P. 130 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 12901-13000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eluz4nn 12901	An integer greater than or equal to 4 is a positive integer. (Contributed by AV, 30-May-2023.)
⊢ (𝑋 ∈ (ℤ≥‘4) → 𝑋 ∈ ℕ)
Theorem	eluzge2nn0 12902	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)
Theorem	eluz2n0 12903	An integer greater than or equal to 2 is not 0. (Contributed by AV, 25-May-2020.)
⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ≠ 0)
Theorem	uzuzle23 12904	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))
Theorem	eluzge3nn 12905	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)
Theorem	uz3m2nn 12906	An integer greater than or equal to 3 decreased by 2 is a positive integer, analogous to uz2m1nn 12938. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑁 − 2) ∈ ℕ)
Theorem	1eluzge0 12907	1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
⊢ 1 ∈ (ℤ≥‘0)
Theorem	2eluzge0 12908	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)
Theorem	2eluzge1 12909	2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
⊢ 2 ∈ (ℤ≥‘1)
Theorem	uznnssnn 12910	The upper integers starting from a natural are a subset of the naturals. (Contributed by Scott Fenton, 29-Jun-2013.)
⊢ (𝑁 ∈ ℕ → (ℤ≥‘𝑁) ⊆ ℕ)
Theorem	raluz 12911*	Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
⊢ (𝑀 ∈ ℤ → (∀𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀 ≤ 𝑛 → 𝜑)))
Theorem	raluz2 12912*	Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
⊢ (∀𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀 ≤ 𝑛 → 𝜑)))
Theorem	rexuz 12913*	Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
⊢ (𝑀 ∈ ℤ → (∃𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ ∃𝑛 ∈ ℤ (𝑀 ≤ 𝑛 ∧ 𝜑)))
Theorem	rexuz2 12914*	Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
⊢ (∃𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ (𝑀 ∈ ℤ ∧ ∃𝑛 ∈ ℤ (𝑀 ≤ 𝑛 ∧ 𝜑)))
Theorem	2rexuz 12915*	Double existential quantification in an upper set of integers. (Contributed by NM, 3-Nov-2005.)
⊢ (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)𝜑 ↔ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑚 ≤ 𝑛 ∧ 𝜑))
Theorem	peano2uz 12916	Second Peano postulate for an upper set of integers. (Contributed by NM, 7-Sep-2005.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 1) ∈ (ℤ≥‘𝑀))
Theorem	peano2uzs 12917	Second Peano postulate for an upper set of integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝑁 ∈ 𝑍 → (𝑁 + 1) ∈ 𝑍)
Theorem	peano2uzr 12918	Reversed second Peano axiom for upper integers. (Contributed by NM, 2-Jan-2006.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑁 ∈ (ℤ≥‘𝑀))
Theorem	uzaddcl 12919	Addition closure law for an upper set of integers. (Contributed by NM, 4-Jun-2006.)
⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℕ0) → (𝑁 + 𝐾) ∈ (ℤ≥‘𝑀))
Theorem	nn0pzuz 12920	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 ∈ ℕ0 ∧ 𝑍 ∈ ℤ) → (𝑁 + 𝑍) ∈ (ℤ≥‘𝑍))
Theorem	uzind4 12921*	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 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑀 ∈ ℤ → 𝜓)    &   ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜒 → 𝜃))    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜏)
Theorem	uzind4ALT 12922*	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 12921 or uzind4ALT 12922 may be used; see comment for nnind 12261. (Contributed by NM, 7-Sep-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝑀 ∈ ℤ → 𝜓)    &   ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜒 → 𝜃))    &   ⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏))    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜏)
Theorem	uzind4s 12923*	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 ∈ (ℤ≥‘𝑀) → (𝜑 → [(𝑘 + 1) / 𝑘]𝜑))    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → [𝑁 / 𝑘]𝜑)
Theorem	uzind4s2 12924*	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 12923 when 𝑗 and 𝑘 must be distinct in [(𝑘 + 1) / 𝑗]𝜑. (Contributed by NM, 16-Nov-2005.)
⊢ (𝑀 ∈ ℤ → [𝑀 / 𝑗]𝜑)    &   ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → ([𝑘 / 𝑗]𝜑 → [(𝑘 + 1) / 𝑗]𝜑))    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → [𝑁 / 𝑗]𝜑)
Theorem	uzind4i 12925*	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 12921 assuming that 𝜓 holds unconditionally. Notice that 𝑁 ∈ (ℤ≥‘𝑀) implies that the lower bound 𝑀 is an integer (𝑀 ∈ ℤ, see eluzel2 12858). (Contributed by NM, 4-Sep-2005.) (Revised by AV, 13-Jul-2022.)
⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜒 → 𝜃))    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜏)
Theorem	uzwo 12926*	Well-ordering principle: any nonempty subset of an upper set of integers has a least element. (Contributed by NM, 8-Oct-2005.)
⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘)
Theorem	uzwo2 12927*	Well-ordering principle: any nonempty subset of an upper set of integers has a unique least element. (Contributed by NM, 8-Oct-2005.)
⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → ∃!𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘)
Theorem	nnwo 12928*	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 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
Theorem	nnwof 12929*	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.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    ⇒   ⊢ ((𝐴 ⊆ ℕ ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
Theorem	nnwos 12930*	Well-ordering principle: any nonempty set of positive integers has a least element (schema form). (Contributed by NM, 17-Aug-2001.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ ℕ 𝜑 → ∃𝑥 ∈ ℕ (𝜑 ∧ ∀𝑦 ∈ ℕ (𝜓 → 𝑥 ≤ 𝑦)))
Theorem	indstr 12931*	Strong Mathematical Induction for positive integers (inference schema). (Contributed by NM, 17-Aug-2001.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦 < 𝑥 → 𝜓) → 𝜑))    ⇒   ⊢ (𝑥 ∈ ℕ → 𝜑)
Theorem	eluznn0 12932	Membership in a nonnegative upper set of integers implies membership in ℕ0. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ ℕ0)
Theorem	eluznn 12933	Membership in a positive upper set of integers implies membership in ℕ. (Contributed by JJ, 1-Oct-2018.)
⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ ℕ)
Theorem	eluz2b1 12934	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))
Theorem	eluz2gt1 12935	An integer greater than or equal to 2 is greater than 1. (Contributed by AV, 24-May-2020.)
⊢ (𝑁 ∈ (ℤ≥‘2) → 1 < 𝑁)
Theorem	eluz2b2 12936	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))
Theorem	eluz2b3 12937	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))
Theorem	uz2m1nn 12938	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)
Theorem	1nuz2 12939	1 is not in (ℤ≥‘2). (Contributed by Paul Chapman, 21-Nov-2012.)
⊢ ¬ 1 ∈ (ℤ≥‘2)
Theorem	elnn1uz2 12940	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)))
Theorem	uz2mulcl 12941	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))
Theorem	indstr2 12942*	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 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜒    &   ⊢ (𝑥 ∈ (ℤ≥‘2) → (∀𝑦 ∈ ℕ (𝑦 < 𝑥 → 𝜓) → 𝜑))    ⇒   ⊢ (𝑥 ∈ ℕ → 𝜑)
Theorem	uzinfi 12943	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((ℤ≥‘𝑀), ℝ, < ) = 𝑀
Theorem	nninf 12944	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
Theorem	nn0inf 12945	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
Theorem	infssuzle 12946	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 ⊆ (ℤ≥‘𝑀) ∧ 𝐴 ∈ 𝑆) → inf(𝑆, ℝ, < ) ≤ 𝐴)
Theorem	infssuzcl 12947	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 ⊆ (ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → inf(𝑆, ℝ, < ) ∈ 𝑆)
Theorem	ublbneg 12948*	The image under negation of a bounded-above set of reals is bounded below. (Contributed by Paul Chapman, 21-Mar-2011.)
⊢ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦)
Theorem	eqreznegel 12949*	Two ways to express the image under negation of a set of integers. (Contributed by Paul Chapman, 21-Mar-2011.)
⊢ (𝐴 ⊆ ℤ → {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} = {𝑧 ∈ ℤ ∣ -𝑧 ∈ 𝐴})
Theorem	supminf 12950*	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 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) = -inf({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}, ℝ, < ))
Theorem	lbzbi 12951*	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 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
Theorem	zsupss 12952*	Any nonempty bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-sup 11217.) (Contributed by Mario Carneiro, 21-Apr-2015.)
⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
Theorem	suprzcl2 12953*	The supremum of a bounded-above set of integers is a member of the set. (This version of suprzcl 12673 avoids ax-pre-sup 11217.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Mario Carneiro, 24-Dec-2016.)
⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈ 𝐴)
Theorem	suprzub 12954*	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 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ∧ 𝐵 ∈ 𝐴) → 𝐵 ≤ sup(𝐴, ℝ, < ))
Theorem	uzsupss 12955*	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 = (ℤ≥‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ 𝑍 ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ 𝑍 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝑍 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
Theorem	nn01to3 12956	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 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))
Theorem	nn0ge2m1nnALT 12957	Alternate proof of nn0ge2m1nn 12572: 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 12859, 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 12572. (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
Theorem	uzwo3 12958*	Well-ordering principle: any nonempty subset of an upper set of integers has a unique least element. This generalization of uzwo2 12927 allows the lower bound 𝐵 to be any real number. See also nnwo 12928 and nnwos 12930. (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 ∣ 𝐵 ≤ 𝑧} ∧ 𝐴 ≠ ∅)) → ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
Theorem	zmin 12959*	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 (𝐴 ≤ 𝑥 ∧ ∀𝑦 ∈ ℤ (𝐴 ≤ 𝑦 → 𝑥 ≤ 𝑦)))
Theorem	zmax 12960*	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 (𝑥 ≤ 𝐴 ∧ ∀𝑦 ∈ ℤ (𝑦 ≤ 𝐴 → 𝑦 ≤ 𝑥)))
Theorem	zbtwnre 12961*	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 (𝐴 ≤ 𝑥 ∧ 𝑥 < (𝐴 + 1)))
Theorem	rebtwnz 12962*	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 (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))
5.4.13  Rational numbers (as a subset of complex numbers)
Syntax	cq 12963	Extend class notation to include the class of rationals.
