P. 215 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21401-21500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cnfldaddOLD 21401	Obsolete version of cnfldadd 21387 as of 27-Apr-2025. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ + = (+g‘ℂfld)
Theorem	cnfldmulOLD 21402	Obsolete version of cnfldmul 21389 as of 27-Apr-2025. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ · = (.r‘ℂfld)
Theorem	cnfldcjOLD 21403	Obsolete version of cnfldcj 21390 as of 27-Apr-2025. (Contributed by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∗ = (*𝑟‘ℂfld)
Theorem	cnfldtsetOLD 21404	Obsolete version of cnfldtset 21391 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 21405	Obsolete version of cnfldle 21392 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 21406	Obsolete version of cnfldds 21393 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 21407	Obsolete version of cnfldunif 21394 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 21408	Obsolete version of cnfldfun 21395 as of 27-Apr-2025. (Contributed by AV, 18-Nov-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Fun ℂfld
Theorem	cnfldfunALTOLD 21409	Obsolete version of cnfldfunALT 21396 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 21410	Obsolete version of cnfldfunALTOLD 21409 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 21411	The extended real structure is a structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ℝ*𝑠 Struct ⟨1, ;12⟩
Theorem	xrsex 21412	The extended real structure is a set. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ℝ*𝑠 ∈ V
Theorem	xrsbas 21413	The base set of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ℝ* = (Base‘ℝ*𝑠)
Theorem	xrsadd 21414	The addition operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ +𝑒 = (+g‘ℝ*𝑠)
Theorem	xrsmul 21415	The multiplication operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ·e = (.r‘ℝ*𝑠)
Theorem	xrstset 21416	The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ (ordTop‘ ≤ ) = (TopSet‘ℝ*𝑠)
Theorem	xrsle 21417	The ordering of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ≤ = (le‘ℝ*𝑠)
Theorem	cncrng 21418	The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.) Avoid ax-mulf 11232. (Revised by GG, 31-Mar-2025.)
⊢ ℂfld ∈ CRing
Theorem	cncrngOLD 21419	Obsolete version of cncrng 21418 as of 30-Apr-2025. (Contributed by Mario Carneiro, 8-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ℂfld ∈ CRing
Theorem	cnring 21420	The complex numbers form a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ ℂfld ∈ Ring
Theorem	xrsmcmn 21421	The "multiplicative group" of the extended reals is a commutative monoid (even though the "additive group" is not a semigroup, see xrsmgmdifsgrp 21438.) (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ (mulGrp‘ℝ*𝑠) ∈ CMnd
Theorem	cnfld0 21422	Zero is the zero element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 0 = (0g‘ℂfld)
Theorem	cnfld1 21423	One is the unity element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) Avoid ax-mulf 11232. (Revised by GG, 31-Mar-2025.)
⊢ 1 = (1r‘ℂfld)
Theorem	cnfld1OLD 21424	Obsolete version of cnfld1 21423 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 21425	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 21426	The functionalized addition operation of the field of complex numbers. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ + = (+𝑓‘ℂfld)
Theorem	cnfldsub 21427	The subtraction operator in the field of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ − = (-g‘ℂfld)
Theorem	cndrng 21428	The complex numbers form a division ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) Avoid ax-mulf 11232. (Revised by GG, 30-Apr-2025.)
⊢ ℂfld ∈ DivRing
Theorem	cndrngOLD 21429	Obsolete version of cndrng 21428 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 21430	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 11232. (Revised by GG, 30-Apr-2025.)
⊢ / = (/r‘ℂfld)
Theorem	cnflddivOLD 21431	Obsolete version of cnflddiv 21430 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 21432	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 21433	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 21434	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 21435	The complex numbers form a *-ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ ℂfld ∈ *-Ring
Theorem	xrsmgm 21436	The "additive group" of the extended reals is a magma. (Contributed by AV, 30-Jan-2020.)
⊢ ℝ*𝑠 ∈ Mgm
Theorem	xrsnsgrp 21437	The "additive group" of the extended reals is not a semigroup. (Contributed by AV, 30-Jan-2020.)
⊢ ℝ*𝑠 ∉ Smgrp
Theorem	xrsmgmdifsgrp 21438	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 21421. (Contributed by AV, 30-Jan-2020.)
⊢ ℝ*𝑠 ∈ (Mgm ∖ Smgrp)
Theorem	xrs1mnd 21439	The extended real numbers, restricted to ℝ* ∖ {-∞}, form an additive monoid - in contrast to the full structure, see xrsmgmdifsgrp 21438. (Contributed by Mario Carneiro, 27-Nov-2014.)
⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))    ⇒   ⊢ 𝑅 ∈ Mnd
Theorem	xrs10 21440	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 21441	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 21442	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 21443	The nonnegative extended real numbers are a monoid. (Contributed by Mario Carneiro, 30-Aug-2015.)
⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd
Theorem	xrsds 21444*	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 21445	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 21446	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 21447	Lemma for xrsdsreclb 21448. (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 21448	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 21449*	Lemma for nn0subm 21457 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 21450*	Lemma for resubdrg 21643 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 21451*	Lemma for resubdrg 21643 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.) Avoid ax-mulf 11232. (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 21452*	Obsolete version of cnsubrglem 21451 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 21453*	Lemma for resubdrg 21643 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 21454	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 21455	The integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ ℤ ∈ (SubRing‘ℂfld)
Theorem	gzsubrg 21456	The gaussian integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ ℤ[i] ∈ (SubRing‘ℂfld)
Theorem	nn0subm 21457	The nonnegative integers form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ ℕ0 ∈ (SubMnd‘ℂfld)
Theorem	rege0subm 21458	The nonnegative reals form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 20-Jun-2015.)
⊢ (0[,)+∞) ∈ (SubMnd‘ℂfld)
Theorem	absabv 21459	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 21460	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 21461	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 21462	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 21463	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 21464	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 21465*	Lemma for rpmsubg 21466 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 21466	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 21467	Lemma for gzrngunit 21468. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝑍 = (ℂfld ↾s ℤ[i])    ⇒   ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴))
Theorem	gzrngunit 21468	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 21469*	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 21470*	Relate a group sum on (ℂfld ↾s ℝ) to a finite sum on the reals. Cf. gsumfsum 21469. (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 21471*	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 21472	The nonnegative integers form a semiring (commutative by subcmn 19869). (Contributed by Thierry Arnoux, 1-May-2018.)
⊢ (ℂfld ↾s ℕ0) ∈ SRing
Theorem	rge0srg 21473	The nonnegative real numbers form a semiring (commutative by subcmn 19869). (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 21477 it is shown that this restriction is a ring (it is actually a principal ideal ring as shown in zringlpir 21495), and zringbas 21481 shows that its base set is the integers. As of June 2019, there is an abbreviation of this expression as Definition df-zring 21475 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) 21475).
Syntax	czring 21474	Extend class notation with the (unital) ring of integers.
class ℤring
Definition	df-zring 21475	The (unital) ring of integers. (Contributed by Alexander van der Vekens, 9-Jun-2019.)
⊢ ℤring = (ℂfld ↾s ℤ)
Theorem	zringcrng 21476	The ring of integers is a commutative ring. (Contributed by AV, 13-Jun-2019.)
⊢ ℤring ∈ CRing
Theorem	zringring 21477	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 21478	The ring of integers is a non-unital ring. (Contributed by AV, 17-Mar-2025.)
⊢ ℤring ∈ Rng
Theorem	zringabl 21479	The ring of integers is an (additive) abelian group. (Contributed by AV, 13-Jun-2019.)
⊢ ℤring ∈ Abel
Theorem	zringgrp 21480	The ring of integers is an (additive) group. (Contributed by AV, 10-Jun-2019.)
⊢ ℤring ∈ Grp
Theorem	zringbas 21481	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 21482	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 21483	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 21484	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 21485	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 21486	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 21487	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 21488	The ring of integers is a nonzero ring. (Contributed by AV, 18-Apr-2020.)
⊢ ℤring ∈ NzRing
Theorem	dvdsrzring 21489	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 21490	Lemma for zringlpir 21495. 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 21491	Lemma for zringlpir 21495. 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 21492	Lemma for zringlpir 21495. 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 21493	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 21494	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 21495	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 21496	The integers are not a division ring, and therefore not a field. (Contributed by AV, 22-Oct-2021.)
⊢ ℤring ∉ DivRing
Theorem	zringcyg 21497	The integers are a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 9-Jun-2019.)
⊢ ℤring ∈ CycGrp
Theorem	zringsubgval 21498	Subtraction in the ring of integers. (Contributed by AV, 3-Aug-2019.)
⊢ − = (-g‘ℤring)    ⇒   ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑋 − 𝑌) = (𝑋 − 𝑌))
Theorem	zringmpg 21499	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 21500	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))