P. 218 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21701-21800   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-xmet 21701*	Define the set of all extended metrics on a given base set. The definition is similar to df-met 21702, but we also allow the metric to take on the value +∞. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ* ↑m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +e (𝑤𝑑𝑧)))})
Definition	df-met 21702*	Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric; see df-ms 24670. However, we will often also call the metric itself a "metric space".) Equivalent to Definition 14-1.1 of [Gleason] p. 223. The 4 properties in Gleason's definition are shown by met0 24692, metgt0 24708, metsym 24699, and mettri 24701. (Contributed by NM, 25-Aug-2006.)
⊢ Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ ↑m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))})
Definition	df-bl 21703*	Define the metric space ball function. See blval 24735 for its value. (Contributed by NM, 30-Aug-2006.) (Revised by Thierry Arnoux, 11-Feb-2018.)
⊢ ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑧 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑧}))
Definition	df-mopn 21704	Define a function whose value is the family of open sets of a metric space. See elmopn 24791 for its main property. (Contributed by NM, 1-Sep-2006.)
⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
Definition	df-fbas 21705*	Define the class of all filter bases. Note that a filter base on one set is also a filter base for any superset, so there is not a unique base set that can be recovered. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
⊢ fBas = (𝑤 ∈ V ↦ {𝑥 ∈ 𝒫 𝒫 𝑤 ∣ (𝑥 ≠ ∅ ∧ ∅ ∉ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧)) ≠ ∅)})
Definition	df-fg 21706*	Define the filter generating function. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
⊢ filGen = (𝑤 ∈ V, 𝑥 ∈ (fBas‘𝑤) ↦ {𝑦 ∈ 𝒫 𝑤 ∣ (𝑥 ∩ 𝒫 𝑦) ≠ ∅})
Definition	df-metu 21707*	Define the function mapping metrics to the uniform structure generated by that metric. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
⊢ metUnif = (𝑑 ∈ ∪ ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (◡𝑑 “ (0[,)𝑎)))))
Syntax	ccnfld 21708	Extend class notation with the field of complex numbers.
class ℂfld
Definition	df-cnfld 21709*	The field of complex numbers. Other number fields and rings can be constructed by applying the ↾s restriction operator, for instance (ℂfld ↾ 𝔸) is the field of algebraic numbers. The contract of this set is defined entirely by cnfldex 21711, cnfldadd 21714, cnfldmul 21716, cnfldcj 21717, cnfldtset 21718, cnfldle 21719, cnfldds 21720, and cnfldbas 21712. We may add additional members to this in the future. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Thierry Arnoux, 15-Dec-2017.) Use maps-to notation for addition and multiplication. (Revised by GG, 31-Mar-2025.) (New usage is discouraged.)
⊢ ℂfld = (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
Theorem	cnfldstr 21710	The field of complex numbers is a structure. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21709. (Revised by GG, 31-Mar-2025.)
⊢ ℂfld Struct ⟨1, ;13⟩
Theorem	cnfldex 21711	The field of complex numbers is a set. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Avoid complex number axioms and ax-pow 5331. (Revised by GG, 16-Mar-2025.)
⊢ ℂfld ∈ V
Theorem	cnfldbas 21712	The base set of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21709. (Revised by GG, 31-Mar-2025.)
⊢ ℂ = (Base‘ℂfld)
Theorem	mpocnfldadd 21713*	The addition operation of the field of complex numbers. Version of cnfldadd 21714 using maps-to notation, which does not require ax-addf 11306. (Contributed by GG, 31-Mar-2025.)
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) = (+g‘ℂfld)
Theorem	cnfldadd 21714	The addition operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21709. (Revised by GG, 27-Apr-2025.)
⊢ + = (+g‘ℂfld)
Theorem	mpocnfldmul 21715*	The multiplication operation of the field of complex numbers. Version of cnfldmul 21716 using maps-to notation, which does not require ax-mulf 11307. (Contributed by GG, 31-Mar-2025.)
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (.r‘ℂfld)
Theorem	cnfldmul 21716	The multiplication operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21709. (Revised by GG, 27-Apr-2025.)
⊢ · = (.r‘ℂfld)
Theorem	cnfldcj 21717	The conjugation operation of the field of complex numbers. (Contributed by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21709. (Revised by GG, 31-Mar-2025.)
⊢ ∗ = (*𝑟‘ℂfld)
Theorem	cnfldtset 21718	The topology component of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21709. (Revised by GG, 31-Mar-2025.)
