P. 214 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21301-21400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cnfldtsetOLD 21301	Obsolete version of cnfldtset 21288 as of 27-Apr-2025. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (MetOpen‘(abs ∘ − )) = (TopSet‘ℂfld)
Theorem	cnfldleOLD 21302	Obsolete version of cnfldle 21289 as of 27-Apr-2025. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ≤ = (le‘ℂfld)
Theorem	cnflddsOLD 21303	Obsolete version of cnfldds 21290 as of 27-Apr-2025. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (abs ∘ − ) = (dist‘ℂfld)
Theorem	cnfldunifOLD 21304	Obsolete version of cnfldunif 21291 as of 27-Apr-2025. (Contributed by Thierry Arnoux, 17-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (metUnif‘(abs ∘ − )) = (UnifSet‘ℂfld)
Theorem	cnfldfunOLD 21305	Obsolete version of cnfldfun 21292 as of 27-Apr-2025. (Contributed by AV, 18-Nov-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Fun ℂfld
Theorem	cnfldfunALTOLD 21306	Obsolete version of cnfldfunALT 21293 as of 27-Apr-2025. (Contributed by AV, 14-Nov-2021.) (Proof shortened by AV, 11-Nov-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Fun ℂfld
Theorem	cnfldfunALTOLDOLD 21307	Obsolete proof of cnfldfunALTOLD 21306 as of 10-Nov-2024. The field of complex numbers is a function. (Contributed by AV, 14-Nov-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Fun ℂfld
Theorem	xrsstr 21308	The extended real structure is a structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ℝ*𝑠 Struct ⟨1, ;12⟩
Theorem	xrsex 21309	The extended real structure is a set. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ℝ*𝑠 ∈ V
Theorem	xrsbas 21310	The base set of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ℝ* = (Base‘ℝ*𝑠)
Theorem	xrsadd 21311	The addition operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ +𝑒 = (+g‘ℝ*𝑠)
Theorem	xrsmul 21312	The multiplication operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ·e = (.r‘ℝ*𝑠)
Theorem	xrstset 21313	The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ (ordTop‘ ≤ ) = (TopSet‘ℝ*𝑠)
Theorem	xrsle 21314	The ordering of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ≤ = (le‘ℝ*𝑠)
Theorem	cncrng 21315	The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.) Avoid ax-mulf 11213. (Revised by GG, 31-Mar-2025.)
⊢ ℂfld ∈ CRing
Theorem	cncrngOLD 21316	Obsolete version of cncrng 21315 as of 30-Apr-2025. (Contributed by Mario Carneiro, 8-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ℂfld ∈ CRing
Theorem	cnring 21317	The complex numbers form a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ ℂfld ∈ Ring
Theorem	xrsmcmn 21318	The "multiplicative group" of the extended reals is a commutative monoid (even though the "additive group" is not a semigroup, see xrsmgmdifsgrp 21335.) (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ (mulGrp‘ℝ*𝑠) ∈ CMnd
Theorem	cnfld0 21319	Zero is the zero element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 0 = (0g‘ℂfld)
Theorem	cnfld1 21320	One is the unity element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) Avoid ax-mulf 11213. (Revised by GG, 31-Mar-2025.)
⊢ 1 = (1r‘ℂfld)
Theorem	cnfld1OLD 21321	Obsolete version of cnfld1 21320 as of 30-Apr-2025. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 1 = (1r‘ℂfld)
Theorem	cnfldneg 21322	The additive inverse in the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ (𝑋 ∈ ℂ → ((invg‘ℂfld)‘𝑋) = -𝑋)
Theorem	cnfldplusf 21323	The functionalized addition operation of the field of complex numbers. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ + = (+𝑓‘ℂfld)
Theorem	cnfldsub 21324	The subtraction operator in the field of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ − = (-g‘ℂfld)
Theorem	cndrng 21325	The complex numbers form a division ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) Avoid ax-mulf 11213. (Revised by GG, 30-Apr-2025.)
⊢ ℂfld ∈ DivRing
Theorem	cndrngOLD 21326	Obsolete version of cndrng 21325 as of 30-Apr-2025. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ℂfld ∈ DivRing
Theorem	cnflddiv 21327	The division operation in the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) Avoid ax-mulf 11213. (Revised by GG, 30-Apr-2025.)
