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Theorem List for Metamath Proof Explorer - 11601-11700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelnn0uz 11601 A nonnegative integer expressed as a member an upper set of integers. (Contributed by NM, 6-Jun-2006.)
(𝑁 ∈ ℕ0𝑁 ∈ (ℤ‘0))

Theoremeluz2nn 11602 An integer is greater than or equal to 2 is a positive integer. (Contributed by AV, 3-Nov-2018.)
(𝐴 ∈ (ℤ‘2) → 𝐴 ∈ ℕ)

Theoremeluzge2nn0 11603 If an integer is greater than or equal to 2, then it is a nonnegative integer. (Contributed by AV, 27-Aug-2018.) (Proof shortened by AV, 3-Nov-2018.)
(𝑁 ∈ (ℤ‘2) → 𝑁 ∈ ℕ0)

Theoremeluz2n0 11604 An integer greater than or equal to 2 is not 0. (Contributed by AV, 25-May-2020.)
(𝑁 ∈ (ℤ‘2) → 𝑁 ≠ 0)

Theoremuzuzle23 11605 An integer in the upper set of integers starting at 3 is element of the upper set of integers starting at 2. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
(𝐴 ∈ (ℤ‘3) → 𝐴 ∈ (ℤ‘2))

Theoremeluzge3nn 11606 If an integer is greater than 3, then it is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
(𝑁 ∈ (ℤ‘3) → 𝑁 ∈ ℕ)

Theoremuz3m2nn 11607 An integer greater than or equal to 3 decreased by 2 is a positive integer, analogous to uz2m1nn 11639. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
(𝑁 ∈ (ℤ‘3) → (𝑁 − 2) ∈ ℕ)

Theorem1eluzge0 11608 1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
1 ∈ (ℤ‘0)

Theorem2eluzge0 11609 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)

Theorem2eluzge1 11610 2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
2 ∈ (ℤ‘1)

Theoremuznnssnn 11611 The upper integers starting from a natural are a subset of the naturals. (Contributed by Scott Fenton, 29-Jun-2013.)
(𝑁 ∈ ℕ → (ℤ𝑁) ⊆ ℕ)

Theoremraluz 11612* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑀 ∈ ℤ → (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))

Theoremraluz2 11613* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))

Theoremrexuz 11614* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑀 ∈ ℤ → (∃𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∃𝑛 ∈ ℤ (𝑀𝑛𝜑)))

Theoremrexuz2 11615* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(∃𝑛 ∈ (ℤ𝑀)𝜑 ↔ (𝑀 ∈ ℤ ∧ ∃𝑛 ∈ ℤ (𝑀𝑛𝜑)))

Theorem2rexuz 11616* Double existential quantification in an upper set of integers. (Contributed by NM, 3-Nov-2005.)
(∃𝑚𝑛 ∈ (ℤ𝑚)𝜑 ↔ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑚𝑛𝜑))

Theorempeano2uz 11617 Second Peano postulate for an upper set of integers. (Contributed by NM, 7-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → (𝑁 + 1) ∈ (ℤ𝑀))

Theorempeano2uzs 11618 Second Peano postulate for an upper set of integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
𝑍 = (ℤ𝑀)       (𝑁𝑍 → (𝑁 + 1) ∈ 𝑍)

Theorempeano2uzr 11619 Reversed second Peano axiom for upper integers. (Contributed by NM, 2-Jan-2006.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → 𝑁 ∈ (ℤ𝑀))

Theoremuzaddcl 11620 Addition closure law for an upper set of integers. (Contributed by NM, 4-Jun-2006.)
((𝑁 ∈ (ℤ𝑀) ∧ 𝐾 ∈ ℕ0) → (𝑁 + 𝐾) ∈ (ℤ𝑀))

Theoremnn0pzuz 11621 The sum of a nonnegative integer and an integer is an integer greater than or equal to that integer. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
((𝑁 ∈ ℕ0𝑍 ∈ ℤ) → (𝑁 + 𝑍) ∈ (ℤ𝑍))

