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Theorem List for Metamath Proof Explorer - 11801-11900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremuz2m1nn 11801 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 11802 1 is not in (ℤ‘2). (Contributed by Paul Chapman, 21-Nov-2012.)
¬ 1 ∈ (ℤ‘2)
 
Theoremelnn1uz2 11803 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 11804 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 11805* 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 11806 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 11807 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 11808 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 11809 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 11810 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 11811* 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 11812* 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 11813* 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 11814* 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 11815* Any nonempty bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-sup 10052.) (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 11816* The supremum of a bounded-above set of integers is a member of the set. (This version of suprzcl 11495 avoids ax-pre-sup 10052.) (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 11817* 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 11818* 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 11819 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 11820 Alternate proof of nn0ge2m1nn 11398: 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 11731, 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 11398. (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 11821* Well-ordering principle: any nonempty subset of an upper set of integers has a unique least element. This generalization of uzwo2 11790 allows the lower bound 𝐵 to be any real number. See also nnwo 11791 and nnwos 11793. (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 11822* 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 11823* 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 11824* 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 11825* 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 11826 Extend class notation to include the class of rationals.
class
 
Definitiondf-q 11827 Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 11828 for the relation "is rational." (Contributed by NM, 8-Jan-2002.)
ℚ = ( / “ (ℤ × ℕ))
 
Theoremelq 11828* 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 11829* 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 11830 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 11831 A rational number is a real number. (Contributed by NM, 14-Nov-2002.)
(𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
 
Theoremzq 11832 An integer is a rational number. (Contributed by NM, 9-Jan-2002.)
(𝐴 ∈ ℤ → 𝐴 ∈ ℚ)
 
Theoremzssq 11833 The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)
ℤ ⊆ ℚ
 
Theoremnn0ssq 11834 The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
0 ⊆ ℚ
 
Theoremnnssq 11835 The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
ℕ ⊆ ℚ
 
Theoremqssre 11836 The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)
ℚ ⊆ ℝ
 
Theoremqsscn 11837 The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
ℚ ⊆ ℂ
 
Theoremqex 11838 The set of rational numbers exists. See also qexALT 11841. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
ℚ ∈ V
 
Theoremnnq 11839 A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
(𝐴 ∈ ℕ → 𝐴 ∈ ℚ)
 
Theoremqcn 11840 A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)
(𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
 
TheoremqexALT 11841 Alternate proof of qex 11838. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
ℚ ∈ V
 
Theoremqaddcl 11842 Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ)
 
Theoremqnegcl 11843 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 11844 Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 · 𝐵) ∈ ℚ)
 
Theoremqsubcl 11845 Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴𝐵) ∈ ℚ)
 
Theoremqreccl 11846 Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℚ)
 
Theoremqdivcl 11847 Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℚ)
 
Theoremqrevaddcl 11848 Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.)
(𝐵 ∈ ℚ → ((𝐴 ∈ ℂ ∧ (𝐴 + 𝐵) ∈ ℚ) ↔ 𝐴 ∈ ℚ))
 
Theoremnnrecq 11849 The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)
(𝐴 ∈ ℕ → (1 / 𝐴) ∈ ℚ)
 
Theoremirradd 11850 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 11851 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))
 
5.4.14  Existence of the set of complex numbers
 
Theoremrpnnen1lem2 11852* Lemma for rpnnen1 11858. (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 11853* Lemma for rpnnen1 11858. (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 11854* Lemma for rpnnen1 11858. (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 11855* Lemma for rpnnen1 11858. (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 11856* Lemma for rpnnen1 11858. (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 11857* Lemma for rpnnen1 11858. (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 11858 One half of rpnnen 15000, 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 11857) 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 11064 and qex 11838, or nnexALT 11060 and qexALT 11841. 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 11859* Lemma for rpnnen1OLD 11863. (Contributed by Mario Carneiro, 12-May-2013.) Obsolete version of rpnnen1lem1 11853 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 11860* Lemma for rpnnen1OLD 11863. (Contributed by Mario Carneiro, 12-May-2013.) Obsolete version of rpnnen1lem3 11854 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 11861* Lemma for rpnnen1OLD 11863. (Contributed by Mario Carneiro, 12-May-2013.) Obsolete version of rpnnen1lem4 11855 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 11862* Lemma for rpnnen1OLD 11863. (Contributed by Mario Carneiro, 12-May-2013.) Obsolete version of rpnnen1lem5 11856 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 11863* One half of rpnnen 15000, 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 11858 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 11864 Alternate proof of reex 10065. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 23-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
ℝ ∈ V
 
