P. 123 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 12201-12300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	negiso 12201	Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ 𝐹 = (𝑥 ∈ ℝ ↦ -𝑥)    ⇒   ⊢ (𝐹 Isom < , ◡ < (ℝ, ℝ) ∧ ◡𝐹 = 𝐹)
Theorem	dfinfre 12202*	The infimum of a set of reals 𝐴. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)
⊢ (𝐴 ⊆ ℝ → inf(𝐴, ℝ, < ) = ∪ {𝑥 ∈ ℝ ∣ (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))})
Theorem	infrecl 12203*	Closure of infimum of a nonempty bounded set of reals. (Contributed by NM, 8-Oct-2005.) (Revised by AV, 4-Sep-2020.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → inf(𝐴, ℝ, < ) ∈ ℝ)
Theorem	infrenegsup 12204*	The infimum of a set of reals 𝐴 is the negative of the supremum of the negatives of its elements. The antecedent ensures that 𝐴 is nonempty and has a lower bound. (Contributed by NM, 14-Jun-2005.) (Revised by AV, 4-Sep-2020.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → inf(𝐴, ℝ, < ) = -sup({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}, ℝ, < ))
Theorem	infregelb 12205*	Any lower bound of a nonempty set of real numbers is less than or equal to its infimum. (Contributed by Jeff Hankins, 1-Sep-2013.) (Revised by AV, 4-Sep-2020.) (Proof modification is discouraged.)
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) ∧ 𝐵 ∈ ℝ) → (𝐵 ≤ inf(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧))
Theorem	infrelb 12206*	If a nonempty set of real numbers has a lower bound, its infimum is less than or equal to any of its elements. (Contributed by Jeff Hankins, 15-Sep-2013.) (Revised by AV, 4-Sep-2020.)
⊢ ((𝐵 ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦 ∧ 𝐴 ∈ 𝐵) → inf(𝐵, ℝ, < ) ≤ 𝐴)
Theorem	infrefilb 12207	The infimum of a finite set of reals is less than or equal to any of its elements. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ ((𝐵 ⊆ ℝ ∧ 𝐵 ∈ Fin ∧ 𝐴 ∈ 𝐵) → inf(𝐵, ℝ, < ) ≤ 𝐴)
Theorem	supfirege 12208	The supremum of a finite set of real numbers is greater than or equal to all the real numbers of the set. (Contributed by AV, 1-Oct-2019.)
⊢ (𝜑 → 𝐵 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 = sup(𝐵, ℝ, < ))    ⇒   ⊢ (𝜑 → 𝐶 ≤ 𝑆)
5.3.9  Imaginary and complex number properties
Theorem	neg1cn 12209	-1 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
⊢ -1 ∈ ℂ
Theorem	neg1rr 12210	-1 is a real number. (Contributed by David A. Wheeler, 5-Dec-2018.)
⊢ -1 ∈ ℝ
Theorem	neg1ne0 12211	-1 is nonzero. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ -1 ≠ 0
Theorem	neg1lt0 12212	-1 is less than 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ -1 < 0
Theorem	negneg1e1 12213	--1 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ --1 = 1
Theorem	inelr 12214	The imaginary unit i is not a real number. (Contributed by NM, 6-May-1999.)
⊢ ¬ i ∈ ℝ
Theorem	rimul 12215	A real number times the imaginary unit is real only if the number is 0. (Contributed by NM, 28-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ (i · 𝐴) ∈ ℝ) → 𝐴 = 0)
Theorem	cru 12216	The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 + (i · 𝐵)) = (𝐶 + (i · 𝐷)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
Theorem	crne0 12217	The real representation of complex numbers is nonzero iff one of its terms is nonzero. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ↔ (𝐴 + (i · 𝐵)) ≠ 0))
Theorem	creur 12218*	The real part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ (𝐴 ∈ ℂ → ∃!𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
Theorem	creui 12219*	The imaginary part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ (𝐴 ∈ ℂ → ∃!𝑦 ∈ ℝ ∃𝑥 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
Theorem	cju 12220*	The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.)