Theoremuzind4 11622* Induction on the upper set of integers that starts at an integer 𝑀. The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction step. (Contributed by NM, 7-Sep-2005.)
(𝑗 = 𝑀 → (𝜑𝜓))    &   (𝑗 = 𝑘 → (𝜑𝜒))    &   (𝑗 = (𝑘 + 1) → (𝜑𝜃))    &   (𝑗 = 𝑁 → (𝜑𝜏))    &   (𝑀 ∈ ℤ → 𝜓)    &   (𝑘 ∈ (ℤ𝑀) → (𝜒𝜃))       (𝑁 ∈ (ℤ𝑀) → 𝜏)

Theoremuzind4ALT 11623* Induction on the upper set of integers that starts at an integer 𝑀. The last four hypotheses give us the substitution instances we need; the first two are the basis and the induction step. Either uzind4 11622 or uzind4ALT 11623 may be used; see comment for nnind 10915. (Contributed by NM, 7-Sep-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑀 ∈ ℤ → 𝜓)    &   (𝑘 ∈ (ℤ𝑀) → (𝜒𝜃))    &   (𝑗 = 𝑀 → (𝜑𝜓))    &   (𝑗 = 𝑘 → (𝜑𝜒))    &   (𝑗 = (𝑘 + 1) → (𝜑𝜃))    &   (𝑗 = 𝑁 → (𝜑𝜏))       (𝑁 ∈ (ℤ𝑀) → 𝜏)

Theoremuzind4s 11624* Induction on the upper set of integers that starts at an integer 𝑀, using explicit substitution. The hypotheses are the basis and the induction step. (Contributed by NM, 4-Nov-2005.)
(𝑀 ∈ ℤ → [𝑀 / 𝑘]𝜑)    &   (𝑘 ∈ (ℤ𝑀) → (𝜑[(𝑘 + 1) / 𝑘]𝜑))       (𝑁 ∈ (ℤ𝑀) → [𝑁 / 𝑘]𝜑)

Theoremuzind4s2 11625* Induction on the upper set of integers that starts at an integer 𝑀, using explicit substitution. The hypotheses are the basis and the induction step. Use this instead of uzind4s 11624 when 𝑗 and 𝑘 must be distinct in [(𝑘 + 1) / 𝑗]𝜑. (Contributed by NM, 16-Nov-2005.)
(𝑀 ∈ ℤ → [𝑀 / 𝑗]𝜑)    &   (𝑘 ∈ (ℤ𝑀) → ([𝑘 / 𝑗]𝜑[(𝑘 + 1) / 𝑗]𝜑))       (𝑁 ∈ (ℤ𝑀) → [𝑁 / 𝑗]𝜑)

Theoremuzind4i 11626* Induction on the upper integers that start at 𝑀. The first hypothesis specifies the lower bound, the next four give us the substitution instances we need, and the last two are the basis and the induction step. (Contributed by NM, 4-Sep-2005.)
𝑀 ∈ ℤ    &   (𝑗 = 𝑀 → (𝜑𝜓))    &   (𝑗 = 𝑘 → (𝜑𝜒))    &   (𝑗 = (𝑘 + 1) → (𝜑𝜃))    &   (𝑗 = 𝑁 → (𝜑𝜏))    &   𝜓    &   (𝑘 ∈ (ℤ𝑀) → (𝜒𝜃))       (𝑁 ∈ (ℤ𝑀) → 𝜏)

Theoremuzwo 11627* Well-ordering principle: any nonempty subset of an upper set of integers has the least element. (Contributed by NM, 8-Oct-2005.)
((𝑆 ⊆ (ℤ𝑀) ∧ 𝑆 ≠ ∅) → ∃𝑗𝑆𝑘𝑆 𝑗𝑘)

Theoremuzwo2 11628* Well-ordering principle: any nonempty subset of an upper set of integers has a unique least element. (Contributed by NM, 8-Oct-2005.)
((𝑆 ⊆ (ℤ𝑀) ∧ 𝑆 ≠ ∅) → ∃!𝑗𝑆𝑘𝑆 𝑗𝑘)

Theoremnnwo 11629* Well-ordering principle: any nonempty set of positive integers has a least element. Theorem I.37 (well-ordering principle) of [Apostol] p. 34. (Contributed by NM, 17-Aug-2001.)
((𝐴 ⊆ ℕ ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 𝑥𝑦)