⊢ / = (/r‘ℂfld)
Theorem	cnflddivOLD 21328	Obsolete version of cnflddiv 21327 as of 30-Apr-2025. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ / = (/r‘ℂfld)
Theorem	cnfldinv 21329	The multiplicative inverse in the field of complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ ((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) → ((invr‘ℂfld)‘𝑋) = (1 / 𝑋))
Theorem	cnfldmulg 21330	The group multiple function in the field of complex numbers. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℂ) → (𝐴(.g‘ℂfld)𝐵) = (𝐴 · 𝐵))
Theorem	cnfldexp 21331	The exponentiation operator in the field of complex numbers (for nonnegative exponents). (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ0) → (𝐵(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝐵))
Theorem	cnsrng 21332	The complex numbers form a *-ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ ℂfld ∈ *-Ring
Theorem	xrsmgm 21333	The "additive group" of the extended reals is a magma. (Contributed by AV, 30-Jan-2020.)
⊢ ℝ*𝑠 ∈ Mgm
Theorem	xrsnsgrp 21334	The "additive group" of the extended reals is not a semigroup. (Contributed by AV, 30-Jan-2020.)
⊢ ℝ*𝑠 ∉ Smgrp
Theorem	xrsmgmdifsgrp 21335	The "additive group" of the extended reals is a magma but not a semigroup, and therefore also not a monoid nor a group, in contrast to the "multiplicative group", see xrsmcmn 21318. (Contributed by AV, 30-Jan-2020.)
⊢ ℝ*𝑠 ∈ (Mgm ∖ Smgrp)
Theorem	xrs1mnd 21336	The extended real numbers, restricted to ℝ* ∖ {-∞}, form an additive monoid - in contrast to the full structure, see xrsmgmdifsgrp 21335. (Contributed by Mario Carneiro, 27-Nov-2014.)
⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))    ⇒   ⊢ 𝑅 ∈ Mnd
Theorem	xrs10 21337	The zero of the extended real number monoid. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))    ⇒   ⊢ 0 = (0g‘𝑅)
Theorem	xrs1cmn 21338	The extended real numbers restricted to ℝ* ∖ {-∞} form a commutative monoid. They are not a group because 1 + +∞ = 2 + +∞ even though 1 ≠ 2. (Contributed by Mario Carneiro, 27-Nov-2014.)
⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))    ⇒   ⊢ 𝑅 ∈ CMnd
Theorem	xrge0subm 21339	The nonnegative extended real numbers are a submonoid of the nonnegative-infinite extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))    ⇒   ⊢ (0[,]+∞) ∈ (SubMnd‘𝑅)
Theorem	xrge0cmn 21340	The nonnegative extended real numbers are a monoid. (Contributed by Mario Carneiro, 30-Aug-2015.)
⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd
Theorem	xrsds 21341*	The metric of the extended real number structure. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ 𝐷 = (dist‘ℝ*𝑠)    ⇒   ⊢ 𝐷 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))
Theorem	xrsdsval 21342	The metric of the extended real number structure. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ 𝐷 = (dist‘ℝ*𝑠)    ⇒   ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴𝐷𝐵) = if(𝐴 ≤ 𝐵, (𝐵 +𝑒 -𝑒𝐴), (𝐴 +𝑒 -𝑒𝐵)))
Theorem	xrsdsreval 21343	The metric of the extended real number structure coincides with the real number metric on the reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝐷 = (dist‘ℝ*𝑠)    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐷𝐵) = (abs‘(𝐴 − 𝐵)))
Theorem	xrsdsreclblem 21344	Lemma for xrsdsreclb 21345. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝐷 = (dist‘ℝ*𝑠)    ⇒   ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≠ 𝐵) ∧ 𝐴 ≤ 𝐵) → ((𝐵 +𝑒 -𝑒𝐴) ∈ ℝ → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)))
Theorem	xrsdsreclb 21345	The metric of the extended real number structure is only real when both arguments are real. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝐷 = (dist‘ℝ*𝑠)    ⇒   ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≠ 𝐵) → ((𝐴𝐷𝐵) ∈ ℝ ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)))
Theorem	cnsubmlem 21346*	Lemma for nn0subm 21354 and friends. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ)    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 + 𝑦) ∈ 𝐴)    &   ⊢ 0 ∈ 𝐴    ⇒   ⊢ 𝐴 ∈ (SubMnd‘ℂfld)
Theorem	cnsubglem 21347*	Lemma for resubdrg 21539 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ)    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 + 𝑦) ∈ 𝐴)    &   ⊢ (𝑥 ∈ 𝐴 → -𝑥 ∈ 𝐴)    &   ⊢ 𝐵 ∈ 𝐴    ⇒   ⊢ 𝐴 ∈ (SubGrp‘ℂfld)
Theorem	cnsubrglem 21348*	Lemma for resubdrg 21539 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.) Avoid ax-mulf 11213. (Revised by GG, 30-Apr-2025.)
⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ)    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 + 𝑦) ∈ 𝐴)    &   ⊢ (𝑥 ∈ 𝐴 → -𝑥 ∈ 𝐴)    &   ⊢ 1 ∈ 𝐴    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴)    ⇒   ⊢ 𝐴 ∈ (SubRing‘ℂfld)
Theorem	cnsubrglemOLD 21349*	Obsolete version of cnsubrglem 21348 as of 30-Apr-2025. (Contributed by Mario Carneiro, 4-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ)    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 + 𝑦) ∈ 𝐴)    &   ⊢ (𝑥 ∈ 𝐴 → -𝑥 ∈ 𝐴)    &   ⊢ 1 ∈ 𝐴    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴)    ⇒   ⊢ 𝐴 ∈ (SubRing‘ℂfld)
Theorem	cnsubdrglem 21350*	Lemma for resubdrg 21539 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ)    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 + 𝑦) ∈ 𝐴)    &   ⊢ (𝑥 ∈ 𝐴 → -𝑥 ∈ 𝐴)    &   ⊢ 1 ∈ 𝐴    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴)    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0) → (1 / 𝑥) ∈ 𝐴)    ⇒   ⊢ (𝐴 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝐴) ∈ DivRing)
Theorem	qsubdrg 21351	The rational numbers form a division subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ (ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing)
Theorem	zsubrg 21352	The integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ ℤ ∈ (SubRing‘ℂfld)
Theorem	gzsubrg 21353	The gaussian integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ ℤ[i] ∈ (SubRing‘ℂfld)
Theorem	nn0subm 21354	The nonnegative integers form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ ℕ0 ∈ (SubMnd‘ℂfld)
Theorem	rege0subm 21355	The nonnegative reals form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 20-Jun-2015.)
⊢ (0[,)+∞) ∈ (SubMnd‘ℂfld)
Theorem	absabv 21356	The regular absolute value function on the complex numbers is in fact an absolute value under our definition. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ abs ∈ (AbsVal‘ℂfld)
Theorem	zsssubrg 21357	The integers are a subset of any subring of the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ (𝑅 ∈ (SubRing‘ℂfld) → ℤ ⊆ 𝑅)
Theorem	qsssubdrg 21358	The rational numbers are a subset of any subfield of the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) → ℚ ⊆ 𝑅)
Theorem	cnsubrg 21359	There are no subrings of the complex numbers strictly between ℝ and ℂ. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ ℝ ⊆ 𝑅) → 𝑅 ∈ {ℝ, ℂ})
Theorem	cnmgpabl 21360	The unit group of the complex numbers is an abelian group. (Contributed by Mario Carneiro, 21-Jun-2015.)
⊢ 𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))    ⇒   ⊢ 𝑀 ∈ Abel
Theorem	cnmgpid 21361	The group identity element of nonzero complex number multiplication is one. (Contributed by Steve Rodriguez, 23-Feb-2007.) (Revised by AV, 26-Aug-2021.)
⊢ 𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))    ⇒   ⊢ (0g‘𝑀) = 1
Theorem	cnmsubglem 21362*	Lemma for rpmsubg 21363 and friends. (Contributed by Mario Carneiro, 21-Jun-2015.)
⊢ 𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))    &   ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ)    &   ⊢ (𝑥 ∈ 𝐴 → 𝑥 ≠ 0)    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴)    &   ⊢ 1 ∈ 𝐴    &   ⊢ (𝑥 ∈ 𝐴 → (1 / 𝑥) ∈ 𝐴)    ⇒   ⊢ 𝐴 ∈ (SubGrp‘𝑀)
Theorem	rpmsubg 21363	The positive reals form a multiplicative subgroup of the complex numbers. (Contributed by Mario Carneiro, 21-Jun-2015.)
