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Theorem List for Intuitionistic Logic Explorer - 8801-8900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeluzp1l 8801 Strict ordering implied by membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 < 𝑁)
 
Theoremeluzp1p1 8802 Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.)
(𝑁 ∈ (ℤ𝑀) → (𝑁 + 1) ∈ (ℤ‘(𝑀 + 1)))
 
Theoremeluzaddi 8803 Membership in a later upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
𝑀 ∈ ℤ    &   𝐾 ∈ ℤ       (𝑁 ∈ (ℤ𝑀) → (𝑁 + 𝐾) ∈ (ℤ‘(𝑀 + 𝐾)))
 
Theoremeluzsubi 8804 Membership in an earlier upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
𝑀 ∈ ℤ    &   𝐾 ∈ ℤ       (𝑁 ∈ (ℤ‘(𝑀 + 𝐾)) → (𝑁𝐾) ∈ (ℤ𝑀))
 
Theoremeluzadd 8805 Membership in a later upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝑁 ∈ (ℤ𝑀) ∧ 𝐾 ∈ ℤ) → (𝑁 + 𝐾) ∈ (ℤ‘(𝑀 + 𝐾)))
 
Theoremeluzsub 8806 Membership in an earlier upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 𝐾))) → (𝑁𝐾) ∈ (ℤ𝑀))
 
Theoremuzm1 8807 Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑁 ∈ (ℤ𝑀) → (𝑁 = 𝑀 ∨ (𝑁 − 1) ∈ (ℤ𝑀)))
 
Theoremuznn0sub 8808 The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → (𝑁𝑀) ∈ ℕ0)
 
Theoremuzin 8809 Intersection of two upper intervals of integers. (Contributed by Mario Carneiro, 24-Dec-2013.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((ℤ𝑀) ∩ (ℤ𝑁)) = (ℤ‘if(𝑀𝑁, 𝑁, 𝑀)))
 
Theoremuzp1 8810 Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑁 ∈ (ℤ𝑀) → (𝑁 = 𝑀𝑁 ∈ (ℤ‘(𝑀 + 1))))
 
Theoremnn0uz 8811 Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
0 = (ℤ‘0)
 
Theoremnnuz 8812 Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
ℕ = (ℤ‘1)
 
Theoremelnnuz 8813 A positive integer expressed as a member of an upper set of integers. (Contributed by NM, 6-Jun-2006.)
(𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ‘1))
 
Theoremelnn0uz 8814 A nonnegative integer expressed as a member an upper set of integers. (Contributed by NM, 6-Jun-2006.)
(𝑁 ∈ ℕ0𝑁 ∈ (ℤ‘0))
 
Theoremeluz2nn 8815 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 8816 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)
 
Theoremuzuzle23 8817 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 8818 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 8819 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 8820 1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
1 ∈ (ℤ‘0)
 
Theorem2eluzge0 8821 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 8822 2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
2 ∈ (ℤ‘1)
 
Theoremuznnssnn 8823 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 8824* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑀 ∈ ℤ → (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
 
Theoremraluz2 8825* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
 
Theoremrexuz 8826* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑀 ∈ ℤ → (∃𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∃𝑛 ∈ ℤ (𝑀𝑛𝜑)))
 
Theoremrexuz2 8827* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(∃𝑛 ∈ (ℤ𝑀)𝜑 ↔ (𝑀 ∈ ℤ ∧ ∃𝑛 ∈ ℤ (𝑀𝑛𝜑)))
 
Theorem2rexuz 8828* Double existential quantification in an upper set of integers. (Contributed by NM, 3-Nov-2005.)
(∃𝑚𝑛 ∈ (ℤ𝑚)𝜑 ↔ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑚𝑛𝜑))
 
Theorempeano2uz 8829 Second Peano postulate for an upper set of integers. (Contributed by NM, 7-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → (𝑁 + 1) ∈ (ℤ𝑀))
 
Theorempeano2uzs 8830 Second Peano postulate for an upper set of integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
𝑍 = (ℤ𝑀)       (𝑁𝑍 → (𝑁 + 1) ∈ 𝑍)
 
Theorempeano2uzr 8831 Reversed second Peano axiom for upper integers. (Contributed by NM, 2-Jan-2006.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → 𝑁 ∈ (ℤ𝑀))
 
Theoremuzaddcl 8832 Addition closure law for an upper set of integers. (Contributed by NM, 4-Jun-2006.)
((𝑁 ∈ (ℤ𝑀) ∧ 𝐾 ∈ ℕ0) → (𝑁 + 𝐾) ∈ (ℤ𝑀))
 
Theoremnn0pzuz 8833 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 8834* 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 8835* 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 8834 or uzind4ALT 8835 may be used; see comment for nnind 8199. (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 8836* 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 8837* 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 8836 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 8838* 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)
 
Theoremindstr 8839* 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 8840* 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 8841* 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 8856. (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 8842* 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 8841. (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 8843* 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, < ))
 