class ℚ
Definition	df-q 12964	Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 12965 for the relation "is rational". (Contributed by NM, 8-Jan-2002.)
⊢ ℚ = ( / “ (ℤ × ℕ))
Theorem	elq 12965*	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))
Theorem	qmulz 12966*	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)
Theorem	znq 12967	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)
Theorem	qre 12968	A rational number is a real number. (Contributed by NM, 14-Nov-2002.)
⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
Theorem	zq 12969	An integer is a rational number. (Contributed by NM, 9-Jan-2002.) (Proof shortened by Steven Nguyen, 23-Mar-2023.)
⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ)
Theorem	qred 12970	A rational number is a real number. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℚ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	zssq 12971	The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)
⊢ ℤ ⊆ ℚ
Theorem	nn0ssq 12972	The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
⊢ ℕ0 ⊆ ℚ
Theorem	nnssq 12973	The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
⊢ ℕ ⊆ ℚ
Theorem	qssre 12974	The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)
⊢ ℚ ⊆ ℝ
Theorem	qsscn 12975	The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
⊢ ℚ ⊆ ℂ
Theorem	qex 12976	The set of rational numbers exists. See also qexALT 12979. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ ℚ ∈ V
Theorem	nnq 12977	A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℚ)
Theorem	qcn 12978	A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)
⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
Theorem	qexALT 12979	Alternate proof of qex 12976. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ℚ ∈ V
Theorem	qaddcl 12980	Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ)
Theorem	qnegcl 12981	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)
Theorem	qmulcl 12982	Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 · 𝐵) ∈ ℚ)
Theorem	qsubcl 12983	Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 − 𝐵) ∈ ℚ)
Theorem	qreccl 12984	Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℚ)
Theorem	qdivcl 12985	Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℚ)
Theorem	qrevaddcl 12986	Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.)
⊢ (𝐵 ∈ ℚ → ((𝐴 ∈ ℂ ∧ (𝐴 + 𝐵) ∈ ℚ) ↔ 𝐴 ∈ ℚ))
Theorem	nnrecq 12987	The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)
⊢ (𝐴 ∈ ℕ → (1 / 𝐴) ∈ ℚ)
Theorem	irradd 12988	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))
Theorem	irrmul 12989	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))
Theorem	elpq 12990*	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))
Theorem	elpqb 12991*	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))
5.4.14  Existence of the set of complex numbers
Theorem	rpnnen1lem2 12992*	Lemma for rpnnen1 12998. (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)
Theorem	rpnnen1lem1 12993*	Lemma for rpnnen1 12998. (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 → (𝐹‘𝑥) ∈ (ℚ ↑m ℕ))
Theorem	rpnnen1lem3 12994*	Lemma for rpnnen1 12998. (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 (𝐹‘𝑥)𝑛 ≤ 𝑥)
Theorem	rpnnen1lem4 12995*	Lemma for rpnnen1 12998. (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 (𝐹‘𝑥), ℝ, < ) ∈ ℝ)
Theorem	rpnnen1lem5 12996*	Lemma for rpnnen1 12998. (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 (𝐹‘𝑥), ℝ, < ) = 𝑥)
Theorem	rpnnen1lem6 12997*	Lemma for rpnnen1 12998. (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)
Theorem	rpnnen1 12998	One half of rpnnen 16204, where we show an injection from the real numbers to sequences of rational numbers. Specifically, we map a real number 𝑥 to the sequence (𝐹‘𝑥):ℕ⟶ℚ (see rpnnen1lem6 12997) such that ((𝐹‘𝑥)‘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 ℕ and ℚ existence hypotheses provide for use with either nnex 12249 and qex 12976, or nnexALT 12245 and qexALT 12979. 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)
Theorem	reexALT 12999	Alternate proof of reex 11230. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 23-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ℝ ∈ V
Theorem	cnref1o 13000*	There is a natural one-to-one mapping from (ℝ × ℝ) to ℂ, where we map ⟨𝑥, 𝑦⟩ to (𝑥 + (i · 𝑦)). In our construction of the complex numbers, this is in fact our definition of ℂ (see df-c 11145), 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