⊢ 𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))    ⇒   ⊢ ℝ+ ∈ (SubGrp‘𝑀)
Theorem	gzrngunitlem 21364	Lemma for gzrngunit 21365. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝑍 = (ℂfld ↾s ℤ[i])    ⇒   ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴))
Theorem	gzrngunit 21365	The units on ℤ[i] are the gaussian integers with norm 1. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝑍 = (ℂfld ↾s ℤ[i])    ⇒   ⊢ (𝐴 ∈ (Unit‘𝑍) ↔ (𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1))
Theorem	gsumfsum 21366*	Relate a group sum on ℂfld to a finite sum on the complex numbers. (Contributed by Mario Carneiro, 28-Dec-2014.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵)
Theorem	regsumfsum 21367*	Relate a group sum on (ℂfld ↾s ℝ) to a finite sum on the reals. Cf. gsumfsum 21366. (Contributed by Thierry Arnoux, 7-Sep-2018.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((ℂfld ↾s ℝ) Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵)
Theorem	expmhm 21368*	Exponentiation is a monoid homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ 𝑁 = (ℂfld ↾s ℕ0)    &   ⊢ 𝑀 = (mulGrp‘ℂfld)    ⇒   ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ0 ↦ (𝐴↑𝑥)) ∈ (𝑁 MndHom 𝑀))
Theorem	nn0srg 21369	The nonnegative integers form a semiring (commutative by subcmn 19791). (Contributed by Thierry Arnoux, 1-May-2018.)
⊢ (ℂfld ↾s ℕ0) ∈ SRing
Theorem	rge0srg 21370	The nonnegative real numbers form a semiring (commutative by subcmn 19791). (Contributed by Thierry Arnoux, 6-Sep-2018.)
⊢ (ℂfld ↾s (0[,)+∞)) ∈ SRing
10.8.2  Ring of integers According to Wikipedia ("Integer", 25-May-2019, https://en.wikipedia.org/wiki/Integer) "The integers form a unital ring which is the most basic one, in the following sense: for any unital ring, there is a unique ring homomorphism from the integers into this ring. This universal property, namely to be an initial object in the category of [unital] rings, characterizes the ring 𝑍." In set.mm, there was no explicit definition for the ring of integers until June 2019, but it was denoted by (ℂfld ↾s ℤ), the field of complex numbers restricted to the integers. In zringring 21374 it is shown that this restriction is a ring (it is actually a principal ideal ring as shown in zringlpir 21392), and zringbas 21378 shows that its base set is the integers. As of June 2019, there is an abbreviation of this expression as Definition df-zring 21372 of the ring of integers. Remark: Instead of using the symbol "ZZrng" analogous to ℂfld used for the field of complex numbers, we have chosen the version with an "i" to indicate that the ring of integers is a unital ring, see also Wikipedia ("Rng (algebra)", 9-Jun-2019, https://en.wikipedia.org/wiki/Rng_(algebra) 21372).
Syntax	czring 21371	Extend class notation with the (unital) ring of integers.
class ℤring
Definition	df-zring 21372	The (unital) ring of integers. (Contributed by Alexander van der Vekens, 9-Jun-2019.)
⊢ ℤring = (ℂfld ↾s ℤ)
Theorem	zringcrng 21373	The ring of integers is a commutative ring. (Contributed by AV, 13-Jun-2019.)
⊢ ℤring ∈ CRing
Theorem	zringring 21374	The ring of integers is a ring. (Contributed by AV, 20-May-2019.) (Revised by AV, 9-Jun-2019.) (Proof shortened by AV, 13-Jun-2019.)
⊢ ℤring ∈ Ring
Theorem	zringrng 21375	The ring of integers is a non-unital ring. (Contributed by AV, 17-Mar-2025.)
⊢ ℤring ∈ Rng
Theorem	zringabl 21376	The ring of integers is an (additive) abelian group. (Contributed by AV, 13-Jun-2019.)
⊢ ℤring ∈ Abel
Theorem	zringgrp 21377	The ring of integers is an (additive) group. (Contributed by AV, 10-Jun-2019.)
⊢ ℤring ∈ Grp
Theorem	zringbas 21378	The integers are the base of the ring of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.) (Revised by AV, 9-Jun-2019.)
⊢ ℤ = (Base‘ℤring)
Theorem	zringplusg 21379	The addition operation of the ring of integers. (Contributed by Thierry Arnoux, 8-Nov-2017.) (Revised by AV, 9-Jun-2019.)
⊢ + = (+g‘ℤring)
Theorem	zringsub 21380	The subtraction of elements in the ring of integers. (Contributed by AV, 24-Mar-2025.)
⊢ − = (-g‘ℤring)    ⇒   ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑋 − 𝑌) = (𝑋 − 𝑌))
Theorem	zringmulg 21381	The multiplication (group power) operation of the group of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.) (Revised by AV, 9-Jun-2019.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴(.g‘ℤring)𝐵) = (𝐴 · 𝐵))
Theorem	zringmulr 21382	The multiplication operation of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
⊢ · = (.r‘ℤring)
Theorem	zring0 21383	The zero element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
⊢ 0 = (0g‘ℤring)
Theorem	zring1 21384	The unity element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
⊢ 1 = (1r‘ℤring)
Theorem	zringnzr 21385	The ring of integers is a nonzero ring. (Contributed by AV, 18-Apr-2020.)