Theoremnnwof 11630* Well-ordering principle: any nonempty set of positive integers has a least element. This version allows 𝑥 and 𝑦 to be present in 𝐴 as long as they are effectively not free. (Contributed by NM, 17-Aug-2001.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴    &   𝑦𝐴       ((𝐴 ⊆ ℕ ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 𝑥𝑦)

Theoremnnwos 11631* Well-ordering principle: any nonempty set of positive integers has a least element (schema form). (Contributed by NM, 17-Aug-2001.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥 ∈ ℕ 𝜑 → ∃𝑥 ∈ ℕ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓𝑥𝑦)))

Theoremindstr 11632* Strong Mathematical Induction for positive integers (inference schema). (Contributed by NM, 17-Aug-2001.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓) → 𝜑))       (𝑥 ∈ ℕ → 𝜑)

Theoremeluznn0 11633 Membership in a nonnegative upper set of integers implies membership in 0. (Contributed by Paul Chapman, 22-Jun-2011.)
((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁)) → 𝑀 ∈ ℕ0)

Theoremeluznn 11634 Membership in a positive upper set of integers implies membership in . (Contributed by JJ, 1-Oct-2018.)
((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ𝑁)) → 𝑀 ∈ ℕ)

Theoremeluz2b1 11635 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)
(𝑁 ∈ (ℤ‘2) ↔ (𝑁 ∈ ℤ ∧ 1 < 𝑁))

Theoremeluz2gt1 11636 An integer greater than or equal to 2 is greater than 1. (Contributed by AV, 24-May-2020.)
(𝑁 ∈ (ℤ‘2) → 1 < 𝑁)

Theoremeluz2b2 11637 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)
(𝑁 ∈ (ℤ‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁))

Theoremeluz2b3 11638 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)
(𝑁 ∈ (ℤ‘2) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1))

Theoremuz2m1nn 11639 One less than an integer greater than or equal to 2 is a positive integer. (Contributed by Paul Chapman, 17-Nov-2012.)
(𝑁 ∈ (ℤ‘2) → (𝑁 − 1) ∈ ℕ)

Theorem1nuz2 11640 1 is not in (ℤ‘2). (Contributed by Paul Chapman, 21-Nov-2012.)
¬ 1 ∈ (ℤ‘2)

Theoremelnn1uz2 11641 A positive integer is either 1 or greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.)
(𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ‘2)))

Theoremuz2mulcl 11642 Closure of multiplication of integers greater than or equal to 2. (Contributed by Paul Chapman, 26-Oct-2012.)
((𝑀 ∈ (ℤ‘2) ∧ 𝑁 ∈ (ℤ‘2)) → (𝑀 · 𝑁) ∈ (ℤ‘2))

Theoremindstr2 11643* Strong Mathematical Induction for positive integers (inference schema). The first two hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 21-Nov-2012.)
(𝑥 = 1 → (𝜑𝜒))    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   𝜒    &   (𝑥 ∈ (ℤ‘2) → (∀𝑦 ∈ ℕ (𝑦 < 𝑥𝜓) → 𝜑))       (𝑥 ∈ ℕ → 𝜑)

Theoremuzinfi 11644 Extract the lower bound of an upper set of integers as its infimum. (Contributed by NM, 7-Oct-2005.) (Revised by AV, 4-Sep-2020.)
𝑀 ∈ ℤ       inf((ℤ𝑀), ℝ, < ) = 𝑀

Theoremnninf 11645 The infimum of the set of positive integers is one. (Contributed by NM, 16-Jun-2005.) (Revised by AV, 5-Sep-2020.)
inf(ℕ, ℝ, < ) = 1

Theoremnn0inf 11646 The infimum of the set of nonnegative integers is zero. (Contributed by NM, 16-Jun-2005.) (Revised by AV, 5-Sep-2020.)
inf(ℕ0, ℝ, < ) = 0

Theoreminfssuzle 11647 The infimum of a subset of an upper set of integers is less than or equal to all members of the subset. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 5-Sep-2020.)
((𝑆 ⊆ (ℤ𝑀) ∧ 𝐴𝑆) → inf(𝑆, ℝ, < ) ≤ 𝐴)

Theoreminfssuzcl 11648 The infimum of a subset of an upper set of integers belongs to the subset. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 5-Sep-2020.)
((𝑆 ⊆ (ℤ𝑀) ∧ 𝑆 ≠ ∅) → inf(𝑆, ℝ, < ) ∈ 𝑆)

