted. (Contributed by Jim Kingdon, 29-Nov-2019.)
    ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0 ∧ 𝐶 ∈ Q0) → ((𝐵 +Q0 𝐶) ·Q0 𝐴) = ((𝐵 ·Q0 𝐴) +Q0 (𝐶 ·Q0 𝐴)))
     
    29-Nov-2019addassnq0 6560 Addition of non-negaative fractions is associative. (Contributed by Jim Kingdon, 29-Nov-2019.)
    ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0 ∧ 𝐶 ∈ Q0) → ((𝐴 +Q0 𝐵) +Q0 𝐶) = (𝐴 +Q0 (𝐵 +Q0 𝐶)))
     
    29-Nov-2019addclnq0 6549 Closure of addition on non-negative fractions. (Contributed by Jim Kingdon, 29-Nov-2019.)
    ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0) → (𝐴 +Q0 𝐵) ∈ Q0)
     
    29-Nov-2019mulcanenq0ec 6543 Lemma for distributive law: cancellation of common factor. (Contributed by Jim Kingdon, 29-Nov-2019.)
    ((𝐴 ∈ N ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ N) → [⟨(𝐴 ·𝑜 𝐵), (𝐴 ·𝑜 𝐶)⟩] ~Q0 = [⟨𝐵, 𝐶⟩] ~Q0 )
     
    28-Nov-2019ax-bdsetind 10093 Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.)
    BOUNDED 𝜑    ⇒   (∀𝑎(∀𝑦 ∈ 𝑎 [𝑦 / 𝑎]𝜑 → 𝜑) → ∀𝑎𝜑)
     
    28-Nov-2019bj-peano4 10080 Remove from peano4 4320 dependency on ax-setind 4262. Therefore, it only requires core constructive axioms (albeit more of them). (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.)
    ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))
     
    28-Nov-2019bj-nnen2lp 10079 A version of en2lp 4278 for natural numbers, which does not require ax-setind 4262.

    Note: using this theorem and bj-nnelirr 10078, one can remove dependency on ax-setind 4262 from nntri2 6073 and nndcel 6078; one can actually remove more dependencies from these. (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.)

    ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
     
    27-Nov-2019mulcomnq0 6558 Multiplication of non-negative fractions is commutative. (Contributed by Jim Kingdon, 27-Nov-2019.)
    ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0) → (𝐴 ·Q0 𝐵) = (𝐵 ·Q0 𝐴))
     
    27-Nov-2019distrnq0 6557 Multiplication of non-negative fractions is distributive. (Contributed by Jim Kingdon, 27-Nov-2019.)
    ((𝐴 ∈ Q0 ∧ 𝐵 ∈ Q0 ∧ 𝐶 ∈ Q0) → (𝐴 ·Q0 (𝐵 +Q0 𝐶)) = ((𝐴 ·Q0 𝐵) +Q0 (𝐴 ·Q0 𝐶)))
     
    25-Nov-2019prcunqu 6583 An upper cut is closed upwards under the positive fractions. (Contributed by Jim Kingdon, 25-Nov-2019.)
    ((⟨𝐿, 𝑈⟩ ∈ P ∧ 𝐶 ∈ 𝑈) → (𝐶 <Q 𝐵 → 𝐵 ∈ 𝑈))
     
    25-Nov-2019prarloclemarch2 6517 Like prarloclemarch 6516 but the integer must be at least two, and there is also 𝐵 added to the right hand side. These details follow straightforwardly but are chosen to be helpful in the proof of prarloc 6601. (Contributed by Jim Kingdon, 25-Nov-2019.)
    ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ∃𝑥 ∈ N (1𝑜 <N 𝑥 ∧ 𝐴 <Q (𝐵 +Q ([⟨𝑥, 1𝑜⟩] ~Q ·Q 𝐶))))
     
    25-Nov-2019subhalfnqq 6512 There is a number which is less than half of any positive fraction. The case where 𝐴 is one is Lemma 11.4 of [BauerTaylor], p. 50, and they use the word "approximate half" for such a number (since there may be constructions, for some structures other than the rationals themselves, which rely on such an approximate half but do not require division by two as seen at halfnqq 6508). (Contributed by Jim Kingdon, 25-Nov-2019.)
    (𝐴 ∈ Q → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) <Q 𝐴)
     
    24-Nov-2019bj-nnelirr 10078 A natural number does not belong to itself. Version of elirr 4266 for natural numbers, which does not require ax-setind 4262. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)
    (𝐴 ∈ ω → ¬ 𝐴 ∈ 𝐴)
     
