P. 217 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21601-21700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	prmidl 21601*	The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) ∧ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐽 (𝑥 · 𝑦) ∈ 𝑃) → (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃))
Theorem	prmidl2 21602*	A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 38972 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ Ring ∧ 𝑃 ∈ (LIdeal‘𝑅)) ∧ (𝑃 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))) → 𝑃 ∈ (PrmIdeal‘𝑅))
Theorem	idlmulssprm 21603	Let 𝑃 be a prime ideal containing the product (𝐼 × 𝐽) of two ideals 𝐼 and 𝐽. Then 𝐼 ⊆ 𝑃 or 𝐽 ⊆ 𝑃. (Contributed by Thierry Arnoux, 13-Apr-2024.)
⊢ × = (LSSum‘(mulGrp‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑃 ∈ (PrmIdeal‘𝑅))    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   ⊢ (𝜑 → 𝐽 ∈ (LIdeal‘𝑅))    &   ⊢ (𝜑 → (𝐼 × 𝐽) ⊆ 𝑃)    ⇒   ⊢ (𝜑 → (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃))
Theorem	pridln1 21604	A proper ideal cannot contain the ring unity. (Contributed by Thierry Arnoux, 9-Apr-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼 ≠ 𝐵) → ¬ 1 ∈ 𝐼)
Theorem	prmidlidl 21605	A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃 ∈ (LIdeal‘𝑅))
Theorem	prmidlssidl 21606	Prime ideals as a subset of ideals. (Contributed by Thierry Arnoux, 2-Jun-2024.)
⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅))
Theorem	cringm4 21607	Commutative/associative law for commutative ring. (Contributed by Thierry Arnoux, 14-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 · 𝑌) · (𝑍 · 𝑊)) = ((𝑋 · 𝑍) · (𝑌 · 𝑊)))
Theorem	isprmidlc 21608*	The predicate "is prime ideal" for commutative rings. Alternate definition for commutative rings. See definition in [Lang] p. 92. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))))
Theorem	prmidlc 21609	Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 12-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ 𝐵 ∧ 𝐽 ∈ 𝐵 ∧ (𝐼 · 𝐽) ∈ 𝑃)) → (𝐼 ∈ 𝑃 ∨ 𝐽 ∈ 𝑃))
Theorem	prmidlc2 21610	Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by AV, 12-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (𝐵 ∖ 𝑃) ∧ 𝐽 ∈ 𝐵 ∧ (𝐼 · 𝐽) ∈ 𝑃)) → 𝐽 ∈ 𝑃)
Theorem	cmprmidlmcl 21611	The complement of a prime ideal is multiplicatively closed. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by AV, 12-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (𝐵 ∖ 𝑃) ∧ 𝐽 ∈ (𝐵 ∖ 𝑃))) → (𝐼 · 𝐽) ∈ (𝐵 ∖ 𝑃))
Theorem	prmidlprop 21612	Property of prime ideals. (Contributed by Thierry Arnoux, 6-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑃 ∈ (PrmIdeal‘𝑅))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝑃 ∨ 𝑌 ∈ 𝑃))
Theorem	0ringprmidl 21613	The trivial ring does not have any prime ideal. (Contributed by Thierry Arnoux, 30-Jun-2024.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → (PrmIdeal‘𝑅) = ∅)
Theorem	prmidl0 21614	The zero ideal of a commutative ring 𝑅 is a prime ideal if and only if 𝑅 is an integral domain. (Contributed by Thierry Arnoux, 30-Jun-2024.)
⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ { 0 } ∈ (PrmIdeal‘𝑅)) ↔ 𝑅 ∈ IDomn)
Theorem	rhmpreimaprmidl 21615	The preimage of a prime ideal by a ring homomorphism is a prime ideal. (Contributed by Thierry Arnoux, 29-Jun-2024.)