⊢ (𝐴 ∈ ℂ → ∃!𝑥 ∈ ℂ ((𝐴 + 𝑥) ∈ ℝ ∧ (i · (𝐴 − 𝑥)) ∈ ℝ))
5.3.10  Function operation analogue theorems
Theorem	ofsubeq0 12221	Function analogue of subeq0 11490. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → ((𝐹 ∘f − 𝐺) = (𝐴 × {0}) ↔ 𝐹 = 𝐺))
Theorem	ofnegsub 12222	Function analogue of negsub 11512. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → (𝐹 ∘f + ((𝐴 × {-1}) ∘f · 𝐺)) = (𝐹 ∘f − 𝐺))
Theorem	ofsubge0 12223	Function analogue of subge0 11733. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ) → ((𝐴 × {0}) ∘r ≤ (𝐹 ∘f − 𝐺) ↔ 𝐺 ∘r ≤ 𝐹))
5.3.11  Indicator Functions According to Wikipedia (https://en.wikipedia.org/wiki/Indicator_function, "Indicator function", 11-Apr-2026): "In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if 𝐴 is a subset of some set 𝑋, then the indicator function of 𝐴 is the function 𝟭A defined by 𝟭A ( x ) = 1 if 𝑥 ∈ 𝐴, and 𝟭A ( x ) = 0 otherwise." See also definition in [Lang2] p. 3: "The characteristic function of a subset S' of S is the function 𝜒 such that 𝜒(x) = 1 if 𝑥 ∈ 𝑆' and 𝜒(x) = 0 if 𝑥 ∉ 𝑆'".
Syntax	cind 12224	Extend class notation with the indicator function generator.
class 𝟭
Definition	df-ind 12225*	Define the indicator function generator. It generates an indicator function ((𝟭‘𝑂)‘𝐴) for a given domain 𝑂 and a given subset 𝐴 of the domain, see indval 12227. In contrast to the definitions and notations in Wikipedia and [Lang2] p. 3, the domain and the subset are always mentioned explicitly. (Contributed by Thierry Arnoux, 20-Jan-2017.)
⊢ 𝟭 = (𝑜 ∈ V ↦ (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥 ∈ 𝑜 ↦ if(𝑥 ∈ 𝑎, 1, 0))))
Theorem	indv 12226*	Value of the indicator function generator with domain 𝑂. (Contributed by Thierry Arnoux, 23-Aug-2017.)
⊢ (𝑂 ∈ 𝑉 → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0))))
Theorem	indval 12227*	Value of the indicator function generator for a set 𝐴 and a domain 𝑂, i.e., an indicator function for a given domain 𝑂 and a given subset 𝐴 of the domain. (Contributed by Thierry Arnoux, 2-Feb-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0)))
Theorem	indval0 12228	The indicator function generator does not generate a (meaningful) indicator function for a class which is not a subset of the domain. (Contributed by AV, 11-Apr-2026.)
⊢ (¬ 𝐴 ⊆ 𝑂 → ((𝟭‘𝑂)‘𝐴) = ∅)
Theorem	indval2 12229	Alternate value of the indicator function generator. (Contributed by Thierry Arnoux, 2-Feb-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = ((𝐴 × {1}) ∪ ((𝑂 ∖ 𝐴) × {0})))
Theorem	indf 12230	An indicator function as a function with domain and codomain. (Contributed by Thierry Arnoux, 13-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴):𝑂⟶{0, 1})
Theorem	indfval 12231	Value of the indicator function. (Contributed by Thierry Arnoux, 13-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = if(𝑋 ∈ 𝐴, 1, 0))
Theorem	fvindre 12232	The range of the indicator function is a subset of ℝ. (Contributed by AV, 10-Apr-2026.)
⊢ (((𝑂 ∈ Fin ∧ 𝐴 ⊆ 𝑂) ∧ 𝑋 ∈ 𝑂) → (((𝟭‘𝑂)‘𝐴)‘𝑋) ∈ ℝ)
Theorem	ind1 12233	Value of the indicator function where it is 1. (Contributed by Thierry Arnoux, 14-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝐴) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = 1)
Theorem	ind0 12234	Value of the indicator function where it is 0. (Contributed by Thierry Arnoux, 14-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ (𝑂 ∖ 𝐴)) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = 0)
Theorem	ind1a 12235	Value of the indicator function where it is 1. (Contributed by Thierry Arnoux, 22-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂) → ((((𝟭‘𝑂)‘𝐴)‘𝑋) = 1 ↔ 𝑋 ∈ 𝐴))
Theorem	indconst0 12236	Indicator of the empty set. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ (𝑂 ∈ 𝑉 → ((𝟭‘𝑂)‘∅) = (𝑂 × {0}))
Theorem	indconst1 12237	Indicator of the whole set. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ (𝑂 ∈ 𝑉 → ((𝟭‘𝑂)‘𝑂) = (𝑂 × {1}))
Theorem	indpi1 12238	Preimage of the singleton {1} by the indicator function. See i1f1lem 25859. (Contributed by Thierry Arnoux, 21-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → (◡((𝟭‘𝑂)‘𝐴) “ {1}) = 𝐴)
5.4  Integer sets
5.4.1  Positive integers (as a subset of complex numbers)
Syntax	cn 12239	Extend class notation to include the class of positive integers.
