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Theorem List for Metamath Proof Explorer - 12301-12400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremuzind4s2 12301* 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 12300 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 12302* 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 12298 assuming that 𝜓 holds unconditionally. Notice that 𝑁 ∈ (ℤ𝑀) implies that the lower bound 𝑀 is an integer (𝑀 ∈ ℤ, see eluzel2 12240). (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)
 
Theoremuzwo 12303* Well-ordering principle: any nonempty subset of an upper set of integers has a least element. (Contributed by NM, 8-Oct-2005.)
((𝑆 ⊆ (ℤ𝑀) ∧ 𝑆 ≠ ∅) → ∃𝑗𝑆𝑘𝑆 𝑗𝑘)
 
Theoremuzwo2 12304* 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 12305* 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 12306* 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 12307* 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 12308* 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 12309 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 12310 Membership in a positive upper set of integers implies membership in . (Contributed by JJ, 1-Oct-2018.)
((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ𝑁)) → 𝑀 ∈ ℕ)
 
Theoremeluz2b1 12311 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 12312 An integer greater than or equal to 2 is greater than 1. (Contributed by AV, 24-May-2020.)
(𝑁 ∈ (ℤ‘2) → 1 < 𝑁)
 
Theoremeluz2b2 12313 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 12314 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 12315 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 12316 1 is not in (ℤ‘2). (Contributed by Paul Chapman, 21-Nov-2012.)
¬ 1 ∈ (ℤ‘2)
 
Theoremelnn1uz2 12317 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 12318 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 12319* 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 12320 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 12321 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 12322 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 12323 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 12324 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 12325* 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 12326* 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 12327* 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 12328* 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 12329* Any nonempty bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-sup 10608.) (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 12330* The supremum of a bounded-above set of integers is a member of the set. (This version of suprzcl 12054 avoids ax-pre-sup 10608.) (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 12331* 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 12332* 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 12333 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 12334 Alternate proof of nn0ge2m1nn 11956: 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 12241, 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 11956. (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 12335* Well-ordering principle: any nonempty subset of an upper set of integers has a unique least element. This generalization of uzwo2 12304 allows the lower bound 𝐵 to be any real number. See also nnwo 12305 and nnwos 12307. (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 12336* 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 12337* 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 12338* 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 12339* 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 12340 Extend class notation to include the class of rationals.
class
 
Definitiondf-q 12341 Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 12342 for the relation "is rational." (Contributed by NM, 8-Jan-2002.)
ℚ = ( / “ (ℤ × ℕ))
 
Theoremelq 12342* 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 12343* 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 12344 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 12345 A rational number is a real number. (Contributed by NM, 14-Nov-2002.)
(𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
 
Theoremzq 12346 An integer is a rational number. (Contributed by NM, 9-Jan-2002.) (Proof shortened by Steven Nguyen, 23-Mar-2023.)
(𝐴 ∈ ℤ → 𝐴 ∈ ℚ)
 
Theoremzssq 12347 The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)
ℤ ⊆ ℚ
 
Theoremnn0ssq 12348 The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
0 ⊆ ℚ
 
Theoremnnssq 12349 The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
ℕ ⊆ ℚ
 
Theoremqssre 12350 The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)
ℚ ⊆ ℝ
 
Theoremqsscn 12351 The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
ℚ ⊆ ℂ
 
Theoremqex 12352 The set of rational numbers exists. See also qexALT 12355. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
ℚ ∈ V
 
Theoremnnq 12353 A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
(𝐴 ∈ ℕ → 𝐴 ∈ ℚ)
 
Theoremqcn 12354 A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)
(𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
 
TheoremqexALT 12355 Alternate proof of qex 12352. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
ℚ ∈ V
 
Theoremqaddcl 12356 Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ)
 
Theoremqnegcl 12357 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 12358 Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 · 𝐵) ∈ ℚ)
 
Theoremqsubcl 12359 Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴𝐵) ∈ ℚ)
 