⊢ 𝑃 = (PrmIdeal‘𝑅)    ⇒   ⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ 𝑃)
Theorem	qsidomlem1 21616	If the quotient ring of a commutative ring relative to an ideal is an integral domain, that ideal must be prime. (Contributed by Thierry Arnoux, 16-Jan-2024.)
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ∈ (PrmIdeal‘𝑅))
Theorem	qsidomlem2 21617	A quotient by a prime ideal is an integral domain. (Contributed by Thierry Arnoux, 16-Jan-2024.)
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ IDomn)
Theorem	qsidom 21618	An ideal 𝐼 in the commutative ring 𝑅 is prime if and only if the factor ring 𝑄 is an integral domain. (Contributed by Thierry Arnoux, 16-Jan-2024.)
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (𝑄 ∈ IDomn ↔ 𝐼 ∈ (PrmIdeal‘𝑅)))
Theorem	qsnzr 21619	A quotient of a nonzero ring by a proper ideal is a nonzero ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑅 ∈ NzRing)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → 𝐼 ≠ 𝐵)    ⇒   ⊢ (𝜑 → 𝑄 ∈ NzRing)
Theorem	ssdifidllem 21620*	Lemma for ssdifidl 21621: The set 𝑃 used in the proof of ssdifidl 21621 satisfies the condition of Zorn's Lemma. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ (𝜑 → (𝑆 ∩ 𝐼) = ∅)    &   ⊢ 𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ 𝐼 ⊆ 𝑝)}    &   ⊢ (𝜑 → 𝑍 ⊆ 𝑃)    &   ⊢ (𝜑 → 𝑍 ≠ ∅)    &   ⊢ (𝜑 → [⊊] Or 𝑍)    ⇒   ⊢ (𝜑 → ∪ 𝑍 ∈ 𝑃)
Theorem	ssdifidl 21621*	Let 𝑅 be a ring, and let 𝐼 be an ideal of 𝑅 disjoint with a set 𝑆. Then there exists an ideal 𝑖, maximal among the set 𝑃 of ideals containing 𝐼 and disjoint with 𝑆. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ (𝜑 → (𝑆 ∩ 𝐼) = ∅)    &   ⊢ 𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ 𝐼 ⊆ 𝑝)}    ⇒   ⊢ (𝜑 → ∃𝑖 ∈ 𝑃 ∀𝑗 ∈ 𝑃 ¬ 𝑖 ⊊ 𝑗)
Theorem	ssdifidlprm 21622*	If the set 𝑆 of ssdifidl 21621 is multiplicatively closed, then the ideal 𝑖 is prime. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝑀))    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ (𝜑 → (𝑆 ∩ 𝐼) = ∅)    &   ⊢ 𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ 𝐼 ⊆ 𝑝)}    ⇒   ⊢ (𝜑 → ∃𝑖 ∈ 𝑃 (𝑖 ∈ (PrmIdeal‘𝑅) ∧ ∀𝑗 ∈ 𝑃 ¬ 𝑖 ⊊ 𝑗))
Theorem	prmidlsubm 21623	The complement of a prime ideal is multiplicatively closed. Converse of ssdifidlprm 21622. (Contributed by Thierry Arnoux, 6-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑃 ∈ (PrmIdeal‘𝑅))    ⇒   ⊢ (𝜑 → (𝐵 ∖ 𝑃) ∈ (SubMnd‘(mulGrp‘𝑅)))
10.7.4  Principal ideal rings. Divisibility in the integers
Syntax	clpidl 21624	Ring left-principal-ideal function.
class LPIdeal
Syntax	clpir 21625	Class of left principal ideal rings.
