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	lidldvgen 21301*	An element generates an ideal iff it is contained in the ideal and all elements are right-divided by it. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) → (𝐼 = (𝐾‘{𝐺}) ↔ (𝐺 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 𝐺 ∥ 𝑥)))
Theorem	lpigen 21302*	An ideal is principal iff it contains an element which right-divides all elements. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Wolf Lammen, 6-Sep-2020.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝑃 = (LPIdeal‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (𝐼 ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 𝑥 ∥ 𝑦))
10.7.5  Principal ideal domains
Syntax	cpid 21303	Class of principal ideal domains.
class PID
Definition	df-pid 21304	A principal ideal domain is an integral domain satisfying the left principal ideal property. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ PID = (IDomn ∩ LPIR)
10.8  The complex numbers as an algebraic extensible structure
10.8.1  Definition and basic properties
Syntax	cpsmet 21305	Extend class notation with the class of all pseudometric spaces.
class PsMet
Syntax	cxmet 21306	Extend class notation with the class of all extended metric spaces.
class ∞Met
Syntax	cmet 21307	Extend class notation with the class of all metrics.
class Met
Syntax	cbl 21308	Extend class notation with the metric space ball function.
class ball
Syntax	cfbas 21309	Extend class definition to include the class of filter bases.
class fBas
Syntax	cfg 21310	Extend class definition to include the filter generating function.
class filGen
Syntax	cmopn 21311	Extend class notation with a function mapping each metric space to the family of its open sets.
class MetOpen
Syntax	cmetu 21312	Extend class notation with the function mapping metrics to the uniform structure generated by that metric.
class metUnif
Definition	df-psmet 21313*	Define the set of all pseudometrics on a given base set. In a pseudo metric, two distinct points may have a distance zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ* ↑m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
Definition	df-xmet 21314*	Define the set of all extended metrics on a given base set. The definition is similar to df-met 21315, 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 21315*	Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric; see df-ms 24277. 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 24299, metgt0 24315, metsym 24306, and mettri 24308. (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 21316*	Define the metric space ball function. See blval 24342 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 21317	Define a function whose value is the family of open sets of a metric space. See elmopn 24398 for its main property. (Contributed by NM, 1-Sep-2006.)
⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
Definition	df-fbas 21318*	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 21319*	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 21320*	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 21321	Extend class notation with the field of complex numbers.
class ℂfld
Definition	df-cnfld 21322*	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 21324, cnfldadd 21327, cnfldmul 21329, cnfldcj 21330, cnfldtset 21331, cnfldle 21332, cnfldds 21333, and cnfldbas 21325. 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 21323	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 21322. (Revised by GG, 31-Mar-2025.)
⊢ ℂfld Struct ⟨1, ;13⟩
Theorem	cnfldex 21324	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 5312. (Revised by GG, 16-Mar-2025.)
⊢ ℂfld ∈ V
Theorem	cnfldbas 21325	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 21322. (Revised by GG, 31-Mar-2025.)
⊢ ℂ = (Base‘ℂfld)
Theorem	mpocnfldadd 21326*	The addition operation of the field of complex numbers. Version of cnfldadd 21327 using maps-to notation, which does not require ax-addf 11117. (Contributed by GG, 31-Mar-2025.)
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) = (+g‘ℂfld)
Theorem	cnfldadd 21327	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 21322. (Revised by GG, 27-Apr-2025.)
⊢ + = (+g‘ℂfld)
Theorem	mpocnfldmul 21328*	The multiplication operation of the field of complex numbers. Version of cnfldmul 21329 using maps-to notation, which does not require ax-mulf 11118. (Contributed by GG, 31-Mar-2025.)
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (.r‘ℂfld)
Theorem	cnfldmul 21329	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 21322. (Revised by GG, 27-Apr-2025.)
⊢ · = (.r‘ℂfld)
Theorem	cnfldcj 21330	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 21322. (Revised by GG, 31-Mar-2025.)
⊢ ∗ = (*𝑟‘ℂfld)
Theorem	cnfldtset 21331	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 21322. (Revised by GG, 31-Mar-2025.)
⊢ (MetOpen‘(abs ∘ − )) = (TopSet‘ℂfld)
Theorem	cnfldle 21332	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 21322. (Revised by GG, 31-Mar-2025.)
⊢ ≤ = (le‘ℂfld)
Theorem	cnfldds 21333	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 21322. (Revised by GG, 31-Mar-2025.)
⊢ (abs ∘ − ) = (dist‘ℂfld)
Theorem	cnfldunif 21334	The uniform structure component of the complex numbers. (Contributed by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21322. (Revised by GG, 31-Mar-2025.)
⊢ (metUnif‘(abs ∘ − )) = (UnifSet‘ℂfld)
Theorem	cnfldfun 21335	The field of complex numbers is a function. The proof is much shorter than the proof of cnfldfunALT 21336 by using cnfldstr 21323 and structn0fun 17090: in addition, it must be shown that ∅ ∉ ℂfld. (Contributed by AV, 18-Nov-2021.) Revise df-cnfld 21322. (Revised by GG, 31-Mar-2025.)