Theoremcnref1o 11865* 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 9980), 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 11866 The set of complex numbers exists. This theorem shows that ax-cnex 10030 is redundant if we assume ax-rep 4804. See also ax-cnex 10030. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
ℂ ∈ V
 
Theoremxrex 11867 The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
* ∈ V
 
Theoremaddex 11868 The addition operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
+ ∈ V
 
Theoremmulex 11869 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 11870 Extend class notation to include the class of positive reals.
class +
 
Definitiondf-rp 11871 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 11872 Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)
(𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
 
Theoremelrpii 11873 Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
𝐴 ∈ ℝ    &   0 < 𝐴       𝐴 ∈ ℝ+
 
Theorem1rp 11874 1 is a positive real. (Contributed by Jeff Hankins, 23-Nov-2008.)
1 ∈ ℝ+
 
Theorem2rp 11875 2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
2 ∈ ℝ+
 
Theorem3rp 11876 3 is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
3 ∈ ℝ+
 
Theoremrpre 11877 A positive real is a real. (Contributed by NM, 27-Oct-2007.)
(𝐴 ∈ ℝ+𝐴 ∈ ℝ)
 
Theoremrpxr 11878 A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)
(𝐴 ∈ ℝ+𝐴 ∈ ℝ*)
 
Theoremrpcn 11879 A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
(𝐴 ∈ ℝ+𝐴 ∈ ℂ)
 
Theoremnnrp 11880 A positive integer is a positive real. (Contributed by NM, 28-Nov-2008.)
(𝐴 ∈ ℕ → 𝐴 ∈ ℝ+)
 
Theoremrpssre 11881 The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)
+ ⊆ ℝ
 
Theoremrpgt0 11882 A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
(𝐴 ∈ ℝ+ → 0 < 𝐴)
 
Theoremrpge0 11883 A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)
(𝐴 ∈ ℝ+ → 0 ≤ 𝐴)
 
Theoremrpregt0 11884 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 11885 A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)
(𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
 
Theoremrpne0 11886 A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)
(𝐴 ∈ ℝ+𝐴 ≠ 0)
 
Theoremrprene0 11887 A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)
(𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0))
 
Theoremrpcnne0 11888 A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)
(𝐴 ∈ ℝ+ → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
 
Theoremrpcndif0 11889 A positive real number is a complex number not being 0. (Contributed by AV, 29-May-2020.)
(𝐴 ∈ ℝ+𝐴 ∈ (ℂ ∖ {0}))
 
Theoremralrp 11890 Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)
(∀𝑥 ∈ ℝ+ 𝜑 ↔ ∀𝑥 ∈ ℝ (0 < 𝑥𝜑))
 
Theoremrexrp 11891 Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)
(∃𝑥 ∈ ℝ+ 𝜑 ↔ ∃𝑥 ∈ ℝ (0 < 𝑥𝜑))
 
Theoremrpaddcl 11892 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 11893 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 11894 Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ+)
 
Theoremrpreccl 11895 Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.)
(𝐴 ∈ ℝ+ → (1 / 𝐴) ∈ ℝ+)
 
Theoremrphalfcl 11896 Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)
(𝐴 ∈ ℝ+ → (𝐴 / 2) ∈ ℝ+)
 
Theoremrpgecl 11897 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ ∧ 𝐴𝐵) → 𝐵 ∈ ℝ+)
 
Theoremrphalflt 11898 Half of a positive real is less than the original number. (Contributed by Mario Carneiro, 21-May-2014.)
(𝐴 ∈ ℝ+ → (𝐴 / 2) < 𝐴)
 
Theoremrerpdivcl 11899 Closure law for division of a real by a positive real. (Contributed by NM, 10-Nov-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ)
 
Theoremge0p1rp 11900 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 5-Oct-2015.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 + 1) ∈ ℝ+)
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