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Theorem List for Intuitionistic Logic Explorer - 9601-9700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeluzge2nn0 9601 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 9602 An integer greater than or equal to 2 is not 0. (Contributed by AV, 25-May-2020.)
(𝑁 ∈ (ℤ‘2) → 𝑁 ≠ 0)
 
Theoremuzuzle23 9603 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 9604 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 9605 An integer greater than or equal to 3 decreased by 2 is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
(𝑁 ∈ (ℤ‘3) → (𝑁 − 2) ∈ ℕ)
 
Theorem1eluzge0 9606 1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
1 ∈ (ℤ‘0)
 
Theorem2eluzge0 9607 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 9608 2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
2 ∈ (ℤ‘1)
 
Theoremuznnssnn 9609 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 9610* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑀 ∈ ℤ → (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
 
Theoremraluz2 9611* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
 
Theoremrexuz 9612* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑀 ∈ ℤ → (∃𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∃𝑛 ∈ ℤ (𝑀𝑛𝜑)))
 
Theoremrexuz2 9613* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(∃𝑛 ∈ (ℤ𝑀)𝜑 ↔ (𝑀 ∈ ℤ ∧ ∃𝑛 ∈ ℤ (𝑀𝑛𝜑)))
 
Theorem2rexuz 9614* Double existential quantification in an upper set of integers. (Contributed by NM, 3-Nov-2005.)
(∃𝑚𝑛 ∈ (ℤ𝑚)𝜑 ↔ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑚𝑛𝜑))
 
Theorempeano2uz 9615 Second Peano postulate for an upper set of integers. (Contributed by NM, 7-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → (𝑁 + 1) ∈ (ℤ𝑀))
 
Theorempeano2uzs 9616 Second Peano postulate for an upper set of integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
𝑍 = (ℤ𝑀)       (𝑁𝑍 → (𝑁 + 1) ∈ 𝑍)
 
Theorempeano2uzr 9617 Reversed second Peano axiom for upper integers. (Contributed by NM, 2-Jan-2006.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → 𝑁 ∈ (ℤ𝑀))
 
Theoremuzaddcl 9618 Addition closure law for an upper set of integers. (Contributed by NM, 4-Jun-2006.)
((𝑁 ∈ (ℤ𝑀) ∧ 𝐾 ∈ ℕ0) → (𝑁 + 𝐾) ∈ (ℤ𝑀))
 
Theoremnn0pzuz 9619 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 9620* 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 9621* 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 9620 or uzind4ALT 9621 may be used; see comment for nnind 8966. (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 9622* 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 9623* 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 9622 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 9624* Induction on the upper integers that start at 𝑀. The first four give us the substitution instances we need, and the last two are the basis and the induction step. This is a stronger version of uzind4 9620 assuming that 𝜓 holds unconditionally. Notice that 𝑁 ∈ (ℤ𝑀) implies that the lower bound 𝑀 is an integer (𝑀 ∈ ℤ, see eluzel2 9564). (Contributed by NM, 4-Sep-2005.) (Revised by AV, 13-Jul-2022.)
(𝑗 = 𝑀 → (𝜑𝜓))    &   (𝑗 = 𝑘 → (𝜑𝜒))    &   (𝑗 = (𝑘 + 1) → (𝜑𝜃))    &   (𝑗 = 𝑁 → (𝜑𝜏))    &   𝜓    &   (𝑘 ∈ (ℤ𝑀) → (𝜒𝜃))       (𝑁 ∈ (ℤ𝑀) → 𝜏)
 
Theoremindstr 9625* 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)
 
Theoreminfrenegsupex 9626* The infimum of a set of reals 𝐴 is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 14-Jan-2022.)
(𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))    &   (𝜑𝐴 ⊆ ℝ)       (𝜑 → inf(𝐴, ℝ, < ) = -sup({𝑧 ∈ ℝ ∣ -𝑧𝐴}, ℝ, < ))
 
Theoremsupinfneg 9627* If a set of real numbers has a least upper bound, the set of the negation of those numbers has a greatest lower bound. For a theorem which is similar but only for the boundedness part, see ublbneg 9645. (Contributed by Jim Kingdon, 15-Jan-2022.)
(𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))    &   (𝜑𝐴 ⊆ ℝ)       (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦)))
 
Theoreminfsupneg 9628* If a set of real numbers has a greatest lower bound, the set of the negation of those numbers has a least upper bound. To go in the other direction see supinfneg 9627. (Contributed by Jim Kingdon, 15-Jan-2022.)
(𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))    &   (𝜑𝐴 ⊆ ℝ)       (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑦 < 𝑧)))
 
