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Theorem List for Metamath Proof Explorer - 28901-29000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnn0sqeq1 28901 Integer square one. (Contributed by Thierry Arnoux, 2-Feb-2020.)
((𝑁 ∈ ℕ0 ∧ (𝑁↑2) = 1) → 𝑁 = 1)

Theorem1neg1t1neg1 28902 An integer unit times itself. (Contributed by Thierry Arnoux, 23-Aug-2020.)
(𝑁 ∈ {-1, 1} → (𝑁 · 𝑁) = 1)

21.3.5.2  Ordering on reals - misc additions

Theoremlt2addrd 28903* If the right-hand side of a 'less than' relationship is an addition, then we can express the left-hand side as an addition, too, where each term is respectively less than each term of the original right side. (Contributed by Thierry Arnoux, 15-Mar-2017.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < (𝐵 + 𝐶))       (𝜑 → ∃𝑏 ∈ ℝ ∃𝑐 ∈ ℝ (𝐴 = (𝑏 + 𝑐) ∧ 𝑏 < 𝐵𝑐 < 𝐶))

21.3.5.3  Extended reals - misc additions

Theoremxgepnf 28904 An extended real which is greater than plus infinity is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝐴 ∈ ℝ* → (+∞ ≤ 𝐴𝐴 = +∞))

Theoremxlemnf 28905 An extended real which is less than minus infinity is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.)
(𝐴 ∈ ℝ* → (𝐴 ≤ -∞ ↔ 𝐴 = -∞))

Theoremxrlelttric 28906 Trichotomy law for extended reals. (Contributed by Thierry Arnoux, 12-Sep-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵𝐵 < 𝐴))

Theoremxaddeq0 28907 Two extended reals which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 13-Jun-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 +𝑒 𝐵) = 0 ↔ 𝐴 = -𝑒𝐵))

Theoreminfxrmnf 28908 The infinimum of a set of extended reals containing minus infinity is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.) (Revised by AV, 28-Sep-2020.)
((𝐴 ⊆ ℝ* ∧ -∞ ∈ 𝐴) → inf(𝐴, ℝ*, < ) = -∞)

Theoremxrinfm 28909 The extended real numbers are unbounded below. (Contributed by Thierry Arnoux, 18-Feb-2018.) (Revised by AV, 28-Sep-2020.)
inf(ℝ*, ℝ*, < ) = -∞

Theoremle2halvesd 28910 A sum is less than the whole if each term is less than half. (Contributed by Thierry Arnoux, 29-Nov-2017.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 ≤ (𝐶 / 2))    &   (𝜑𝐵 ≤ (𝐶 / 2))       (𝜑 → (𝐴 + 𝐵) ≤ 𝐶)

Theoremxraddge02 28911 A number is less than or equal to itself plus a nonnegative number. (Contributed by Thierry Arnoux, 28-Dec-2016.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (0 ≤ 𝐵𝐴 ≤ (𝐴 +𝑒 𝐵)))

Theoremxrge0addge 28912 A number is less than or equal to itself plus a nonnegative number. (Contributed by Thierry Arnoux, 19-Jul-2020.)
((𝐴 ∈ ℝ*𝐵 ∈ (0[,]+∞)) → 𝐴 ≤ (𝐴 +𝑒 𝐵))

Theoremxlt2addrd 28913* If the right-hand side of a 'less than' relationship is an addition, then we can express the left-hand side as an addition, too, where each term is respectively less than each term of the original right side. (Contributed by Thierry Arnoux, 15-Mar-2017.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐵 ≠ -∞)    &   (𝜑𝐶 ≠ -∞)    &   (𝜑𝐴 < (𝐵 +𝑒 𝐶))       (𝜑 → ∃𝑏 ∈ ℝ*𝑐 ∈ ℝ* (𝐴 = (𝑏 +𝑒 𝑐) ∧ 𝑏 < 𝐵𝑐 < 𝐶))