⊢ ℤring ∈ NzRing
Theorem	dvdsrzring 21386	Ring divisibility in the ring of integers corresponds to ordinary divisibility in ℤ. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.)
⊢ ∥ = (∥r‘ℤring)
Theorem	zringlpirlem1 21387	Lemma for zringlpir 21392. A nonzero ideal of integers contains some positive integers. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.)
⊢ (𝜑 → 𝐼 ∈ (LIdeal‘ℤring))    &   ⊢ (𝜑 → 𝐼 ≠ {0})    ⇒   ⊢ (𝜑 → (𝐼 ∩ ℕ) ≠ ∅)
Theorem	zringlpirlem2 21388	Lemma for zringlpir 21392. A nonzero ideal of integers contains the least positive element. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.) (Revised by AV, 27-Sep-2020.)
⊢ (𝜑 → 𝐼 ∈ (LIdeal‘ℤring))    &   ⊢ (𝜑 → 𝐼 ≠ {0})    &   ⊢ 𝐺 = inf((𝐼 ∩ ℕ), ℝ, < )    ⇒   ⊢ (𝜑 → 𝐺 ∈ 𝐼)
Theorem	zringlpirlem3 21389	Lemma for zringlpir 21392. All elements of a nonzero ideal of integers are divided by the least one. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.) (Proof shortened by AV, 27-Sep-2020.)
⊢ (𝜑 → 𝐼 ∈ (LIdeal‘ℤring))    &   ⊢ (𝜑 → 𝐼 ≠ {0})    &   ⊢ 𝐺 = inf((𝐼 ∩ ℕ), ℝ, < )    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    ⇒   ⊢ (𝜑 → 𝐺 ∥ 𝑋)
Theorem	zringinvg 21390	The additive inverse of an element of the ring of integers. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ (𝐴 ∈ ℤ → -𝐴 = ((invg‘ℤring)‘𝐴))
Theorem	zringunit 21391	The units of ℤ are the integers with norm 1, i.e. 1 and -1. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.)
⊢ (𝐴 ∈ (Unit‘ℤring) ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1))
Theorem	zringlpir 21392	The integers are a principal ideal ring. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.) (Proof shortened by AV, 27-Sep-2020.)
⊢ ℤring ∈ LPIR
Theorem	zringndrg 21393	The integers are not a division ring, and therefore not a field. (Contributed by AV, 22-Oct-2021.)
⊢ ℤring ∉ DivRing
Theorem	zringcyg 21394	The integers are a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 9-Jun-2019.)
⊢ ℤring ∈ CycGrp
Theorem	zringsubgval 21395	Subtraction in the ring of integers. (Contributed by AV, 3-Aug-2019.)
⊢ − = (-g‘ℤring)    ⇒   ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑋 − 𝑌) = (𝑋 − 𝑌))
Theorem	zringmpg 21396	The multiplicative group of the ring of integers is the restriction of the multiplicative group of the complex numbers to the integers. (Contributed by AV, 15-Jun-2019.)
⊢ ((mulGrp‘ℂfld) ↾s ℤ) = (mulGrp‘ℤring)
Theorem	prmirredlem 21397	A positive integer is irreducible over ℤ iff it is a prime number. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.)
⊢ 𝐼 = (Irred‘ℤring)    ⇒   ⊢ (𝐴 ∈ ℕ → (𝐴 ∈ 𝐼 ↔ 𝐴 ∈ ℙ))
Theorem	dfprm2 21398	The positive irreducible elements of ℤ are the prime numbers. This is an alternative way to define ℙ. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.)
⊢ 𝐼 = (Irred‘ℤring)    ⇒   ⊢ ℙ = (ℕ ∩ 𝐼)
Theorem	prmirred 21399	The irreducible elements of ℤ are exactly the prime numbers (and their negatives). (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.)
⊢ 𝐼 = (Irred‘ℤring)    ⇒   ⊢ (𝐴 ∈ 𝐼 ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) ∈ ℙ))
Theorem	expghm 21400*	Exponentiation is a group homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑀 = (mulGrp‘ℂfld)    &   ⊢ 𝑈 = (𝑀 ↾s (ℂ ∖ {0}))    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) ∈ (ℤring GrpHom 𝑈))