Theoremublbneg 11649* The image under negation of a bounded-above set of reals is bounded below. (Contributed by Paul Chapman, 21-Mar-2011.)
(∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧𝐴}𝑥𝑦)

Theoremeqreznegel 11650* Two ways to express the image under negation of a set of integers. (Contributed by Paul Chapman, 21-Mar-2011.)
(𝐴 ⊆ ℤ → {𝑧 ∈ ℝ ∣ -𝑧𝐴} = {𝑧 ∈ ℤ ∣ -𝑧𝐴})

Theoremsupminf 11651* The supremum of a bounded-above set of reals is the negation of the infimum of that set's image under negation. (Contributed by Paul Chapman, 21-Mar-2011.) ( Revised by AV, 13-Sep-2020.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥) → sup(𝐴, ℝ, < ) = -inf({𝑧 ∈ ℝ ∣ -𝑧𝐴}, ℝ, < ))

Theoremlbzbi 11652* If a set of reals is bounded below, it is bounded below by an integer. (Contributed by Paul Chapman, 21-Mar-2011.)
(𝐴 ⊆ ℝ → (∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑥𝑦 ↔ ∃𝑥 ∈ ℤ ∀𝑦𝐴 𝑥𝑦))

Theoremzsupss 11653* Any nonempty bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-sup 9893.) (Contributed by Mario Carneiro, 21-Apr-2015.)
((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦𝐴 𝑦𝑥) → ∃𝑥𝐴 (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦𝐵 (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))

Theoremsuprzcl2 11654* The supremum of a bounded-above set of integers is a member of the set. (This version of suprzcl 11333 avoids ax-pre-sup 9893.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Mario Carneiro, 24-Dec-2016.)
((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦𝐴 𝑦𝑥) → sup(𝐴, ℝ, < ) ∈ 𝐴)

Theoremsuprzub 11655* The supremum of a bounded-above set of integers is greater than any member of the set. (Contributed by Mario Carneiro, 21-Apr-2015.)
((𝐴 ⊆ ℤ ∧ ∃𝑥 ∈ ℤ ∀𝑦𝐴 𝑦𝑥𝐵𝐴) → 𝐵 ≤ sup(𝐴, ℝ, < ))

Theoremuzsupss 11656* Any bounded subset of an upper set of integers has a supremum. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 21-Apr-2015.)
𝑍 = (ℤ𝑀)       ((𝑀 ∈ ℤ ∧ 𝐴𝑍 ∧ ∃𝑥 ∈ ℤ ∀𝑦𝐴 𝑦𝑥) → ∃𝑥𝑍 (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦𝑍 (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))

Theoremnn01to3 11657 A (nonnegative) integer between 1 and 3 must be 1, 2 or 3. (Contributed by Alexander van der Vekens, 13-Sep-2018.)
((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁𝑁 ≤ 3) → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))

Theoremnn0ge2m1nnALT 11658 Alternate proof of nn0ge2m1nn 11237: If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. This version is proved using eluz2 11569, a theorem for upper sets of integers, which are defined later than the positive and nonnegative integers. This proof is, however, much shorter than the proof of nn0ge2m1nn 11237. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ)

5.4.12  Well-ordering principle for bounded-below sets of integers

Theoremuzwo3 11659* Well-ordering principle: any nonempty subset of an upper set of integers has a unique least element. This generalization of uzwo2 11628 allows the lower bound 𝐵 to be any real number. See also nnwo 11629 and nnwos 11631. (Contributed by NM, 12-Nov-2004.) (Proof shortened by Mario Carneiro, 2-Oct-2015.) (Proof shortened by AV, 27-Sep-2020.)
((𝐵 ∈ ℝ ∧ (𝐴 ⊆ {𝑧 ∈ ℤ ∣ 𝐵𝑧} ∧ 𝐴 ≠ ∅)) → ∃!𝑥𝐴𝑦𝐴 𝑥𝑦)

Theoremzmin 11660* There is a unique smallest integer greater than or equal to a given real number. (Contributed by NM, 12-Nov-2004.) (Revised by Mario Carneiro, 13-Jun-2014.)
(𝐴 ∈ ℝ → ∃!𝑥 ∈ ℤ (𝐴𝑥 ∧ ∀𝑦 ∈ ℤ (𝐴𝑦𝑥𝑦)))