Theoremsupminfex 9629* A supremum is the negation of the infimum of that set's image under negation. (Contributed by Jim Kingdon, 14-Jan-2022.)
(𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))    &   (𝜑𝐴 ⊆ ℝ)       (𝜑 → sup(𝐴, ℝ, < ) = -inf({𝑤 ∈ ℝ ∣ -𝑤𝐴}, ℝ, < ))
 
Theoreminfregelbex 9630* Any lower bound of a set of real numbers with an infimum is less than or equal to the infimum. (Contributed by Jim Kingdon, 27-Sep-2024.)
(𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐵 ≤ inf(𝐴, ℝ, < ) ↔ ∀𝑧𝐴 𝐵𝑧))
 
Theoremeluznn0 9631 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 9632 Membership in a positive upper set of integers implies membership in . (Contributed by JJ, 1-Oct-2018.)
((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ𝑁)) → 𝑀 ∈ ℕ)
 
Theoremeluz2b1 9633 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 9634 An integer greater than or equal to 2 is greater than 1. (Contributed by AV, 24-May-2020.)
(𝑁 ∈ (ℤ‘2) → 1 < 𝑁)
 
Theoremeluz2b2 9635 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 9636 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 9637 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 9638 1 is not in (ℤ‘2). (Contributed by Paul Chapman, 21-Nov-2012.)
¬ 1 ∈ (ℤ‘2)
 
Theoremelnn1uz2 9639 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 9640 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 9641* 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)
 
Theoremeluzdc 9642 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 ∈ (ℤ𝑀))
 
Theoremelnn0dc 9643 Membership of an integer in 0 is decidable. (Contributed by Jim Kingdon, 8-Oct-2024.)
(𝑁 ∈ ℤ → DECID 𝑁 ∈ ℕ0)
 
Theoremelnndc 9644 Membership of an integer in is decidable. (Contributed by Jim Kingdon, 17-Oct-2024.)
(𝑁 ∈ ℤ → DECID 𝑁 ∈ ℕ)
 
Theoremublbneg 9645* The image under negation of a bounded-above set of reals is bounded below. For a theorem which is similar but also adds that the bounds need to be the tightest possible, see supinfneg 9627. (Contributed by Paul Chapman, 21-Mar-2011.)
(∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧𝐴}𝑥𝑦)
 
Theoremeqreznegel 9646* 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})
 
Theoremnegm 9647* The image under negation of an inhabited set of reals is inhabited. (Contributed by Jim Kingdon, 10-Apr-2020.)
((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥𝐴) → ∃𝑦 𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧𝐴})
 
Theoremlbzbi 9648* 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))
 
Theoremnn01to3 9649 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 9650 Alternate proof of nn0ge2m1nn 9267: 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 9565, 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 9267. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ)
 
4.4.12  Rational numbers (as a subset of complex numbers)
 
Syntaxcq 9651 Extend class notation to include the class of rationals.
class
 
Definitiondf-q 9652 Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 9654 for the relation "is rational". (Contributed by NM, 8-Jan-2002.)
ℚ = ( / “ (ℤ × ℕ))
 
Theoremdivfnzn 9653 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 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)
 
Theoremelq 9654* 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 9655* 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 9656 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 9657 A rational number is a real number. (Contributed by NM, 14-Nov-2002.)
(𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
 
Theoremzq 9658 An integer is a rational number. (Contributed by NM, 9-Jan-2002.)
(𝐴 ∈ ℤ → 𝐴 ∈ ℚ)
 
Theoremzssq 9659 The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)
ℤ ⊆ ℚ
 
Theoremnn0ssq 9660 The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
0 ⊆ ℚ
 
Theoremnnssq 9661 The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
ℕ ⊆ ℚ
 
Theoremqssre 9662 The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)
ℚ ⊆ ℝ
 
Theoremqsscn 9663 The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
ℚ ⊆ ℂ
 
Theoremqex 9664 The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
ℚ ∈ V
 
Theoremnnq 9665 A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
(𝐴 ∈ ℕ → 𝐴 ∈ ℚ)
 
Theoremqcn 9666 A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)
(𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
 
Theoremqaddcl 9667 Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ)
 
Theoremqnegcl 9668 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 9669 Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 · 𝐵) ∈ ℚ)
 
Theoremqsubcl 9670 Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴𝐵) ∈ ℚ)
 
Theoremqapne 9671 Apartness is equivalent to not equal for rationals. (Contributed by Jim Kingdon, 20-Mar-2020.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 # 𝐵𝐴𝐵))
 