Theoremeluznn0 8844 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 8845 Membership in a positive upper set of integers implies membership in . (Contributed by JJ, 1-Oct-2018.)
((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ𝑁)) → 𝑀 ∈ ℕ)
 
Theoremeluz2b1 8846 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 8847 An integer greater than or equal to 2 is greater than 1. (Contributed by AV, 24-May-2020.)
(𝑁 ∈ (ℤ‘2) → 1 < 𝑁)
 
Theoremeluz2b2 8848 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 8849 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 8850 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 8851 1 is not in (ℤ‘2). (Contributed by Paul Chapman, 21-Nov-2012.)
¬ 1 ∈ (ℤ‘2)
 
Theoremelnn1uz2 8852 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 8853 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 8854* 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 8855 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))
 
Theoremublbneg 8856* 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 8841. (Contributed by Paul Chapman, 21-Mar-2011.)
(∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧𝐴}𝑥𝑦)
 
Theoremeqreznegel 8857* 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 8858* 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 8859* 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 8860 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 8861 Alternate proof of nn0ge2m1nn 8492: 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 8783, 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 8492. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ)
 
3.4.12  Rational numbers (as a subset of complex numbers)
 
Syntaxcq 8862 Extend class notation to include the class of rationals.
class
 
Definitiondf-q 8863 Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 8865 for the relation "is rational." (Contributed by NM, 8-Jan-2002.)
ℚ = ( / “ (ℤ × ℕ))
 
Theoremdivfnzn 8864 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 8865* 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 8866* 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 8867 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 8868 A rational number is a real number. (Contributed by NM, 14-Nov-2002.)
(𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
 
Theoremzq 8869 An integer is a rational number. (Contributed by NM, 9-Jan-2002.)
(𝐴 ∈ ℤ → 𝐴 ∈ ℚ)
 
Theoremzssq 8870 The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)
ℤ ⊆ ℚ
 
Theoremnn0ssq 8871 The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
0 ⊆ ℚ
 
Theoremnnssq 8872 The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
ℕ ⊆ ℚ
 
Theoremqssre 8873 The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)
ℚ ⊆ ℝ
 
Theoremqsscn 8874 The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
ℚ ⊆ ℂ
 
Theoremqex 8875 The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
ℚ ∈ V
 
Theoremnnq 8876 A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
(𝐴 ∈ ℕ → 𝐴 ∈ ℚ)
 
Theoremqcn 8877 A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)
(𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
 
Theoremqaddcl 8878 Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ)
 
Theoremqnegcl 8879 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 8880 Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 · 𝐵) ∈ ℚ)
 
Theoremqsubcl 8881 Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴𝐵) ∈ ℚ)
 
Theoremqapne 8882 Apartness is equivalent to not equal for rationals. (Contributed by Jim Kingdon, 20-Mar-2020.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 # 𝐵𝐴𝐵))
 
Theoremqltlen 8883 Rational 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 7874 which is a similar result for real numbers. (Contributed by Jim Kingdon, 11-Oct-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 < 𝐵 ↔ (𝐴𝐵𝐵𝐴)))
 
Theoremqlttri2 8884 Apartness is equivalent to not equal for rationals. (Contributed by Jim Kingdon, 9-Nov-2021.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴𝐵 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
 
Theoremqreccl 8885 Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℚ)
 
Theoremqdivcl 8886 Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℚ)
 
Theoremqrevaddcl 8887 Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.)
(𝐵 ∈ ℚ → ((𝐴 ∈ ℂ ∧ (𝐴 + 𝐵) ∈ ℚ) ↔ 𝐴 ∈ ℚ))
 
Theoremnnrecq 8888 The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)
(𝐴 ∈ ℕ → (1 / 𝐴) ∈ ℚ)
 
Theoremirradd 8889 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 8890 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))
 
3.4.13  Complex numbers as pairs of reals
 
Theoremcnref1o 8891* 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 7126), 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
 
3.5  Order sets
 
3.5.1  Positive reals (as a subset of complex numbers)
 
Syntaxcrp 8892 Extend class notation to include the class of positive reals.
class +
 
Definitiondf-rp 8893 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 8894 Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)
(𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
 
Theoremelrpii 8895 Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
𝐴 ∈ ℝ    &   0 < 𝐴       𝐴 ∈ ℝ+
 
Theorem1rp 8896 1 is a positive real. (Contributed by Jeff Hankins, 23-Nov-2008.)
1 ∈ ℝ+
 
Theorem2rp 8897 2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
2 ∈ ℝ+
 
Theoremrpre 8898 A positive real is a real. (Contributed by NM, 27-Oct-2007.)
(𝐴 ∈ ℝ+𝐴 ∈ ℝ)
 
Theoremrpxr 8899 A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)
(𝐴 ∈ ℝ+𝐴 ∈ ℝ*)
 
Theoremrpcn 8900 A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
(𝐴 ∈ ℝ+𝐴 ∈ ℂ)
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