⊢ (MetOpen‘(abs ∘ − )) = (TopSet‘ℂfld)
Theorem	cnfldle 21719	The ordering of the field of complex numbers. Note that this is not actually an ordering on ℂ, but we put it in the structure anyway because restricting to ℝ does not affect this component, so that (ℂfld ↾s ℝ) is an ordered field even though ℂfld itself is not. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21709. (Revised by GG, 31-Mar-2025.)
⊢ ≤ = (le‘ℂfld)
Theorem	cnfldds 21720	The metric of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21709. (Revised by GG, 31-Mar-2025.)
⊢ (abs ∘ − ) = (dist‘ℂfld)
Theorem	cnfldunif 21721	The uniform structure component of the complex numbers. (Contributed by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21709. (Revised by GG, 31-Mar-2025.)
⊢ (metUnif‘(abs ∘ − )) = (UnifSet‘ℂfld)
Theorem	cnfldfun 21722	The field of complex numbers is a function. The proof is much shorter than the proof of cnfldfunALT 21723 by using cnfldstr 21710 and structn0fun 17358: in addition, it must be shown that ∅ ∉ ℂfld. (Contributed by AV, 18-Nov-2021.) Revise df-cnfld 21709. (Revised by GG, 31-Mar-2025.)
⊢ Fun ℂfld
Theorem	cnfldfunALT 21723	The field of complex numbers is a function. Alternate proof of cnfldfun 21722 not requiring that the index set of the components is ordered, but using quadratically many inequalities for the indices. (Contributed by AV, 14-Nov-2021.) (Proof shortened by AV, 11-Nov-2024.) Revise df-cnfld 21709. (Revised by GG, 31-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Fun ℂfld
Theorem	xrsstr 21724	The extended real structure is a structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ℝ*𝑠 Struct ⟨1, ;12⟩
Theorem	xrsex 21725	The extended real structure is a set. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ℝ*𝑠 ∈ V
Theorem	xrsadd 21726	The addition operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ +e = (+g‘ℝ*𝑠)
Theorem	xrsmul 21727	The multiplication operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ·e = (.r‘ℝ*𝑠)
Theorem	xrstset 21728	The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ (ordTop‘ ≤ ) = (TopSet‘ℝ*𝑠)
Theorem	cncrng 21729	The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.) Avoid ax-mulf 11307. (Revised by GG, 31-Mar-2025.)
⊢ ℂfld ∈ CRing
Theorem	cnring 21730	The complex numbers form a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ ℂfld ∈ Ring
Theorem	xrsmcmn 21731	The "multiplicative group" of the extended reals is a commutative monoid (even though the "additive group" is not a semigroup, see xrsmgmdifsgrp 21745.) (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ (mulGrp‘ℝ*𝑠) ∈ CMnd
Theorem	cnfld0 21732	Zero is the zero element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 0 = (0g‘ℂfld)
Theorem	cnfld1 21733	One is the unity element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) Avoid ax-mulf 11307. (Revised by GG, 31-Mar-2025.)
⊢ 1 = (1r‘ℂfld)
Theorem	cnfldneg 21734	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 21735	The functionalized addition operation of the field of complex numbers. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ + = (+𝑓‘ℂfld)
Theorem	cnfldsub 21736	The subtraction operator in the field of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ − = (-g‘ℂfld)
Theorem	cndrng 21737	The complex numbers form a division ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) Avoid ax-mulf 11307. (Revised by GG, 30-Apr-2025.)
⊢ ℂfld ∈ DivRing
Theorem	cnflddiv 21738	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 11307. (Revised by GG, 30-Apr-2025.)
⊢ / = (/r‘ℂfld)
Theorem	cnfldinv 21739	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 21740	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 21741	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 21742	The complex numbers form a *-ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ ℂfld ∈ *-Ring
Theorem	xrsmgm 21743	The "additive group" of the extended reals is a magma. (Contributed by AV, 30-Jan-2020.)
⊢ ℝ*𝑠 ∈ Mgm
Theorem	xrsnsgrp 21744	The "additive group" of the extended reals is not a semigroup. (Contributed by AV, 30-Jan-2020.)
⊢ ℝ*𝑠 ∉ Smgrp
Theorem	xrsmgmdifsgrp 21745	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 21731. (Contributed by AV, 30-Jan-2020.)