Theoremxrsupssd 28914 Inequality deduction for supremum of an extended real subset. (Contributed by Thierry Arnoux, 21-Mar-2017.)
(𝜑𝐵𝐶)    &   (𝜑𝐶 ⊆ ℝ*)       (𝜑 → sup(𝐵, ℝ*, < ) ≤ sup(𝐶, ℝ*, < ))

Theoremxrge0infss 28915* Any subset of nonnegative extended reals has an infimum. (Contributed by Thierry Arnoux, 16-Sep-2019.) (Revised by AV, 4-Oct-2020.)
(𝐴 ⊆ (0[,]+∞) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))

Theoremxrge0infssd 28916 Inequality deduction for infimum of a nonnegative extended real subset. (Contributed by Thierry Arnoux, 16-Sep-2019.) (Revised by AV, 4-Oct-2020.)
(𝜑𝐶𝐵)    &   (𝜑𝐵 ⊆ (0[,]+∞))       (𝜑 → inf(𝐵, (0[,]+∞), < ) ≤ inf(𝐶, (0[,]+∞), < ))

Theoremxrge0addcld 28917 Nonnegative extended reals are closed under addition. (Contributed by Thierry Arnoux, 16-Sep-2019.)
(𝜑𝐴 ∈ (0[,]+∞))    &   (𝜑𝐵 ∈ (0[,]+∞))       (𝜑 → (𝐴 +𝑒 𝐵) ∈ (0[,]+∞))

Theoremxrge0subcld 28918 Condition for closure of nonnegative extended reals under subtraction. (Contributed by Thierry Arnoux, 27-May-2020.)
(𝜑𝐴 ∈ (0[,]+∞))    &   (𝜑𝐵 ∈ (0[,]+∞))    &   (𝜑𝐵𝐴)       (𝜑 → (𝐴 +𝑒 -𝑒𝐵) ∈ (0[,]+∞))

Theoreminfxrge0lb 28919 A member of a set of nonnegative extended reals is greater than or equal to the set's infimum. (Contributed by Thierry Arnoux, 19-Jul-2020.) (Revised by AV, 4-Oct-2020.)
(𝜑𝐴 ⊆ (0[,]+∞))    &   (𝜑𝐵𝐴)       (𝜑 → inf(𝐴, (0[,]+∞), < ) ≤ 𝐵)

Theoreminfxrge0glb 28920* The infimum of a set of nonnegative extended reals is the greatest lower bound. (Contributed by Thierry Arnoux, 19-Jul-2020.) (Revised by AV, 4-Oct-2020.)
(𝜑𝐴 ⊆ (0[,]+∞))    &   (𝜑𝐵 ∈ (0[,]+∞))       (𝜑 → (inf(𝐴, (0[,]+∞), < ) < 𝐵 ↔ ∃𝑥𝐴 𝑥 < 𝐵))

Theoreminfxrge0gelb 28921* The infimum of a set of nonnegative extended reals is greater than or equal to a lower bound. (Contributed by Thierry Arnoux, 19-Jul-2020.) (Revised by AV, 4-Oct-2020.)
(𝜑𝐴 ⊆ (0[,]+∞))    &   (𝜑𝐵 ∈ (0[,]+∞))       (𝜑 → (𝐵 ≤ inf(𝐴, (0[,]+∞), < ) ↔ ∀𝑥𝐴 𝐵𝑥))

Theoremdfrp2 28922 Alternate definition of the positive real numbers. (Contributed by Thierry Arnoux, 4-May-2020.)
+ = (0(,)+∞)

Theoremxrofsup 28923 The supremum is preserved by extended addition set operation. (Provided minus infinity is not involved as it does not behave well with addition.) (Contributed by Thierry Arnoux, 20-Mar-2017.)
(𝜑𝑋 ⊆ ℝ*)    &   (𝜑𝑌 ⊆ ℝ*)    &   (𝜑 → sup(𝑋, ℝ*, < ) ≠ -∞)    &   (𝜑 → sup(𝑌, ℝ*, < ) ≠ -∞)    &   (𝜑𝑍 = ( +𝑒 “ (𝑋 × 𝑌)))       (𝜑 → sup(𝑍, ℝ*, < ) = (sup(𝑋, ℝ*, < ) +𝑒 sup(𝑌, ℝ*, < )))

