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Theorem List for Intuitionistic Logic Explorer - 8701-8800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-uz 8701* Define a function whose value at 𝑗 is the semi-infinite set of contiguous integers starting at 𝑗, which we will also call the upper integers starting at 𝑗. Read "𝑀 " as "the set of integers greater than or equal to 𝑀." See uzval 8702 for its value, uzssz 8719 for its relationship to , nnuz 8735 and nn0uz 8734 for its relationships to and 0, and eluz1 8704 and eluz2 8706 for its membership relations. (Contributed by NM, 5-Sep-2005.)
= (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗𝑘})

Theoremuzval 8702* The value of the upper integers function. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
(𝑁 ∈ ℤ → (ℤ𝑁) = {𝑘 ∈ ℤ ∣ 𝑁𝑘})

Theoremuzf 8703 The domain and range of the upper integers function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 3-Nov-2013.)
:ℤ⟶𝒫 ℤ

Theoremeluz1 8704 Membership in the upper set of integers starting at 𝑀. (Contributed by NM, 5-Sep-2005.)
(𝑀 ∈ ℤ → (𝑁 ∈ (ℤ𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀𝑁)))

Theoremeluzel2 8705 Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
(𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)

Theoremeluz2 8706 Membership in an upper set of integers. We use the fact that a function's value (under our function value definition) is empty outside of its domain to show 𝑀 ∈ ℤ. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
(𝑁 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀𝑁))

Theoremeluz1i 8707 Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.)
𝑀 ∈ ℤ       (𝑁 ∈ (ℤ𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀𝑁))

Theoremeluzuzle 8708 An integer in an upper set of integers is an element of an upper set of integers with a smaller bound. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
((𝐵 ∈ ℤ ∧ 𝐵𝐴) → (𝐶 ∈ (ℤ𝐴) → 𝐶 ∈ (ℤ𝐵)))

Theoremeluzelz 8709 A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)

Theoremeluzelre 8710 A member of an upper set of integers is a real. (Contributed by Mario Carneiro, 31-Aug-2013.)
(𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℝ)

Theoremeluzelcn 8711 A member of an upper set of integers is a complex number. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℂ)

Theoremeluzle 8712 Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → 𝑀𝑁)

Theoremeluz 8713 Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ𝑀) ↔ 𝑀𝑁))

Theoremuzid 8714 Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.)
(𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))

Theoremuzn0 8715 The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
(𝑀 ∈ ran ℤ𝑀 ≠ ∅)

Theoremuztrn 8716 Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)
((𝑀 ∈ (ℤ𝐾) ∧ 𝐾 ∈ (ℤ𝑁)) → 𝑀 ∈ (ℤ𝑁))

Theoremuztrn2 8717 Transitive law for sets of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
𝑍 = (ℤ𝐾)       ((𝑁𝑍𝑀 ∈ (ℤ𝑁)) → 𝑀𝑍)

Theoremuzneg 8718 Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
(𝑁 ∈ (ℤ𝑀) → -𝑀 ∈ (ℤ‘-𝑁))

Theoremuzssz 8719 An upper set of integers is a subset of all integers. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
(ℤ𝑀) ⊆ ℤ

Theoremuzss 8720 Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → (ℤ𝑁) ⊆ (ℤ𝑀))

Theoremuztric 8721 Trichotomy of the ordering relation on integers, stated in terms of upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jun-2013.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ𝑀) ∨ 𝑀 ∈ (ℤ𝑁)))

Theoremuz11 8722 The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)
(𝑀 ∈ ℤ → ((ℤ𝑀) = (ℤ𝑁) ↔ 𝑀 = 𝑁))

Theoremeluzp1m1 8723 Membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → (𝑁 − 1) ∈ (ℤ𝑀))

Theoremeluzp1l 8724 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 8725 Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.)
(𝑁 ∈ (ℤ𝑀) → (𝑁 + 1) ∈ (ℤ‘(𝑀 + 1)))

Theoremeluzaddi 8726 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 8727 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 8728 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 8729 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 8730 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 8731 The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → (𝑁𝑀) ∈ ℕ0)

Theoremuzin 8732 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 8733 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 8734 Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
0 = (ℤ‘0)

Theoremnnuz 8735 Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
ℕ = (ℤ‘1)

Theoremelnnuz 8736 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 8737 A nonnegative integer expressed as a member an upper set of integers. (Contributed by NM, 6-Jun-2006.)
(𝑁 ∈ ℕ0𝑁 ∈ (ℤ‘0))