⊢ ℝ*𝑠 ∈ (Mgm ∖ Smgrp)
Theorem	xrsds 21746*	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 21747	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 21748	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 21749	Lemma for xrsdsreclb 21750. (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 21750	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 21751*	Lemma for nn0subm 21758 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 21752*	Lemma for resubdrg 21944 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 21753*	Lemma for resubdrg 21944 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.) Avoid ax-mulf 11307. (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	cnsubdrglem 21754*	Lemma for resubdrg 21944 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 21755	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 21756	The integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ ℤ ∈ (SubRing‘ℂfld)
Theorem	gzsubrg 21757	The gaussian integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ ℤ[i] ∈ (SubRing‘ℂfld)
Theorem	nn0subm 21758	The nonnegative integers form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ ℕ0 ∈ (SubMnd‘ℂfld)
Theorem	rege0subm 21759	The nonnegative reals form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 20-Jun-2015.)
⊢ (0[,)+∞) ∈ (SubMnd‘ℂfld)
Theorem	absabv 21760	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 21761	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 21762	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 21763	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 21764	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 21765	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 21766*	Lemma for rpmsubg 21767 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 21767	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 21768	Lemma for gzrngunit 21769. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝑍 = (ℂfld ↾s ℤ[i])    ⇒   ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴))
Theorem	gzrngunit 21769	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 21770*	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 21771*	Relate a group sum on (ℂfld ↾s ℝ) to a finite sum on the reals. Cf. gsumfsum 21770. (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 21772*	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 21773	The nonnegative integers form a semiring (commutative by subcmn 20081). (Contributed by Thierry Arnoux, 1-May-2018.)
⊢ (ℂfld ↾s ℕ0) ∈ SRing
Theorem	rge0srg 21774	The nonnegative real numbers form a semiring (commutative by subcmn 20081). (Contributed by Thierry Arnoux, 6-Sep-2018.)
⊢ (ℂfld ↾s (0[,)+∞)) ∈ SRing
Theorem	xrge0plusg 21775	The additive law of the extended nonnegative real numbers monoid is the addition in the extended real numbers. (Contributed by Thierry Arnoux, 20-Mar-2017.)
⊢ +e = (+g‘(ℝ*𝑠 ↾s (0[,]+∞)))
Theorem	xrs1mnd 21776	The extended real numbers, restricted to ℝ* ∖ {-∞}, form an additive monoid - in contrast to the full structure, see xrsmgmdifsgrp 21745. (Contributed by Mario Carneiro, 27-Nov-2014.)
⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))    ⇒   ⊢ 𝑅 ∈ Mnd
Theorem	xrs10 21777	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 21778	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 21779	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 21780	The nonnegative extended real numbers are a monoid. (Contributed by Mario Carneiro, 30-Aug-2015.)
⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd
Theorem	xrge0omnd 21781	The nonnegative extended real numbers form an ordered monoid. (Contributed by Thierry Arnoux, 22-Mar-2018.)
⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ oMnd
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 21785 it is shown that this restriction is a ring (it is actually a principal ideal ring as shown in zringlpir 21803), and zringbas 21789 shows that its base set is the integers. As of June 2019, there is an abbreviation of this expression as Definition df-zring 21783 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) 21783).
Syntax	czring 21782	Extend class notation with the (unital) ring of integers.
class ℤring
Definition	df-zring 21783	The (unital) ring of integers. (Contributed by Alexander van der Vekens, 9-Jun-2019.)
⊢ ℤring = (ℂfld ↾s ℤ)
Theorem	zringcrng 21784	The ring of integers is a commutative ring. (Contributed by AV, 13-Jun-2019.)
⊢ ℤring ∈ CRing
Theorem	zringring 21785	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 21786	The ring of integers is a non-unital ring. (Contributed by AV, 17-Mar-2025.)
⊢ ℤring ∈ Rng
Theorem	zringabl 21787	The ring of integers is an (additive) abelian group. (Contributed by AV, 13-Jun-2019.)
⊢ ℤring ∈ Abel
Theorem	zringgrp 21788	The ring of integers is an (additive) group. (Contributed by AV, 10-Jun-2019.)
⊢ ℤring ∈ Grp
Theorem	zringbas 21789	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 21790	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 21791	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 21792	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 21793	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 21794	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 21795	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 21796	The ring of integers is a nonzero ring. (Contributed by AV, 18-Apr-2020.)
⊢ ℤring ∈ NzRing
Theorem	dvdsrzring 21797	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 21798	Lemma for zringlpir 21803. 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 21799	Lemma for zringlpir 21803. 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 21800	Lemma for zringlpir 21803. 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)