Theoremsupxrnemnf 28924 The supremum of a nonempty set of extended reals which does not contain minus infinity is not minus infinity. (Contributed by Thierry Arnoux, 21-Mar-2017.)
((𝐴 ⊆ ℝ*𝐴 ≠ ∅ ∧ ¬ -∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) ≠ -∞)

Theoremxrhaus 28925 The topology of the extended reals is Hausdorff. (Contributed by Thierry Arnoux, 24-Mar-2017.)
(ordTop‘ ≤ ) ∈ Haus

21.3.5.4  Real number intervals - misc additions

Theoremjoiniooico 28926 Disjoint joining an open interval with a closed-below, open-above interval to form a closed-below, open-above interval. (Contributed by Thierry Arnoux, 26-Sep-2017.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵𝐵𝐶)) → (((𝐴(,)𝐵) ∩ (𝐵[,)𝐶)) = ∅ ∧ ((𝐴(,)𝐵) ∪ (𝐵[,)𝐶)) = (𝐴(,)𝐶)))

Theoremubico 28927 A right-open interval does not contain its right endpoint. (Contributed by Thierry Arnoux, 5-Apr-2017.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*) → ¬ 𝐵 ∈ (𝐴[,)𝐵))

Theoremxeqlelt 28928 Equality in terms of 'less than or equal to', 'less than'. (Contributed by Thierry Arnoux, 5-Jul-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴 < 𝐵)))

Theoremeliccelico 28929 Relate elementhood to a closed interval with elementhood to the same closed-below, open-above interval or to its upper bound. (Contributed by Thierry Arnoux, 3-Jul-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ (𝐴[,)𝐵) ∨ 𝐶 = 𝐵)))

Theoremelicoelioo 28930 Relate elementhood to a closed-below, open-above interval with elementhood to the same open interval or to its lower bound. (Contributed by Thierry Arnoux, 6-Jul-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 = 𝐴𝐶 ∈ (𝐴(,)𝐵))))

Theoremiocinioc2 28931 Intersection between two open-below, closed-above intervals sharing the same upper bound. (Contributed by Thierry Arnoux, 7-Aug-2017.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ 𝐴𝐵) → ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = (𝐵(,]𝐶))

Theoremxrdifh 28932 Class difference of a half-open interval in the extended reals. (Contributed by Thierry Arnoux, 1-Aug-2017.)
𝐴 ∈ ℝ*       (ℝ* ∖ (𝐴[,]+∞)) = (-∞[,)𝐴)

Theoremiocinif 28933 Relate intersection of two open-below, closed-above intervals with the same upper bound with a conditional construct. (Contributed by Thierry Arnoux, 7-Aug-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = if(𝐴 < 𝐵, (𝐵(,]𝐶), (𝐴(,]𝐶)))

Theoremdifioo 28934 The difference between two open intervals sharing the same lower bound. (Contributed by Thierry Arnoux, 26-Sep-2017.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐶) ∖ (𝐴(,)𝐵)) = (𝐵[,)𝐶))

Theoremdifico 28935 The difference between two closed-below, open-above intervals sharing the same upper bound. (Contributed by Thierry Arnoux, 13-Oct-2017.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴𝐵𝐵𝐶)) → ((𝐴[,)𝐶) ∖ (𝐵[,)𝐶)) = (𝐴[,)𝐵))

21.3.5.5  Finite intervals of integers - misc additions

Theoremnndiffz1 28936 Upper set of the positive integers. (Contributed by Thierry Arnoux, 22-Aug-2017.)
(𝑁 ∈ ℕ0 → (ℕ ∖ (1...𝑁)) = (ℤ‘(𝑁 + 1)))