    24-Nov-2019bdcriota 10003 A class given by a restricted definition binder is bounded, under the given hypotheses. (Contributed by BJ, 24-Nov-2019.)
    BOUNDED 𝜑    &   ∃!𝑥 ∈ 𝑦 𝜑    ⇒   BOUNDED (℩𝑥 ∈ 𝑦 𝜑)
     
    24-Nov-2019nq0nn 6540 Decomposition of a non-negative fraction into numerator and denominator. (Contributed by Jim Kingdon, 24-Nov-2019.)
    (𝐴 ∈ Q0 → ∃𝑤∃𝑣((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ 𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ))
     
    24-Nov-2019enq0eceq 6535 Equivalence class equality of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 24-Nov-2019.)
    (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ↔ (𝐴 ·𝑜 𝐷) = (𝐵 ·𝑜 𝐶)))
     
    24-Nov-2019dfplq0qs 6528 Addition on non-negative fractions. This definition is similar to df-plq0 6525 but expands Q0 (Contributed by Jim Kingdon, 24-Nov-2019.)
    +Q0 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑓) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑓)⟩] ~Q0 ))}
     
    23-Nov-2019addnq0mo 6545 There is at most one result from adding non-negative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.)
    ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))
     
    23-Nov-2019nnnq0lem1 6544 Decomposing non-negative fractions into natural numbers. Lemma for addnnnq0 6547 and mulnnnq0 6548. (Contributed by Jim Kingdon, 23-Nov-2019.)
    (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [𝐶] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ) ∧ 𝑞 = [𝐷] ~Q0 ))) → ((((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ (𝑠 ∈ ω ∧ 𝑓 ∈ N)) ∧ ((𝑢 ∈ ω ∧ 𝑡 ∈ N) ∧ (𝑔 ∈ ω ∧ ℎ ∈ N))) ∧ ((𝑤 ·𝑜 𝑓) = (𝑣 ·𝑜 𝑠) ∧ (𝑢 ·𝑜 ℎ) = (𝑡 ·𝑜 𝑔))))
     
    23-Nov-2019addcmpblnq0 6541 Lemma showing compatibility of addition on non-negative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.)
    ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → (((𝐴 ·𝑜 𝐷) = (𝐵 ·𝑜 𝐶) ∧ (𝐹 ·𝑜 𝑆) = (𝐺 ·𝑜 𝑅)) → ⟨((𝐴 ·𝑜 𝐺) +𝑜 (𝐵 ·𝑜 𝐹)), (𝐵 ·𝑜 𝐺)⟩ ~Q0 ⟨((𝐶 ·𝑜 𝑆) +𝑜 (𝐷 ·𝑜 𝑅)), (𝐷 ·𝑜 𝑆)⟩))
     
    23-Nov-2019ee8anv 1810 Rearrange existential quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.)
    (∃𝑥∃𝑦∃𝑧∃𝑤∃𝑣∃𝑢∃𝑡∃𝑠(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ∧ ∃𝑣∃𝑢∃𝑡∃𝑠𝜓))
     
    23-Nov-201919.42vvvv 1790 Theorem 19.42 of [Margaris] p. 90 with 4 quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.)
    (∃𝑤∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑤∃𝑥∃𝑦∃𝑧𝜓))
     
    22-Nov-2019bdsetindis 10094 Axiom of bounded set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
    BOUNDED 𝜑    &   Ⅎ𝑥𝜓    &   Ⅎ𝑥𝜒    &   Ⅎ𝑦𝜑    &   Ⅎ𝑦𝜓    &   (𝑥 = 𝑧 → (𝜑 → 𝜓))    &   (𝑥 = 𝑦 → (𝜒 → 𝜑))    ⇒   (∀𝑦(∀𝑧 ∈ 𝑦 𝜓 → 𝜒) → ∀𝑥𝜑)
     
    22-Nov-2019setindis 10092 Axiom of set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.)
    Ⅎ𝑥𝜓    &   Ⅎ𝑥𝜒    &   Ⅎ𝑦𝜑    &   Ⅎ𝑦𝜓    &   (𝑥 = 𝑧 → (𝜑 → 𝜓))    &   (𝑥 = 𝑦 → (𝜒 → 𝜑))    ⇒   (∀𝑦(∀𝑧 ∈ 𝑦 𝜓 → 𝜒) → ∀𝑥𝜑)
     
    22-Nov-2019setindf 10091 Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.)
    Ⅎ𝑦𝜑    ⇒   (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥𝜑)
     
    22-Nov-2019setindft 10090 Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.)
    (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥𝜑))
     
    22-Nov-2019bj-nntrans2 10077 A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
    (𝐴 ∈ ω → Tr 𝐴)
     
    22-Nov-2019bj-nntrans 10076 A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
    (𝐴 ∈ ω → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
     