class LPIR
Definition	df-lpidl 21626*	Define the class of left principal ideals of a ring, which are ideals with a single generator. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ LPIdeal = (𝑤 ∈ Ring ↦ ∪ 𝑔 ∈ (Base‘𝑤){((RSpan‘𝑤)‘{𝑔})})
Definition	df-lpir 21627	Define the class of left principal ideal rings, rings where every left ideal has a single generator. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ LPIR = {𝑤 ∈ Ring ∣ (LIdeal‘𝑤) = (LPIdeal‘𝑤)}
Theorem	lpival 21628*	Value of the set of principal ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝑃 = (LPIdeal‘𝑅)    &   ⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝑃 = ∪ 𝑔 ∈ 𝐵 {(𝐾‘{𝑔})})
Theorem	islpidl 21629*	Property of being a principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝑃 = (LPIdeal‘𝑅)    &   ⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐼 ∈ 𝑃 ↔ ∃𝑔 ∈ 𝐵 𝐼 = (𝐾‘{𝑔})))
Theorem	lpi0 21630	The zero ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝑃 = (LPIdeal‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → { 0 } ∈ 𝑃)
Theorem	lpi1 21631	The unit ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝑃 = (LPIdeal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐵 ∈ 𝑃)
Theorem	islpir 21632	Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝑃 = (LPIdeal‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑅)    ⇒   ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝑈 = 𝑃))
Theorem	lpiss 21633	Principal ideals are a subclass of ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝑃 = (LPIdeal‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝑃 ⊆ 𝑈)
Theorem	islpir2 21634	Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ 𝑃 = (LPIdeal‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑅)    ⇒   ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝑈 ⊆ 𝑃))
Theorem	lpirring 21635	Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ (𝑅 ∈ LPIR → 𝑅 ∈ Ring)
Theorem	drnglpir 21636	Division rings are principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ LPIR)
Theorem	rspsn 21637*	Membership in principal ideals is closely related to divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝐾‘{𝐺}) = {𝑥 ∣ 𝐺 ∥ 𝑥})
Theorem	lidldvgen 21638*	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 21639*	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 21640	Class of principal ideal domains.
class PID
Definition	df-pid 21641	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 21642	Extend class notation with the class of all pseudometric spaces.
class PsMet
Syntax	cxmet 21643	Extend class notation with the class of all extended metric spaces.
class ∞Met
Syntax	cmet 21644	Extend class notation with the class of all metrics.
class Met
Syntax	cbl 21645	Extend class notation with the metric space ball function.
class ball
Syntax	cfbas 21646	Extend class definition to include the class of filter bases.
class fBas
Syntax	cfg 21647	Extend class definition to include the filter generating function.
class filGen
Syntax	cmopn 21648	Extend class notation with a function mapping each metric space to the family of its open sets.
class MetOpen
Syntax	cmetu 21649	Extend class notation with the function mapping metrics to the uniform structure generated by that metric.
class metUnif
Definition	df-psmet 21650*	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 21651*	Define the set of all extended metrics on a given base set. The definition is similar to df-met 21652, 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 21652*	Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric; see df-ms 24620. 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 24642, metgt0 24658, metsym 24649, and mettri 24651. (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 21653*	Define the metric space ball function. See blval 24685 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 21654	Define a function whose value is the family of open sets of a metric space. See elmopn 24741 for its main property. (Contributed by NM, 1-Sep-2006.)
⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
Definition	df-fbas 21655*	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 21656*	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 21657*	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 21658	Extend class notation with the field of complex numbers.
class ℂfld
Definition	df-cnfld 21659*	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 21661, cnfldadd 21664, cnfldmul 21666, cnfldcj 21667, cnfldtset 21668, cnfldle 21669, cnfldds 21670, and cnfldbas 21662. 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 21660	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 21659. (Revised by GG, 31-Mar-2025.)
⊢ ℂfld Struct ⟨1, ;13⟩
Theorem	cnfldex 21661	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 5327. (Revised by GG, 16-Mar-2025.)
⊢ ℂfld ∈ V
Theorem	cnfldbas 21662	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 21659. (Revised by GG, 31-Mar-2025.)
⊢ ℂ = (Base‘ℂfld)
Theorem	mpocnfldadd 21663*	The addition operation of the field of complex numbers. Version of cnfldadd 21664 using maps-to notation, which does not require ax-addf 11260. (Contributed by GG, 31-Mar-2025.)