Theoremssnnssfz 28937* For any finite subset of , find a superset in the form of a set of sequential integers. (Contributed by Thierry Arnoux, 13-Sep-2017.)
(𝐴 ∈ (𝒫 ℕ ∩ Fin) → ∃𝑛 ∈ ℕ 𝐴 ⊆ (1...𝑛))

Theoremfzspl 28938 Split the last element of a finite set of sequential integers. (more generic than fzsuc 12258) (Contributed by Thierry Arnoux, 7-Nov-2016.)
(𝑁 ∈ (ℤ𝑀) → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁}))

Theoremfzdif2 28939 Split the last element of a finite set of sequential integers. (more generic than fzsuc 12258) (Contributed by Thierry Arnoux, 22-Aug-2020.)
(𝑁 ∈ (ℤ𝑀) → ((𝑀...𝑁) ∖ {𝑁}) = (𝑀...(𝑁 − 1)))

Theoremfzsplit3 28940 Split a finite interval of integers into two parts. (Contributed by Thierry Arnoux, 2-May-2017.)
(𝐾 ∈ (𝑀...𝑁) → (𝑀...𝑁) = ((𝑀...(𝐾 − 1)) ∪ (𝐾...𝑁)))

Theorembcm1n 28941 The proportion of one binomial coefficient to another with 𝑁 decreased by 1. (Contributed by Thierry Arnoux, 9-Nov-2016.)
((𝐾 ∈ (0...(𝑁 − 1)) ∧ 𝑁 ∈ ℕ) → (((𝑁 − 1)C𝐾) / (𝑁C𝐾)) = ((𝑁𝐾) / 𝑁))

21.3.5.6  Half-open integer ranges - misc additions

Theoremiundisjfi 28942* Rewrite a countable union as a disjoint union, finite version. Cf. iundisj 23123. (Contributed by Thierry Arnoux, 15-Feb-2017.)
𝑛𝐵    &   (𝑛 = 𝑘𝐴 = 𝐵)        𝑛 ∈ (1..^𝑁)𝐴 = 𝑛 ∈ (1..^𝑁)(𝐴 𝑘 ∈ (1..^𝑛)𝐵)

Theoremiundisj2fi 28943* A disjoint union is disjoint, finite version. Cf. iundisj2 23124. (Contributed by Thierry Arnoux, 16-Feb-2017.)
𝑛𝐵    &   (𝑛 = 𝑘𝐴 = 𝐵)       Disj 𝑛 ∈ (1..^𝑁)(𝐴 𝑘 ∈ (1..^𝑛)𝐵)

Theoremiundisjcnt 28944* Rewrite a countable union as a disjoint union. (Contributed by Thierry Arnoux, 16-Feb-2017.)
𝑛𝐵    &   (𝑛 = 𝑘𝐴 = 𝐵)    &   (𝜑 → (𝑁 = ℕ ∨ 𝑁 = (1..^𝑀)))       (𝜑 𝑛𝑁 𝐴 = 𝑛𝑁 (𝐴 𝑘 ∈ (1..^𝑛)𝐵))

Theoremiundisj2cnt 28945* A countable disjoint union is disjoint. Cf. iundisj2 23124. (Contributed by Thierry Arnoux, 16-Feb-2017.)
𝑛𝐵    &   (𝑛 = 𝑘𝐴 = 𝐵)    &   (𝜑 → (𝑁 = ℕ ∨ 𝑁 = (1..^𝑀)))       (𝜑Disj 𝑛𝑁 (𝐴 𝑘 ∈ (1..^𝑛)𝐵))

Theoremf1ocnt 28946* Given a countable set 𝐴, number its elements by providing a one-to-one mapping either with or an integer range starting from 1. The domain of the function can then be used with iundisjcnt 28944 or iundisj2cnt 28945. (Contributed by Thierry Arnoux, 25-Jul-2020.)
(𝐴 ≼ ω → ∃𝑓(𝑓:dom 𝑓1-1-onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((#‘𝐴) + 1)))))

