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Theorem List for Metamath Proof Explorer - 11501-11600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeluz2b2 11501 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 11502 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 11503 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 11504 1 is not in (ℤ‘2). (Contributed by Paul Chapman, 21-Nov-2012.)
¬ 1 ∈ (ℤ‘2)
 
Theoremelnn1uz2 11505 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 11506 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 11507* 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 11508 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 11509 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 11510 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 11511 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 11512 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)
 
TheoreminfmssuzleOLD 11513 The infimum of a subset of an upper set of integers is less than or equal to all members of the subset. Note that the " < " argument turns supremum into infimum (for which we do not currently have a separate notation). (Contributed by NM, 11-Oct-2005.) Obsolete version of infssuzle 11511 as of 5-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
((𝑆 ⊆ (ℤ𝑀) ∧ 𝐴𝑆) → sup(𝑆, ℝ, < ) ≤ 𝐴)
 
TheoreminfmssuzclOLD 11514 The infimum of a subset of an upper set of integers belongs to the subset. (Contributed by NM, 11-Oct-2005.) Obsolete version of infssuzcl 11512 as of 5-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
((𝑆 ⊆ (ℤ𝑀) ∧ 𝑆 ≠ ∅) → sup(𝑆, ℝ, < ) ∈ 𝑆)
 
Theoremublbneg 11515* 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 11516* 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 11517* 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 11518* 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 11519* Any nonempty bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-sup 9769.) (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 11520* The supremum of a bounded-above set of integers is a member of the set. (This version of suprzcl 11197 avoids ax-pre-sup 9769.) (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 11521* 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 11522* 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 11523 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 11524 Alternate proof of nn0ge2m1nn 11115: 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 11433, 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 11115. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ)
 
5.4.11  Well-ordering principle for bounded-below sets of integers
 
Theoremuzwo3 11525* Well-ordering principle: any nonempty subset of an upper set of integers has a unique least element. This generalization of uzwo2 11492 allows the lower bound 𝐵 to be any real number. See also nnwo 11493 and nnwos 11495. (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 11526* 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 11527* 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 11528* 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 11529* 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)))
 
5.4.12  Rational numbers (as a subset of complex numbers)
 
Syntaxcq 11530 Extend class notation to include the class of rationals.
class
 
Definitiondf-q 11531 Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 11532 for the relation "is rational." (Contributed by NM, 8-Jan-2002.)
ℚ = ( / “ (ℤ × ℕ))
 
Theoremelq 11532* 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 11533* 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 11534 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 11535 A rational number is a real number. (Contributed by NM, 14-Nov-2002.)
(𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
 
Theoremzq 11536 An integer is a rational number. (Contributed by NM, 9-Jan-2002.)
(𝐴 ∈ ℤ → 𝐴 ∈ ℚ)
 
Theoremzssq 11537 The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)
ℤ ⊆ ℚ
 
Theoremnn0ssq 11538 The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
0 ⊆ ℚ
 
Theoremnnssq 11539 The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
ℕ ⊆ ℚ
 
Theoremqssre 11540 The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)
ℚ ⊆ ℝ
 
Theoremqsscn 11541 The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
ℚ ⊆ ℂ
 
Theoremqex 11542 The set of rational numbers exists. See also qexALT 11545. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
ℚ ∈ V
 
Theoremnnq 11543 A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
(𝐴 ∈ ℕ → 𝐴 ∈ ℚ)
 
Theoremqcn 11544 A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)
(𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
 
TheoremqexALT 11545 Alternate proof of qex 11542. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
ℚ ∈ V
 
Theoremqaddcl 11546 Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ)
 
Theoremqnegcl 11547 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 11548 Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 · 𝐵) ∈ ℚ)
 
Theoremqsubcl 11549 Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴𝐵) ∈ ℚ)
 
Theoremqreccl 11550 Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℚ)
 
Theoremqdivcl 11551 Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℚ)
 
Theoremqrevaddcl 11552 Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.)
(𝐵 ∈ ℚ → ((𝐴 ∈ ℂ ∧ (𝐴 + 𝐵) ∈ ℚ) ↔ 𝐴 ∈ ℚ))
 
Theoremnnrecq 11553 The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)
(𝐴 ∈ ℕ → (1 / 𝐴) ∈ ℚ)
 
Theoremirradd 11554 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 11555 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))
 
5.4.13  Existence of the set of complex numbers
 
Theoremrpnnen1lem2 11556* Lemma for rpnnen1 11562. (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 11557* Lemma for rpnnen1 11562. (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 11558* Lemma for rpnnen1 11562. (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 11559* Lemma for rpnnen1 11562. (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 11560* Lemma for rpnnen1 11562. (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 11561* Lemma for rpnnen1 11562. (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 11562 One half of rpnnen 14664, 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 11561) 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 10781 and qex 11542, or nnexALT 10777 and qexALT 11545. 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)
 
Theoremrpnnen1lem1OLD 11563* Lemma for rpnnen1OLD 11567. (Contributed by Mario Carneiro, 12-May-2013.) Obsolete version of rpnnen1lem1 11557 as of 13-Aug-2021. (New usage is discouraged.) (Proof modification is discouraged.)
𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))       (𝑥 ∈ ℝ → (𝐹𝑥) ∈ (ℚ ↑𝑚 ℕ))
 