Theoremzmax 11661* There is a unique largest integer less than or equal to a given real number. (Contributed by NM, 15-Nov-2004.)
(𝐴 ∈ ℝ → ∃!𝑥 ∈ ℤ (𝑥𝐴 ∧ ∀𝑦 ∈ ℤ (𝑦𝐴𝑦𝑥)))

Theoremzbtwnre 11662* There is a unique integer between a real number and the number plus one. Exercise 5 of [Apostol] p. 28. (Contributed by NM, 13-Nov-2004.)
(𝐴 ∈ ℝ → ∃!𝑥 ∈ ℤ (𝐴𝑥𝑥 < (𝐴 + 1)))

Theoremrebtwnz 11663* There is a unique greatest integer less than or equal to a real number. Exercise 4 of [Apostol] p. 28. (Contributed by NM, 15-Nov-2004.)
(𝐴 ∈ ℝ → ∃!𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1)))

5.4.13  Rational numbers (as a subset of complex numbers)

Syntaxcq 11664 Extend class notation to include the class of rationals.
class

Definitiondf-q 11665 Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 11666 for the relation "is rational." (Contributed by NM, 8-Jan-2002.)
ℚ = ( / “ (ℤ × ℕ))

Theoremelq 11666* Membership in the set of rationals. (Contributed by NM, 8-Jan-2002.) (Revised by Mario Carneiro, 28-Jan-2014.)
(𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦))

Theoremqmulz 11667* If 𝐴 is rational, then some integer multiple of it is an integer. (Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro, 22-Jul-2014.)
(𝐴 ∈ ℚ → ∃𝑥 ∈ ℕ (𝐴 · 𝑥) ∈ ℤ)

Theoremznq 11668 The ratio of an integer and a positive integer is a rational number. (Contributed by NM, 12-Jan-2002.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ∈ ℚ)

Theoremqre 11669 A rational number is a real number. (Contributed by NM, 14-Nov-2002.)
(𝐴 ∈ ℚ → 𝐴 ∈ ℝ)

Theoremzq 11670 An integer is a rational number. (Contributed by NM, 9-Jan-2002.)
(𝐴 ∈ ℤ → 𝐴 ∈ ℚ)

Theoremzssq 11671 The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)
ℤ ⊆ ℚ

Theoremnn0ssq 11672 The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
0 ⊆ ℚ

Theoremnnssq 11673 The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
ℕ ⊆ ℚ

Theoremqssre 11674 The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)
ℚ ⊆ ℝ

Theoremqsscn 11675 The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
ℚ ⊆ ℂ

Theoremqex 11676 The set of rational numbers exists. See also qexALT 11679. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
ℚ ∈ V

Theoremnnq 11677 A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
(𝐴 ∈ ℕ → 𝐴 ∈ ℚ)

Theoremqcn 11678 A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)
(𝐴 ∈ ℚ → 𝐴 ∈ ℂ)

TheoremqexALT 11679 Alternate proof of qex 11676. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
ℚ ∈ V

((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ)

Theoremqnegcl 11681 Closure law for the negative of a rational. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 15-Sep-2014.)
(𝐴 ∈ ℚ → -𝐴 ∈ ℚ)

Theoremqmulcl 11682 Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 · 𝐵) ∈ ℚ)

Theoremqsubcl 11683 Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴𝐵) ∈ ℚ)

Theoremqreccl 11684 Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℚ)

Theoremqdivcl 11685 Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℚ)

Theoremqrevaddcl 11686 Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.)
(𝐵 ∈ ℚ → ((𝐴 ∈ ℂ ∧ (𝐴 + 𝐵) ∈ ℚ) ↔ 𝐴 ∈ ℚ))

Theoremnnrecq 11687 The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)
(𝐴 ∈ ℕ → (1 / 𝐴) ∈ ℚ)

Theoremirradd 11688 The sum of an irrational number and a rational number is irrational. (Contributed by NM, 7-Nov-2008.)
((𝐴 ∈ (ℝ ∖ ℚ) ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ (ℝ ∖ ℚ))