Theoremfz1nnct 28947 NN and integer ranges starting from 1 are countable. (Contributed by Thierry Arnoux, 25-Jul-2020.)
((𝐴 = ℕ ∨ 𝐴 = (1..^𝑀)) → 𝐴 ≼ ω)

Theoremfz1nntr 28948 NN and integer ranges starting from 1 are a transitive family of set. (Contributed by Thierry Arnoux, 25-Jul-2020.)
(((𝐴 = ℕ ∨ 𝐴 = (1..^𝑀)) ∧ 𝑁𝐴) → (1..^𝑁) ⊆ 𝐴)

21.3.5.7  The ` # ` (set size) function - misc additions

Theoremhashunif 28949* The cardinality of a disjoint finite union of finite sets. Cf. hashuni 14397. (Contributed by Thierry Arnoux, 17-Feb-2017.)
𝑥𝜑    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ⊆ Fin)    &   (𝜑Disj 𝑥𝐴 𝑥)       (𝜑 → (#‘ 𝐴) = Σ𝑥𝐴 (#‘𝑥))

21.3.5.8  The greatest common divisor operator - misc. add

Theoremnumdenneg 28950 Numerator and denominator of the negative. (Contributed by Thierry Arnoux, 27-Oct-2017.)
(𝑄 ∈ ℚ → ((numer‘-𝑄) = -(numer‘𝑄) ∧ (denom‘-𝑄) = (denom‘𝑄)))

Theoremdivnumden2 28951 Calculate the reduced form of a quotient using gcd. This version extends divnumden 15294 for the negative integers. (Contributed by Thierry Arnoux, 25-Oct-2017.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ -𝐵 ∈ ℕ) → ((numer‘(𝐴 / 𝐵)) = -(𝐴 / (𝐴 gcd 𝐵)) ∧ (denom‘(𝐴 / 𝐵)) = -(𝐵 / (𝐴 gcd 𝐵))))

21.3.5.9  Integers

Theoremnnindf 28952* Principle of Mathematical Induction, using a bound-variable hypothesis instead of distinct variables. (Contributed by Thierry Arnoux, 6-May-2018.)
𝑦𝜑    &   (𝑥 = 1 → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 + 1) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   (𝑦 ∈ ℕ → (𝜒𝜃))       (𝐴 ∈ ℕ → 𝜏)

Theoremnnindd 28953* Principle of Mathematical Induction (inference schema) on integers, a deduction version. (Contributed by Thierry Arnoux, 19-Jul-2020.)
(𝑥 = 1 → (𝜓𝜒))    &   (𝑥 = 𝑦 → (𝜓𝜃))    &   (𝑥 = (𝑦 + 1) → (𝜓𝜏))    &   (𝑥 = 𝐴 → (𝜓𝜂))    &   (𝜑𝜒)    &   (((𝜑𝑦 ∈ ℕ) ∧ 𝜃) → 𝜏)       ((𝜑𝐴 ∈ ℕ) → 𝜂)

Theoremnn0min 28954* Extracting the minimum positive integer for which a property 𝜒 does not hold. This uses substitutions similar to nn0ind 11348. (Contributed by Thierry Arnoux, 6-May-2018.)
(𝑛 = 0 → (𝜓𝜒))    &   (𝑛 = 𝑚 → (𝜓𝜃))    &   (𝑛 = (𝑚 + 1) → (𝜓𝜏))    &   (𝜑 → ¬ 𝜒)    &   (𝜑 → ∃𝑛 ∈ ℕ 𝜓)       (𝜑 → ∃𝑚 ∈ ℕ0𝜃𝜏))

Theoremltesubnnd 28955 Subtracting an integer number from another number decreases it. See ltsubrpd 11780. (Contributed by Thierry Arnoux, 18-Apr-2017.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → ((𝑀 + 1) − 𝑁) ≤ 𝑀)