Theoremrpnnen1lem3OLD 11564* Lemma for rpnnen1OLD 11567. (Contributed by Mario Carneiro, 12-May-2013.) Obsolete version of rpnnen1lem3 11558 as of 13-Aug-2021. (New usage is discouraged.) (Proof modification is discouraged.)
𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))       (𝑥 ∈ ℝ → ∀𝑛 ∈ ran (𝐹𝑥)𝑛𝑥)
 
Theoremrpnnen1lem4OLD 11565* Lemma for rpnnen1OLD 11567. (Contributed by Mario Carneiro, 12-May-2013.) Obsolete version of rpnnen1lem4 11559 as of 13-Aug-2021. (New usage is discouraged.) (Proof modification is discouraged.)
𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))       (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) ∈ ℝ)
 
Theoremrpnnen1lem5OLD 11566* Lemma for rpnnen1OLD 11567. (Contributed by Mario Carneiro, 12-May-2013.) Obsolete version of rpnnen1lem5 11560 as of 13-Aug-2021. (New usage is discouraged.) (Proof modification is discouraged.)
𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))       (𝑥 ∈ ℝ → sup(ran (𝐹𝑥), ℝ, < ) = 𝑥)
 
Theoremrpnnen1OLD 11567* One half of rpnnen 14664, 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 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. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 16-Jun-2013.) Obsolete version of rpnnen1 11562 as of 13-Aug-2021. (New usage is discouraged.) (Proof modification is discouraged.)
𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))       ℝ ≼ (ℚ ↑𝑚 ℕ)
 
TheoremreexALT 11568 Alternate proof of reex 9782. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 23-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
ℝ ∈ V
 
Theoremcnref1o 11569* 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 9697), 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 11570 The set of complex numbers exists. This theorem shows that ax-cnex 9747 is redundant if we assume ax-rep 4597. See also ax-cnex 9747. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
ℂ ∈ V
 
Theoremxrex 11571 The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
* ∈ V
 
Theoremaddex 11572 The addition operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
+ ∈ V
 
Theoremmulex 11573 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 11574 Extend class notation to include the class of positive reals.
class +
 
Definitiondf-rp 11575 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 11576 Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)
(𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
 
Theoremelrpii 11577 Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
𝐴 ∈ ℝ    &   0 < 𝐴       𝐴 ∈ ℝ+
 
Theorem1rp 11578 1 is a positive real. (Contributed by Jeff Hankins, 23-Nov-2008.)
1 ∈ ℝ+
 
Theorem2rp 11579 2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
2 ∈ ℝ+
 
Theorem3rp 11580 3 is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
3 ∈ ℝ+
 
Theoremrpre 11581 A positive real is a real. (Contributed by NM, 27-Oct-2007.)
(𝐴 ∈ ℝ+𝐴 ∈ ℝ)
 
Theoremrpxr 11582 A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)
(𝐴 ∈ ℝ+𝐴 ∈ ℝ*)
 
Theoremrpcn 11583 A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
(𝐴 ∈ ℝ+𝐴 ∈ ℂ)
 
Theoremnnrp 11584 A positive integer is a positive real. (Contributed by NM, 28-Nov-2008.)
(𝐴 ∈ ℕ → 𝐴 ∈ ℝ+)
 
Theoremrpssre 11585 The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)
+ ⊆ ℝ
 
Theoremrpgt0 11586 A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
(𝐴 ∈ ℝ+ → 0 < 𝐴)
 
Theoremrpge0 11587 A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)
(𝐴 ∈ ℝ+ → 0 ≤ 𝐴)
 
Theoremrpregt0 11588 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 11589 A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)
(𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
 
Theoremrpne0 11590 A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)
(𝐴 ∈ ℝ+𝐴 ≠ 0)
 
Theoremrprene0 11591 A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)
(𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0))
 
Theoremrpcnne0 11592 A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)
(𝐴 ∈ ℝ+ → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
 
Theoremrpcndif0 11593 A positive real number is a complex number not being 0. (Contributed by AV, 29-May-2020.)
(𝐴 ∈ ℝ+𝐴 ∈ (ℂ ∖ {0}))
 
Theoremralrp 11594 Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)
(∀𝑥 ∈ ℝ+ 𝜑 ↔ ∀𝑥 ∈ ℝ (0 < 𝑥𝜑))
 
Theoremrexrp 11595 Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)
(∃𝑥 ∈ ℝ+ 𝜑 ↔ ∃𝑥 ∈ ℝ (0 < 𝑥𝜑))
 
Theoremrpaddcl 11596 Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → (𝐴 + 𝐵) ∈ ℝ+)
 
Theoremrpmulcl 11597 Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → (𝐴 · 𝐵) ∈ ℝ+)
 
Theoremrpdivcl 11598 Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ+)
 
Theoremrpreccl 11599 Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.)
(𝐴 ∈ ℝ+ → (1 / 𝐴) ∈ ℝ+)
 
Theoremrphalfcl 11600 Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)
(𝐴 ∈ ℝ+ → (𝐴 / 2) ∈ ℝ+)
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