Theoremqreccl 12360 Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℚ)
 
Theoremqdivcl 12361 Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℚ)
 
Theoremqrevaddcl 12362 Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.)
(𝐵 ∈ ℚ → ((𝐴 ∈ ℂ ∧ (𝐴 + 𝐵) ∈ ℚ) ↔ 𝐴 ∈ ℚ))
 
Theoremnnrecq 12363 The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)
(𝐴 ∈ ℕ → (1 / 𝐴) ∈ ℚ)
 
Theoremirradd 12364 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 12365 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))
 
Theoremelpq 12366* 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 12367* 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
 
Theoremrpnnen1lem2 12368* Lemma for rpnnen1 12374. (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 12369* Lemma for rpnnen1 12374. (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 12370* Lemma for rpnnen1 12374. (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 12371* Lemma for rpnnen1 12374. (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 12372* Lemma for rpnnen1 12374. (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 12373* Lemma for rpnnen1 12374. (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 12374 One half of rpnnen 15576, 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 12373) 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 11635 and qex 12352, or nnexALT 11631 and qexALT 12355. 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)
 
TheoremreexALT 12375 Alternate proof of reex 10621. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 23-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
ℝ ∈ V
 
Theoremcnref1o 12376* 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 10536), 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
 
TheoremcnexALT 12377 The set of complex numbers exists. This theorem shows that ax-cnex 10586 is redundant if we assume ax-rep 5157. See also ax-cnex 10586. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
ℂ ∈ V
 
Theoremxrex 12378 The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
* ∈ V
 
Theoremaddex 12379 The addition operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
+ ∈ V
 
Theoremmulex 12380 The multiplication operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
· ∈ V
 
5.5  Order sets
 
5.5.1  Positive reals (as a subset of complex numbers)
 
Syntaxcrp 12381 Extend class notation to include the class of positive reals.
class +
 
Definitiondf-rp 12382 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 12383 Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)
(𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
 
Theoremelrpii 12384 Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
𝐴 ∈ ℝ    &   0 < 𝐴       𝐴 ∈ ℝ+
 
Theorem1rp 12385 1 is a positive real. (Contributed by Jeff Hankins, 23-Nov-2008.)
1 ∈ ℝ+
 
Theorem2rp 12386 2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
2 ∈ ℝ+
 
Theorem3rp 12387 3 is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
3 ∈ ℝ+
 
Theoremrpssre 12388 The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)
+ ⊆ ℝ
 
Theoremrpre 12389 A positive real is a real. (Contributed by NM, 27-Oct-2007.) (Proof shortened by Steven Nguyen, 8-Oct-2022.)
(𝐴 ∈ ℝ+𝐴 ∈ ℝ)
 
Theoremrpxr 12390 A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)
(𝐴 ∈ ℝ+𝐴 ∈ ℝ*)
 
Theoremrpcn 12391 A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
(𝐴 ∈ ℝ+𝐴 ∈ ℂ)
 
Theoremnnrp 12392 A positive integer is a positive real. (Contributed by NM, 28-Nov-2008.)
(𝐴 ∈ ℕ → 𝐴 ∈ ℝ+)
 
Theoremrpgt0 12393 A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
(𝐴 ∈ ℝ+ → 0 < 𝐴)
 
Theoremrpge0 12394 A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)
(𝐴 ∈ ℝ+ → 0 ≤ 𝐴)
 
Theoremrpregt0 12395 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 12396 A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)
(𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
 
Theoremrpne0 12397 A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)
(𝐴 ∈ ℝ+𝐴 ≠ 0)
 
Theoremrprene0 12398 A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)
(𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0))
 
Theoremrpcnne0 12399 A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)
(𝐴 ∈ ℝ+ → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
 
Theoremrpcndif0 12400 A positive real number is a complex number not being 0. (Contributed by AV, 29-May-2020.)
(𝐴 ∈ ℝ+𝐴 ∈ (ℂ ∖ {0}))
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