Theoremirrmul 11689 The product of an irrational with a nonzero rational is irrational. (Contributed by NM, 7-Nov-2008.)
((𝐴 ∈ (ℝ ∖ ℚ) ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 · 𝐵) ∈ (ℝ ∖ ℚ))

5.4.14  Existence of the set of complex numbers

Theoremrpnnen1lem2 11690* Lemma for rpnnen1 11696. (Contributed by Mario Carneiro, 12-May-2013.)
𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))       ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℤ)

Theoremrpnnen1lem1 11691* Lemma for rpnnen1 11696. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 13-Aug-2021.) (Proof modification is discouraged.)
𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))    &   ℕ ∈ V    &   ℚ ∈ V       (𝑥 ∈ ℝ → (𝐹𝑥) ∈ (ℚ ↑𝑚 ℕ))

Theoremrpnnen1lem3 11692* Lemma for rpnnen1 11696. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 13-Aug-2021.) (Proof modification is discouraged.)
𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))    &   ℕ ∈ V    &   ℚ ∈ V       (𝑥 ∈ ℝ → ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑥)

Theoremrpnnen1lem4 11693* Lemma for rpnnen1 11696. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 13-Aug-2021.) (Proof modification is discouraged.)
𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))    &   ℕ ∈ V    &   ℚ ∈ V       (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ)

Theoremrpnnen1lem5 11694* Lemma for rpnnen1 11696. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 13-Aug-2021.) (Proof modification is discouraged.)
𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))    &   ℕ ∈ V    &   ℚ ∈ V       (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) = 𝑥)

Theoremrpnnen1lem6 11695* Lemma for rpnnen1 11696. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 15-Aug-2021.) (Proof modification is discouraged.)
𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))    &   ℕ ∈ V    &   ℚ ∈ V       ℝ ≼ (ℚ ↑𝑚 ℕ)

Theoremrpnnen1 11696 One half of rpnnen 14795, where we show an injection from the real numbers to sequences of rational numbers. Specifically, we map a real number 𝑥 to the sequence (𝐹𝑥):ℕ⟶ℚ (see rpnnen1lem6 11695) such that ((𝐹𝑥)‘𝑘) is the largest rational number with denominator 𝑘 that is strictly less than 𝑥. In this manner, we get a monotonically increasing sequence that converges to 𝑥, and since each sequence converges to a unique real number, this mapping from reals to sequences of rational numbers is injective. Note: The and existence hypotheses provide for use with either nnex 10903 and qex 11676, or nnexALT 10899 and qexALT 11679. The proof should not be modified to use any of those 4 theorems. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 16-Jun-2013.) (Revised by NM, 15-Aug-2021.) (Proof modification is discouraged.)
ℕ ∈ V    &   ℚ ∈ V       ℝ ≼ (ℚ ↑𝑚 ℕ)

Theoremrpnnen1lem1OLD 11697* Lemma for rpnnen1OLD 11701. (Contributed by Mario Carneiro, 12-May-2013.) Obsolete version of rpnnen1lem1 11691 as of 13-Aug-2021. (New usage is discouraged.) (Proof modification is discouraged.)
𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))       (𝑥 ∈ ℝ → (𝐹𝑥) ∈ (ℚ ↑𝑚 ℕ))

Theoremrpnnen1lem3OLD 11698* Lemma for rpnnen1OLD 11701. (Contributed by Mario Carneiro, 12-May-2013.) Obsolete version of rpnnen1lem3 11692 as of 13-Aug-2021. (New usage is discouraged.) (Proof modification is discouraged.)
𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))       (𝑥 ∈ ℝ → ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑥)

Theoremrpnnen1lem4OLD 11699* Lemma for rpnnen1OLD 11701. (Contributed by Mario Carneiro, 12-May-2013.) Obsolete version of rpnnen1lem4 11693 as of 13-Aug-2021. (New usage is discouraged.) (Proof modification is discouraged.)
𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))       (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ)

Theoremrpnnen1lem5OLD 11700* Lemma for rpnnen1OLD 11701. (Contributed by Mario Carneiro, 12-May-2013.) Obsolete version of rpnnen1lem5 11694 as of 13-Aug-2021. (New usage is discouraged.) (Proof modification is discouraged.)
𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))       (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) = 𝑥)

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