class ℕ
Definition	df-nn 12240	Define the set of positive integers. Some authors, especially in analysis books, call these the natural numbers, whereas other authors choose to include 0 in their definition of natural numbers. Note that ℕ is a subset of complex numbers (nnsscn 12244), in contrast to the more elementary ordinal natural numbers ω, df-om 7861). See nnind 12257 for the principle of mathematical induction. See df-n0 12511 for the set of nonnegative integers ℕ0. See dfn2 12523 for ℕ defined in terms of ℕ0. This is a technical definition that helps us avoid the Axiom of Infinity ax-inf2 9608 in certain proofs. For a more conventional and intuitive definition ("the smallest set of reals containing 1 as well as the successor of every member") see dfnn3 12253 (or its slight variant dfnn2 12252). (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 3-May-2014.)
⊢ ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “ ω)
Theorem	nnexALT 12241	Alternate proof of nnex 12245, more direct, that makes use of ax-rep 5237. (Contributed by Mario Carneiro, 3-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ℕ ∈ V
Theorem	peano5nni 12242*	Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ ((1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 + 1) ∈ 𝐴) → ℕ ⊆ 𝐴)
Theorem	nnssre 12243	The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
⊢ ℕ ⊆ ℝ
Theorem	nnsscn 12244	The positive integers are a subset of the complex numbers. Remark: this could also be proven from nnssre 12243 and ax-resscn 11163 at the cost of using more axioms. (Contributed by NM, 2-Aug-2004.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ ℕ ⊆ ℂ
Theorem	nnex 12245	The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ ℕ ∈ V
Theorem	nnre 12246	A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
Theorem	nncn 12247	A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
Theorem	nnrei 12248	A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
⊢ 𝐴 ∈ ℕ    ⇒   ⊢ 𝐴 ∈ ℝ
Theorem	nncni 12249	A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
⊢ 𝐴 ∈ ℕ    ⇒   ⊢ 𝐴 ∈ ℂ
Theorem	1nn 12250	Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ 1 ∈ ℕ
Theorem	peano2nn 12251	Peano postulate: a successor of a positive integer is a positive integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ (𝐴 ∈ ℕ → (𝐴 + 1) ∈ ℕ)
Theorem	dfnn2 12252*	Alternate definition of the set of positive integers. This was our original definition, before the current df-nn 12240 replaced it. This definition requires the axiom of infinity to ensure it has the properties we expect. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.)
⊢ ℕ = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}
Theorem	dfnn3 12253*	Alternate definition of the set of positive integers. Definition of positive integers in [Apostol] p. 22. (Contributed by NM, 3-Jul-2005.)
⊢ ℕ = ∩ {𝑥 ∣ (𝑥 ⊆ ℝ ∧ 1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}
Theorem	nnred 12254	A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	nncnd 12255	A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℂ)
Theorem	peano2nnd 12256	Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ)
5.4.2  Principle of mathematical induction
Theorem	nnind 12257*	Principle of Mathematical Induction (inference schema). The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. See nnaddcl 12262 for an example of its use. See nn0ind 12697 for induction on nonnegative integers and uzind 12694, uzind4 12936 for induction on an arbitrary upper set of integers. See indstr 12946 for strong induction. See also nnindALT 12258. This is an alternative for Metamath 100 proof #74. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃))    ⇒   ⊢ (𝐴 ∈ ℕ → 𝜏)
Theorem	nnindALT 12258*	Principle of Mathematical Induction (inference schema). The last four hypotheses give us the substitution instances we need; the first two are the induction step and the basis. This ALT version of nnind 12257 has a different hypothesis order. It may be easier to use with the Metamath program Proof Assistant, because "MM-PA> ASSIGN LAST" will be applied to the substitution instances first. We may eventually use this one as the official version. You may use either version. After the proof is complete, the ALT version can be changed to the non-ALT version with "MM-PA> MINIMIZE_WITH nnind / MAYGROW";. (Contributed by NM, 7-Dec-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝑦 ∈ ℕ → (𝜒 → 𝜃))    &   ⊢ 𝜓    &   ⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    ⇒   ⊢ (𝐴 ∈ ℕ → 𝜏)
Theorem	nnindd 12259*	Principle of Mathematical Induction (inference schema) on integers, a deduction version. (Contributed by Thierry Arnoux, 19-Jul-2020.)