    22-Nov-2019bdfind 10071 Bounded induction (principle of induction when 𝐴 is assumed to be bounded), proved from basic constructive axioms. See find 4322 for a nonconstructive proof of the general case. See findset 10070 for a proof when 𝐴 is assumed to be a set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
    BOUNDED 𝐴    ⇒   ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → 𝐴 = ω)
     
    22-Nov-2019findset 10070 Bounded induction (principle of induction when 𝐴 is assumed to be a set) allowing a proof from basic constructive axioms. See find 4322 for a nonconstructive proof of the general case. See bdfind 10071 for a proof when 𝐴 is assumed to be bounded. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
    (𝐴 ∈ 𝑉 → ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → 𝐴 = ω))
     
    22-Nov-2019cbvrald 9927 Rule used to change bound variables, using implicit substitution. (Contributed by BJ, 22-Nov-2019.)
    Ⅎ𝑥𝜑    &   Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐴 𝜒))
     
    22-Nov-2019addnnnq0 6547 Addition of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 22-Nov-2019.)
    (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → ([⟨𝐴, 𝐵⟩] ~Q0 +Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )
     
    22-Nov-2019dfmq0qs 6527 Multiplication on non-negative fractions. This definition is similar to df-mq0 6526 but expands Q0 (Contributed by Jim Kingdon, 22-Nov-2019.)
    ·Q0 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑓⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑓)⟩] ~Q0 ))}
     
    22-Nov-2019neqned 2213 If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2226. One-way deduction form of df-ne 2206. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2255. (Revised by Wolf Lammen, 22-Nov-2019.)
    (𝜑 → ¬ 𝐴 = 𝐵)    ⇒   (𝜑 → 𝐴 ≠ 𝐵)
     
    21-Nov-2019bj-findes 10106 Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 10104 for explanations. From this version, it is easy to prove findes 4326. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
    (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)
     
    21-Nov-2019bj-findisg 10105 Version of bj-findis 10104 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 10104 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
    Ⅎ𝑥𝜓    &   Ⅎ𝑥𝜒    &   Ⅎ𝑥𝜃    &   (𝑥 = ∅ → (𝜓 → 𝜑))    &   (𝑥 = 𝑦 → (𝜑 → 𝜒))    &   (𝑥 = suc 𝑦 → (𝜃 → 𝜑))    &   Ⅎ𝑥𝐴    &   Ⅎ𝑥𝜏    &   (𝑥 = 𝐴 → (𝜑 → 𝜏))    ⇒   ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → (𝐴 ∈ ω → 𝜏))
     
    21-Nov-2019bj-bdfindes 10074 Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 10072 for explanations. From this version, it is easy to prove the bounded version of findes 4326. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
    BOUNDED 𝜑    ⇒   (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑)
     
    21-Nov-2019bj-bdfindisg 10073 Version of bj-bdfindis 10072 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 10072 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
    BOUNDED 𝜑    &   Ⅎ𝑥𝜓    &   Ⅎ𝑥𝜒    &   Ⅎ𝑥𝜃    &   (𝑥 = ∅ → (𝜓 → 𝜑))    &   (𝑥 = 𝑦 → (𝜑 → 𝜒))    &   (𝑥 = suc 𝑦 → (𝜃 → 𝜑))    &   Ⅎ𝑥𝐴    &   Ⅎ𝑥𝜏    &   (𝑥 = 𝐴 → (𝜑 → 𝜏))    ⇒   ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → (𝐴 ∈ ω → 𝜏))
     
    21-Nov-2019bj-bdfindis 10072 Bounded induction (principle of induction for bounded formulas), using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See finds 4323 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4323, finds2 4324, finds1 4325. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
    BOUNDED 𝜑    &   Ⅎ𝑥𝜓    &   Ⅎ𝑥𝜒    &   Ⅎ𝑥𝜃    &   (𝑥 = ∅ → (𝜓 → 𝜑))    &   (𝑥 = 𝑦 → (𝜑 → 𝜒))    &   (𝑥 = suc 𝑦 → (𝜃 → 𝜑))    ⇒   ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑)
     
    21-Nov-2019bdeqsuc 10001 Boundedness of the formula expressing that a setvar is equal to the successor of another. (Contributed by BJ, 21-Nov-2019.)
    BOUNDED 𝑥 = suc 𝑦
     
    21-Nov-2019bdop 9995 The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.)
    BOUNDED ⟨𝑥, 𝑦⟩
     
    21-Nov-2019bdeq0 9987 Boundedness of the formula expressing that a setvar is equal to the empty class. (Contributed by BJ, 21-Nov-2019.)
    BOUNDED 𝑥 = ∅
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