Theoremeluz2nn 8738 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 8739 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 8740 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 8741 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 8742 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 8743 1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
1 ∈ (ℤ‘0)

Theorem2eluzge0 8744 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 8745 2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
2 ∈ (ℤ‘1)

Theoremuznnssnn 8746 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 8747* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑀 ∈ ℤ → (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))

Theoremraluz2 8748* Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))

Theoremrexuz 8749* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(𝑀 ∈ ℤ → (∃𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∃𝑛 ∈ ℤ (𝑀𝑛𝜑)))

Theoremrexuz2 8750* Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
(∃𝑛 ∈ (ℤ𝑀)𝜑 ↔ (𝑀 ∈ ℤ ∧ ∃𝑛 ∈ ℤ (𝑀𝑛𝜑)))

Theorem2rexuz 8751* Double existential quantification in an upper set of integers. (Contributed by NM, 3-Nov-2005.)
(∃𝑚𝑛 ∈ (ℤ𝑚)𝜑 ↔ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑚𝑛𝜑))

Theorempeano2uz 8752 Second Peano postulate for an upper set of integers. (Contributed by NM, 7-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) → (𝑁 + 1) ∈ (ℤ𝑀))

Theorempeano2uzs 8753 Second Peano postulate for an upper set of integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
𝑍 = (ℤ𝑀)       (𝑁𝑍 → (𝑁 + 1) ∈ 𝑍)

Theorempeano2uzr 8754 Reversed second Peano axiom for upper integers. (Contributed by NM, 2-Jan-2006.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → 𝑁 ∈ (ℤ𝑀))

Theoremuzaddcl 8755 Addition closure law for an upper set of integers. (Contributed by NM, 4-Jun-2006.)
((𝑁 ∈ (ℤ𝑀) ∧ 𝐾 ∈ ℕ0) → (𝑁 + 𝐾) ∈ (ℤ𝑀))

Theoremnn0pzuz 8756 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 8757* 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 8758* 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 8757 or uzind4ALT 8758 may be used; see comment for nnind 8122. (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 8759* 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 8760* 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 8759 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 8761* 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 8762* 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 8763* 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 8764* 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 8779. (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 8765* 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 8764. (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 8766* 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 8767 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 8768 Membership in a positive upper set of integers implies membership in . (Contributed by JJ, 1-Oct-2018.)
((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ𝑁)) → 𝑀 ∈ ℕ)

Theoremeluz2b1 8769 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 8770 An integer greater than or equal to 2 is greater than 1. (Contributed by AV, 24-May-2020.)
(𝑁 ∈ (ℤ‘2) → 1 < 𝑁)

Theoremeluz2b2 8771 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 8772 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 8773 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 8774 1 is not in (ℤ‘2). (Contributed by Paul Chapman, 21-Nov-2012.)
¬ 1 ∈ (ℤ‘2)

Theoremelnn1uz2 8775 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 8776 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 8777* 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 8778 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 8779* 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 8764. (Contributed by Paul Chapman, 21-Mar-2011.)
(∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧𝐴}𝑥𝑦)

Theoremeqreznegel 8780* 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 8781* 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 8782* 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 8783 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 8784 Alternate proof of nn0ge2m1nn 8415: 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 8706, 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 8415. (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 8785 Extend class notation to include the class of rationals.
class

Definitiondf-q 8786 Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 8788 for the relation "is rational." (Contributed by NM, 8-Jan-2002.)
ℚ = ( / “ (ℤ × ℕ))

Theoremdivfnzn 8787 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 8788* 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 8789* 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 8790 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 8791 A rational number is a real number. (Contributed by NM, 14-Nov-2002.)
(𝐴 ∈ ℚ → 𝐴 ∈ ℝ)

Theoremzq 8792 An integer is a rational number. (Contributed by NM, 9-Jan-2002.)
(𝐴 ∈ ℤ → 𝐴 ∈ ℚ)

Theoremzssq 8793 The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)
ℤ ⊆ ℚ

Theoremnn0ssq 8794 The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
0 ⊆ ℚ

Theoremnnssq 8795 The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
ℕ ⊆ ℚ

Theoremqssre 8796 The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)
ℚ ⊆ ℝ

Theoremqsscn 8797 The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
ℚ ⊆ ℂ

Theoremqex 8798 The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
ℚ ∈ V

Theoremnnq 8799 A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
(𝐴 ∈ ℕ → 𝐴 ∈ ℚ)

Theoremqcn 8800 A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)
(𝐴 ∈ ℚ → 𝐴 ∈ ℂ)

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