21.3.5.10  Division in the extended real number system

Syntaxcxdiv 28956 Extend class notation to include division of extended reals.
class /𝑒

Definitiondf-xdiv 28957* Define division over extended real numbers. (Contributed by Thierry Arnoux, 17-Dec-2016.)
/𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ (ℝ ∖ {0}) ↦ (𝑧 ∈ ℝ* (𝑦 ·e 𝑧) = 𝑥))

Theoremxdivval 28958* Value of division: the (unique) element 𝑥 such that (𝐵 · 𝑥) = 𝐴. This is meaningful only when 𝐵 is nonzero. (Contributed by Thierry Arnoux, 17-Dec-2016.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) = (𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))

Theoremxrecex 28959* Existence of reciprocal of nonzero real number. (Contributed by Thierry Arnoux, 17-Dec-2016.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 ·e 𝑥) = 1)

Theoremxmulcand 28960 Cancellation law for extended multiplication. (Contributed by Thierry Arnoux, 17-Dec-2016.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐶 ·e 𝐴) = (𝐶 ·e 𝐵) ↔ 𝐴 = 𝐵))

Theoremxreceu 28961* Existential uniqueness of reciprocals. Theorem I.8 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 17-Dec-2016.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → ∃!𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴)

Theoremxdivcld 28962 Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐵 ≠ 0)       (𝜑 → (𝐴 /𝑒 𝐵) ∈ ℝ*)

Theoremxdivcl 28963 Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) ∈ ℝ*)

Theoremxdivmul 28964 Relationship between division and multiplication. (Contributed by Thierry Arnoux, 24-Dec-2016.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≠ 0)) → ((𝐴 /𝑒 𝐶) = 𝐵 ↔ (𝐶 ·e 𝐵) = 𝐴))

Theoremrexdiv 28965 The extended real division operation when both arguments are real. (Contributed by Thierry Arnoux, 18-Dec-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) = (𝐴 / 𝐵))

Theoremxdivrec 28966 Relationship between division and reciprocal. (Contributed by Thierry Arnoux, 5-Jul-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) = (𝐴 ·e (1 /𝑒 𝐵)))

Theoremxdivid 28967 A number divided by itself is one. (Contributed by Thierry Arnoux, 18-Dec-2016.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (𝐴 /𝑒 𝐴) = 1)

Theoremxdiv0 28968 Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (0 /𝑒 𝐴) = 0)

Theoremxdiv0rp 28969 Division into zero is zero. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝐴 ∈ ℝ+ → (0 /𝑒 𝐴) = 0)

Theoremeliccioo 28970 Membership in a closed interval of extended reals vs. the same open interval. (Contributed by Thierry Arnoux, 18-Dec-2016.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 = 𝐴𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐵)))

Theoremelxrge02 28971 Elementhood in the set of nonnegative extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝐴 ∈ (0[,]+∞) ↔ (𝐴 = 0 ∨ 𝐴 ∈ ℝ+𝐴 = +∞))

Theoremxdivpnfrp 28972 Plus infinity divided by a positive real number is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝐴 ∈ ℝ+ → (+∞ /𝑒 𝐴) = +∞)

Theoremrpxdivcld 28973 Closure law for extended division of positive reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴 /𝑒 𝐵) ∈ ℝ+)

Theoremxrpxdivcld 28974 Closure law for extended division of positive extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016.)
(𝜑𝐴 ∈ (0[,]+∞))    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴 /𝑒 𝐵) ∈ (0[,]+∞))

21.3.6  Prime Number Theory

21.3.6.1  Fermat's two square theorem

Theorembhmafibid1 28975 The Brahmagupta-Fibonacci identity. Express the product of two sums of two squares as a sum of two squares. First result. (Contributed by Thierry Arnoux, 1-Feb-2020.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → (((𝐴↑2) + (𝐵↑2)) · ((𝐶↑2) + (𝐷↑2))) = ((((𝐴 · 𝐶) − (𝐵 · 𝐷))↑2) + (((𝐴 · 𝐷) + (𝐵 · 𝐶))↑2)))