Theoremqltlen 9672 Rational 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 8620 which is a similar result for real numbers. (Contributed by Jim Kingdon, 11-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 < 𝐵 ↔ (𝐴𝐵𝐵𝐴)))
 
Theoremqlttri2 9673 Apartness is equivalent to not equal for rationals. (Contributed by Jim Kingdon, 9-Nov-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴𝐵 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
 
Theoremqreccl 9674 Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℚ)
 
Theoremqdivcl 9675 Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℚ)
 
Theoremqrevaddcl 9676 Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.)
(𝐵 ∈ ℚ → ((𝐴 ∈ ℂ ∧ (𝐴 + 𝐵) ∈ ℚ) ↔ 𝐴 ∈ ℚ))
 
Theoremnnrecq 9677 The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)
(𝐴 ∈ ℕ → (1 / 𝐴) ∈ ℚ)
 
Theoremirradd 9678 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 9679 The product of a real which is not rational with a nonzero rational is not rational. Note that by "not rational" we mean the negation of "is rational" (whereas "irrational" is often defined to mean apart from any rational number - given excluded middle these two definitions would be equivalent). (Contributed by NM, 7-Nov-2008.)
((𝐴 ∈ (ℝ ∖ ℚ) ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 · 𝐵) ∈ (ℝ ∖ ℚ))
 
Theoremelpq 9680* A positive rational is the quotient of two positive integers. (Contributed by AV, 29-Dec-2022.)
((𝐴 ∈ ℚ ∧ 0 < 𝐴) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦))
 
Theoremelpqb 9681* A class is a positive rational iff it is the quotient of two positive integers. (Contributed by AV, 30-Dec-2022.)
((𝐴 ∈ ℚ ∧ 0 < 𝐴) ↔ ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦))
 
4.4.13  Complex numbers as pairs of reals
 
Theoremcnref1o 9682* There is a natural one-to-one mapping from (ℝ × ℝ) to , where we map 𝑥, 𝑦 to (𝑥 + (i · 𝑦)). In our construction of the complex numbers, this is in fact our definition of (see df-c 7848), but in the axiomatic treatment we can only show that there is the expected mapping between these two sets. (Contributed by Mario Carneiro, 16-Jun-2013.) (Revised by Mario Carneiro, 17-Feb-2014.)
𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))       𝐹:(ℝ × ℝ)–1-1-onto→ℂ
 
Theoremaddex 9683 The addition operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
+ ∈ V
 
Theoremmulex 9684 The multiplication operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
· ∈ V
 
4.5  Order sets
 
4.5.1  Positive reals (as a subset of complex numbers)
 
Syntaxcrp 9685 Extend class notation to include the class of positive reals.
class +
 
Definitiondf-rp 9686 Define the set of positive reals. Definition of positive numbers in [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
+ = {𝑥 ∈ ℝ ∣ 0 < 𝑥}
 
Theoremelrp 9687 Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)
(𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
 
Theoremelrpii 9688 Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
𝐴 ∈ ℝ    &   0 < 𝐴       𝐴 ∈ ℝ+
 
Theorem1rp 9689 1 is a positive real. (Contributed by Jeff Hankins, 23-Nov-2008.)
1 ∈ ℝ+
 
Theorem2rp 9690 2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
2 ∈ ℝ+
 
Theorem3rp 9691 3 is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
3 ∈ ℝ+
 
Theoremrpre 9692 A positive real is a real. (Contributed by NM, 27-Oct-2007.)
(𝐴 ∈ ℝ+𝐴 ∈ ℝ)
 
Theoremrpxr 9693 A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)
(𝐴 ∈ ℝ+𝐴 ∈ ℝ*)
 
Theoremrpcn 9694 A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
(𝐴 ∈ ℝ+𝐴 ∈ ℂ)
 
Theoremnnrp 9695 A positive integer is a positive real. (Contributed by NM, 28-Nov-2008.)
(𝐴 ∈ ℕ → 𝐴 ∈ ℝ+)
 
Theoremrpssre 9696 The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)
+ ⊆ ℝ
 
Theoremrpgt0 9697 A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
(𝐴 ∈ ℝ+ → 0 < 𝐴)
 
Theoremrpge0 9698 A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)
(𝐴 ∈ ℝ+ → 0 ≤ 𝐴)
 
Theoremrpregt0 9699 A positive real is a positive real number. (Contributed by NM, 11-Nov-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
(𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
 
Theoremrprege0 9700 A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)
(𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
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