⊢ (𝑥 = 1 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜓 ↔ 𝜏))    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂))    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ) ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → 𝜂)
Theorem	nn1m1nn 12260	Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ))
Theorem	nn1suc 12261*	If a statement holds for 1 and also holds for a successor, it holds for all positive integers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 16-May-2014.)
⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜃))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ ℕ → 𝜒)    ⇒   ⊢ (𝐴 ∈ ℕ → 𝜃)
Theorem	nnaddcl 12262	Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) ∈ ℕ)
Theorem	nnmulcl 12263	Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.) Remove dependency on ax-mulcom 11170 and ax-mulass 11172. (Revised by Steven Nguyen, 24-Sep-2022.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ)
Theorem	nnmulcli 12264	Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ    ⇒   ⊢ (𝐴 · 𝐵) ∈ ℕ
Theorem	nnadd1com 12265	Addition with 1 is commutative for natural numbers. (Contributed by Steven Nguyen, 9-Dec-2022.)
⊢ (𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴))
Theorem	nnaddcom 12266	Addition is commutative for natural numbers. Uses fewer axioms than addcom 11402. (Contributed by Steven Nguyen, 9-Dec-2022.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Theorem	nnaddcomli 12267	Version of addcomli 11408 for natural numbers. (Contributed by Steven Nguyen, 1-Aug-2023.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ    &   ⊢ (𝐴 + 𝐵) = 𝐶    ⇒   ⊢ (𝐵 + 𝐴) = 𝐶
Theorem	nnmtmip 12268	"Minus times minus is plus, The reason for this we need not discuss." (W. H. Auden, as quoted in M. Guillen "Bridges to Infinity", p. 64, see also Metamath Book, section 1.1.1, p. 5) This statement, formalized to "The product of two negative integers is a positive integer", is proved by the following theorem, therefore it actually need not be discussed anymore. "The reason for this" is that (-𝐴 · -𝐵) = (𝐴 · 𝐵) for all complex numbers 𝐴 and 𝐵 because of mul2neg 11659, 𝐴 and 𝐵 are complex numbers because of nncn 12247, and (𝐴 · 𝐵) ∈ ℕ because of nnmulcl 12263. This also holds for positive reals, see rpmtmip 13048. Note that the opposites -𝐴 and -𝐵 of the positive integers 𝐴 and 𝐵 are negative integers. (Contributed by AV, 23-Dec-2022.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (-𝐴 · -𝐵) ∈ ℕ)
Theorem	nn2ge 12269*	There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ ℕ (𝐴 ≤ 𝑥 ∧ 𝐵 ≤ 𝑥))
Theorem	nnge1 12270	A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴)
Theorem	nngt1ne1 12271	A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.)
⊢ (𝐴 ∈ ℕ → (1 < 𝐴 ↔ 𝐴 ≠ 1))
Theorem	nnle1eq1 12272	A positive integer is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.)
⊢ (𝐴 ∈ ℕ → (𝐴 ≤ 1 ↔ 𝐴 = 1))
Theorem	nngt0 12273	A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
⊢ (𝐴 ∈ ℕ → 0 < 𝐴)
Theorem	nnnlt1 12274	A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (𝐴 ∈ ℕ → ¬ 𝐴 < 1)
Theorem	nnnle0 12275	A positive integer is not less than or equal to zero. (Contributed by AV, 13-May-2020.)
⊢ (𝐴 ∈ ℕ → ¬ 𝐴 ≤ 0)
Theorem	nnne0 12276	A positive integer is nonzero. See nnne0ALT 12280 for a shorter proof using ax-pre-mulgt0 11183. This proof avoids 0lt1 11742, and thus ax-pre-mulgt0 11183, by splitting ax-1ne0 11175 into the two separate cases 0 < 1 and 1 < 0. (Contributed by NM, 27-Sep-1999.) Remove dependency on ax-pre-mulgt0 11183. (Revised by Steven Nguyen, 30-Jan-2023.)
⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0)
Theorem	nnneneg 12277	No positive integer is equal to its negation. (Contributed by AV, 20-Jun-2023.)
⊢ (𝐴 ∈ ℕ → 𝐴 ≠ -𝐴)
Theorem	0nnn 12278	Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.) Remove dependency on ax-pre-mulgt0 11183. (Revised by Steven Nguyen, 30-Jan-2023.)
⊢ ¬ 0 ∈ ℕ
Theorem	0nnnALT 12279	Alternate proof of 0nnn 12278, which requires ax-pre-mulgt0 11183 but is not based on nnne0 12276 (and which can therefore be used in nnne0ALT 12280). (Contributed by NM, 25-Aug-1999.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ¬ 0 ∈ ℕ
Theorem	nnne0ALT 12280	Alternate version of nnne0 12276. A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0)
Theorem	nngt0i 12281	A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
⊢ 𝐴 ∈ ℕ    ⇒   ⊢ 0 < 𝐴
Theorem	nnne0i 12282	A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
⊢ 𝐴 ∈ ℕ    ⇒   ⊢ 𝐴 ≠ 0
Theorem	nndivre 12283	The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ) → (𝐴 / 𝑁) ∈ ℝ)
Theorem	nnrecre 12284	The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
⊢ (𝑁 ∈ ℕ → (1 / 𝑁) ∈ ℝ)
Theorem	nnrecgt0 12285	The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
⊢ (𝐴 ∈ ℕ → 0 < (1 / 𝐴))
Theorem	nnsub 12286	Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 ↔ (𝐵 − 𝐴) ∈ ℕ))
Theorem	nnsubi 12287	Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 ∈ ℕ    ⇒   ⊢ (𝐴 < 𝐵 ↔ (𝐵 − 𝐴) ∈ ℕ)
Theorem	nndiv 12288*	Two ways to express "𝐴 divides 𝐵 " for positive integers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (∃𝑥 ∈ ℕ (𝐴 · 𝑥) = 𝐵 ↔ (𝐵 / 𝐴) ∈ ℕ))
Theorem	nndivtr 12289	Transitive property of divisibility: if 𝐴 divides 𝐵 and 𝐵 divides 𝐶, then 𝐴 divides 𝐶. Typically, 𝐶 would be an integer, although the theorem holds for complex 𝐶. (Contributed by NM, 3-May-2005.)
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℂ) ∧ ((𝐵 / 𝐴) ∈ ℕ ∧ (𝐶 / 𝐵) ∈ ℕ)) → (𝐶 / 𝐴) ∈ ℕ)
Theorem	nnge1d 12290	A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 1 ≤ 𝐴)
Theorem	nngt0d 12291	A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 0 < 𝐴)
Theorem	nnne0d 12292	A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 0)
Theorem	nnrecred 12293	The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
Theorem	nnaddcld 12294	Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℕ)
Theorem	nnmulcld 12295	Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ)
Theorem	nndivred 12296	A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ)
Theorem	1t1e1ALT 12297	Alternate proof of 1t1e1 12408 using a different set of axioms (add ax-mulrcl 11169, ax-i2m1 11174, ax-1ne0 11175, ax-rrecex 11178 and remove ax-resscn 11163, ax-mulcom 11170, ax-mulass 11172, ax-distr 11173). (Contributed by Steven Nguyen, 20-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (1 · 1) = 1
Theorem	nnadddir 12298	Right-distributivity for natural numbers without ax-mulcom 11170. (Contributed by SN, 5-Feb-2024.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))
Theorem	nnmul1com 12299	Multiplication with 1 is commutative for natural numbers, without ax-mulcom 11170. Since (𝐴 · 1) is 𝐴 by ax-1rid 11176, this is equivalent to remullid 43223 for natural numbers, but using fewer axioms (avoiding ax-resscn 11163, ax-addass 11171, ax-mulass 11172, ax-rnegex 11177, ax-pre-lttri 11180, ax-pre-lttrn 11181, ax-pre-ltadd 11182). (Contributed by SN, 5-Feb-2024.)
⊢ (𝐴 ∈ ℕ → (1 · 𝐴) = (𝐴 · 1))
Theorem	nnmulcom 12300	Multiplication is commutative for natural numbers. (Contributed by SN, 5-Feb-2024.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))