Theorembhmafibid2 28976 The Brahmagupta-Fibonacci identity. Express the product of two sums of two squares as a sum of two squares. Second result. (Contributed by Thierry Arnoux, 1-Feb-2020.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → (((𝐴↑2) + (𝐵↑2)) · ((𝐶↑2) + (𝐷↑2))) = ((((𝐴 · 𝐶) + (𝐵 · 𝐷))↑2) + (((𝐴 · 𝐷) − (𝐵 · 𝐶))↑2)))

Theorem2sqn0 28977 If the sum of two squares is prime, none of the original number is zero. (Contributed by Thierry Arnoux, 4-Feb-2020.)
(𝜑𝑃 ∈ ℙ)    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) = 𝑃)       (𝜑𝐴 ≠ 0)

Theorem2sqcoprm 28978 If the sum of two squares is prime, the two original numbers are coprime. (Contributed by Thierry Arnoux, 2-Feb-2020.)
(𝜑𝑃 ∈ ℙ)    &   (𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) = 𝑃)       (𝜑 → (𝐴 gcd 𝐵) = 1)

Theorem2sqmod 28979 Given two decompositions of a prime as a sum of two squares, show that they are equal. (Contributed by Thierry Arnoux, 2-Feb-2020.)
(𝜑𝑃 ∈ ℙ)    &   (𝜑𝐴 ∈ ℕ0)    &   (𝜑𝐵 ∈ ℕ0)    &   (𝜑𝐶 ∈ ℕ0)    &   (𝜑𝐷 ∈ ℕ0)    &   (𝜑𝐴𝐵)    &   (𝜑𝐶𝐷)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) = 𝑃)    &   (𝜑 → ((𝐶↑2) + (𝐷↑2)) = 𝑃)       (𝜑 → (𝐴 = 𝐶𝐵 = 𝐷))

Theorem2sqmo 28980* There exists at most one decomposition of a prime as a sum of two squares. See 2sqb 24957 for the existence of such a decomposition. (Contributed by Thierry Arnoux, 2-Feb-2020.)
(𝑃 ∈ ℙ → ∃*𝑎 ∈ ℕ0𝑏 ∈ ℕ0 (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))

21.3.7  Extensible Structures

21.3.7.1  Structure restriction operator

Theoremressplusf 28981 The group operation function +𝑓 of a structure's restriction is the operation function's restriction to the new base. (Contributed by Thierry Arnoux, 26-Mar-2017.)
𝐵 = (Base‘𝐺)    &   𝐻 = (𝐺s 𝐴)    &    = (+g𝐺)    &    Fn (𝐵 × 𝐵)    &   𝐴𝐵       (+𝑓𝐻) = ( ↾ (𝐴 × 𝐴))

Theoremressnm 28982 The norm in a restricted structure. (Contributed by Thierry Arnoux, 8-Oct-2017.)
𝐻 = (𝐺s 𝐴)    &   𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝑁 = (norm‘𝐺)       ((𝐺 ∈ Mnd ∧ 0𝐴𝐴𝐵) → (𝑁𝐴) = (norm‘𝐻))

Theoremabvpropd2 28983 Weaker version of abvpropd 18665. (Contributed by Thierry Arnoux, 8-Nov-2017.)
(𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   (𝜑 → (+g𝐾) = (+g𝐿))    &   (𝜑 → (.r𝐾) = (.r𝐿))       (𝜑 → (AbsVal‘𝐾) = (AbsVal‘𝐿))

21.3.7.2  The opposite group

Theoremoppgle 28984 less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝑂 = (oppg𝑅)    &    = (le‘𝑅)        = (le‘𝑂)

Theoremoppglt 28985 less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝑂 = (oppg𝑅)    &    < = (lt‘𝑅)       (𝑅𝑉< = (lt‘𝑂))