⊢ Fun ℂfld
Theorem	cnfldfunALT 21336	The field of complex numbers is a function. Alternate proof of cnfldfun 21335 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 21322. (Revised by GG, 31-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Fun ℂfld
Theorem	dfcnfldOLD 21337	Obsolete version of df-cnfld 21322 as of 27-Apr-2025. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Thierry Arnoux, 15-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ℂfld = (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
Theorem	cnfldstrOLD 21338	Obsolete version of cnfldstr 21323 as of 27-Apr-2025. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ℂfld Struct ⟨1, ;13⟩
Theorem	cnfldexOLD 21339	Obsolete version of cnfldex 21324 as of 27-Apr-2025. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ℂfld ∈ V
Theorem	cnfldbasOLD 21340	Obsolete version of cnfldbas 21325 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.)
⊢ ℂ = (Base‘ℂfld)
Theorem	cnfldaddOLD 21341	Obsolete version of cnfldadd 21327 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 21342	Obsolete version of cnfldmul 21329 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 21343	Obsolete version of cnfldcj 21330 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 21344	Obsolete version of cnfldtset 21331 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 21345	Obsolete version of cnfldle 21332 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 21346	Obsolete version of cnfldds 21333 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 21347	Obsolete version of cnfldunif 21334 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 21348	Obsolete version of cnfldfun 21335 as of 27-Apr-2025. (Contributed by AV, 18-Nov-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Fun ℂfld
Theorem	cnfldfunALTOLD 21349	Obsolete version of cnfldfunALT 21336 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	xrsstr 21350	The extended real structure is a structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ℝ*𝑠 Struct ⟨1, ;12⟩
Theorem	xrsex 21351	The extended real structure is a set. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ℝ*𝑠 ∈ V
Theorem	xrsadd 21352	The addition operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ +𝑒 = (+g‘ℝ*𝑠)
Theorem	xrsmul 21353	The multiplication operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ·e = (.r‘ℝ*𝑠)
Theorem	xrstset 21354	The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ (ordTop‘ ≤ ) = (TopSet‘ℝ*𝑠)
Theorem	cncrng 21355	The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.) Avoid ax-mulf 11118. (Revised by GG, 31-Mar-2025.)
⊢ ℂfld ∈ CRing
Theorem	cncrngOLD 21356	Obsolete version of cncrng 21355 as of 30-Apr-2025. (Contributed by Mario Carneiro, 8-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ℂfld ∈ CRing
Theorem	cnring 21357	The complex numbers form a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ ℂfld ∈ Ring
Theorem	xrsmcmn 21358	The "multiplicative group" of the extended reals is a commutative monoid (even though the "additive group" is not a semigroup, see xrsmgmdifsgrp 21375.) (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ (mulGrp‘ℝ*𝑠) ∈ CMnd
Theorem	cnfld0 21359	Zero is the zero element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 0 = (0g‘ℂfld)
Theorem	cnfld1 21360	One is the unity element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) Avoid ax-mulf 11118. (Revised by GG, 31-Mar-2025.)
⊢ 1 = (1r‘ℂfld)
Theorem	cnfld1OLD 21361	Obsolete version of cnfld1 21360 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 21362	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 21363	The functionalized addition operation of the field of complex numbers. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ + = (+𝑓‘ℂfld)
Theorem	cnfldsub 21364	The subtraction operator in the field of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ − = (-g‘ℂfld)
Theorem	cndrng 21365	The complex numbers form a division ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) Avoid ax-mulf 11118. (Revised by GG, 30-Apr-2025.)
⊢ ℂfld ∈ DivRing
Theorem	cndrngOLD 21366	Obsolete version of cndrng 21365 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 21367	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 11118. (Revised by GG, 30-Apr-2025.)
⊢ / = (/r‘ℂfld)
Theorem	cnflddivOLD 21368	Obsolete version of cnflddiv 21367 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 21369	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 21370	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 21371	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 21372	The complex numbers form a *-ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ ℂfld ∈ *-Ring
Theorem	xrsmgm 21373	The "additive group" of the extended reals is a magma. (Contributed by AV, 30-Jan-2020.)
⊢ ℝ*𝑠 ∈ Mgm
Theorem	xrsnsgrp 21374	The "additive group" of the extended reals is not a semigroup. (Contributed by AV, 30-Jan-2020.)
⊢ ℝ*𝑠 ∉ Smgrp
Theorem	xrsmgmdifsgrp 21375	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 21358. (Contributed by AV, 30-Jan-2020.)
⊢ ℝ*𝑠 ∈ (Mgm ∖ Smgrp)
Theorem	xrsds 21376*	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 21377	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 21378	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 21379	Lemma for xrsdsreclb 21380. (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 21380	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 21381*	Lemma for nn0subm 21389 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 21382*	Lemma for resubdrg 21575 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 21383*	Lemma for resubdrg 21575 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.) Avoid ax-mulf 11118. (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 21384*	Obsolete version of cnsubrglem 21383 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 21385*	Lemma for resubdrg 21575 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 21386	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 21387	The integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ ℤ ∈ (SubRing‘ℂfld)
Theorem	gzsubrg 21388	The gaussian integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ ℤ[i] ∈ (SubRing‘ℂfld)
Theorem	nn0subm 21389	The nonnegative integers form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ ℕ0 ∈ (SubMnd‘ℂfld)
Theorem	rege0subm 21390	The nonnegative reals form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 20-Jun-2015.)
⊢ (0[,)+∞) ∈ (SubMnd‘ℂfld)
Theorem	absabv 21391	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 21392	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 21393	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 21394	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 21395	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 21396	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 21397*	Lemma for rpmsubg 21398 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 21398	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 21399	Lemma for gzrngunit 21400. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝑍 = (ℂfld ↾s ℤ[i])    ⇒   ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴))
Theorem	gzrngunit 21400	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))