21.3.7.3  Posets

Theoremressprs 28986 The restriction of a preordered set is still a preordered set. (Contributed by Thierry Arnoux, 11-Sep-2015.)
𝐵 = (Base‘𝐾)       ((𝐾 ∈ Preset ∧ 𝐴𝐵) → (𝐾s 𝐴) ∈ Preset )

Theoremoduprs 28987 Being a preset is a self-dual property. (Contributed by Thierry Arnoux, 13-Sep-2018.)
𝐷 = (ODual‘𝐾)       (𝐾 ∈ Preset → 𝐷 ∈ Preset )

Theoremposrasymb 28988 A poset ordering is asymetric. (Contributed by Thierry Arnoux, 13-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))       ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌𝑌 𝑋) ↔ 𝑋 = 𝑌))

Theoremtospos 28989 A Toset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
(𝐹 ∈ Toset → 𝐹 ∈ Poset)

Theoremresspos 28990 The restriction of a Poset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
((𝐹 ∈ Poset ∧ 𝐴𝑉) → (𝐹s 𝐴) ∈ Poset)

Theoremresstos 28991 The restriction of a Toset is a Toset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
((𝐹 ∈ Toset ∧ 𝐴𝑉) → (𝐹s 𝐴) ∈ Toset)

Theoremtleile 28992 In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 11-Feb-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)       ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 𝑋))

Theoremtltnle 28993 In a Toset, less-than is equivalent to not inverse less-than-or-equal see pltnle 16789. (Contributed by Thierry Arnoux, 11-Feb-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ ¬ 𝑌 𝑋))

Theoremodutos 28994 Being a toset is a self-dual property. (Contributed by Thierry Arnoux, 13-Sep-2018.)
𝐷 = (ODual‘𝐾)       (𝐾 ∈ Toset → 𝐷 ∈ Toset)

Theoremtlt2 28995 In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 < 𝑋))

Theoremtlt3 28996 In a Toset, two elements must compare strictly, or be equal. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)       ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌𝑋 < 𝑌𝑌 < 𝑋))

Theoremtrleile 28997 In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))       ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 𝑋))

Theoremtoslublem 28998* Lemma for toslub 28999 and xrsclat 29011. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by NM, 15-Sep-2018.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   (𝜑𝐾 ∈ Toset)    &   (𝜑𝐴𝐵)    &    = (le‘𝐾)       ((𝜑𝑎𝐵) → ((∀𝑏𝐴 𝑏 𝑎 ∧ ∀𝑐𝐵 (∀𝑏𝐴 𝑏 𝑐𝑎 𝑐)) ↔ (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏𝐵 (𝑏 < 𝑎 → ∃𝑑𝐴 𝑏 < 𝑑))))

Theoremtoslub 28999 In a toset, the lowest upper bound lub, defined for partial orders is the supremum, sup(𝐴, 𝐵, < ), defined for total orders. (these are the set.mm definitions: lowest upper bound and supremum are normally synonymous). Note that those two values are also equal if such a supremum does not exist: in that case, both are equal to the empty set. (Contributed by Thierry Arnoux, 15-Feb-2018.) (Revised by Thierry Arnoux, 24-Sep-2018.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   (𝜑𝐾 ∈ Toset)    &   (𝜑𝐴𝐵)       (𝜑 → ((lub‘𝐾)‘𝐴) = sup(𝐴, 𝐵, < ))

Theoremtosglblem 29000* Lemma for tosglb 29001 and xrsclat 29011. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by NM, 15-Sep-2018.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   (𝜑𝐾 ∈ Toset)    &   (𝜑𝐴𝐵)    &    = (le‘𝐾)       ((𝜑𝑎𝐵) → ((∀𝑏𝐴 𝑎 𝑏 ∧ ∀𝑐𝐵 (∀𝑏𝐴 𝑐 𝑏𝑐 𝑎)) ↔ (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏𝐵 (𝑏 < 𝑎 → ∃𝑑𝐴 𝑏 < 𝑑))))

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