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) = (+g‘ℂfld)
Theorem	cnfldadd 21664	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 21659. (Revised by GG, 27-Apr-2025.)
⊢ + = (+g‘ℂfld)
Theorem	mpocnfldmul 21665*	The multiplication operation of the field of complex numbers. Version of cnfldmul 21666 using maps-to notation, which does not require ax-mulf 11261. (Contributed by GG, 31-Mar-2025.)
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (.r‘ℂfld)
Theorem	cnfldmul 21666	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 21659. (Revised by GG, 27-Apr-2025.)
⊢ · = (.r‘ℂfld)
Theorem	cnfldcj 21667	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 21659. (Revised by GG, 31-Mar-2025.)
⊢ ∗ = (*𝑟‘ℂfld)
Theorem	cnfldtset 21668	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 21659. (Revised by GG, 31-Mar-2025.)
⊢ (MetOpen‘(abs ∘ − )) = (TopSet‘ℂfld)
Theorem	cnfldle 21669	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 21659. (Revised by GG, 31-Mar-2025.)
⊢ ≤ = (le‘ℂfld)
Theorem	cnfldds 21670	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 21659. (Revised by GG, 31-Mar-2025.)
⊢ (abs ∘ − ) = (dist‘ℂfld)
Theorem	cnfldunif 21671	The uniform structure component of the complex numbers. (Contributed by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21659. (Revised by GG, 31-Mar-2025.)
⊢ (metUnif‘(abs ∘ − )) = (UnifSet‘ℂfld)
Theorem	cnfldfun 21672	The field of complex numbers is a function. The proof is much shorter than the proof of cnfldfunALT 21673 by using cnfldstr 21660 and structn0fun 17309: in addition, it must be shown that ∅ ∉ ℂfld. (Contributed by AV, 18-Nov-2021.) Revise df-cnfld 21659. (Revised by GG, 31-Mar-2025.)
⊢ Fun ℂfld
Theorem	cnfldfunALT 21673	The field of complex numbers is a function. Alternate proof of cnfldfun 21672 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 21659. (Revised by GG, 31-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Fun ℂfld
Theorem	xrsstr 21674	The extended real structure is a structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ℝ*𝑠 Struct ⟨1, ;12⟩
Theorem	xrsex 21675	The extended real structure is a set. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ℝ*𝑠 ∈ V
Theorem	xrsadd 21676	The addition operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ +𝑒 = (+g‘ℝ*𝑠)
Theorem	xrsmul 21677	The multiplication operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ·e = (.r‘ℝ*𝑠)
Theorem	xrstset 21678	The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ (ordTop‘ ≤ ) = (TopSet‘ℝ*𝑠)
Theorem	cncrng 21679	The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.) Avoid ax-mulf 11261. (Revised by GG, 31-Mar-2025.)
⊢ ℂfld ∈ CRing
Theorem	cnring 21680	The complex numbers form a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ ℂfld ∈ Ring
Theorem	xrsmcmn 21681	The "multiplicative group" of the extended reals is a commutative monoid (even though the "additive group" is not a semigroup, see xrsmgmdifsgrp 21695.) (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ (mulGrp‘ℝ*𝑠) ∈ CMnd
Theorem	cnfld0 21682	Zero is the zero element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 0 = (0g‘ℂfld)
Theorem	cnfld1 21683	One is the unity element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) Avoid ax-mulf 11261. (Revised by GG, 31-Mar-2025.)
⊢ 1 = (1r‘ℂfld)
Theorem	cnfldneg 21684	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 21685	The functionalized addition operation of the field of complex numbers. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ + = (+𝑓‘ℂfld)
Theorem	cnfldsub 21686	The subtraction operator in the field of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ − = (-g‘ℂfld)
Theorem	cndrng 21687	The complex numbers form a division ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) Avoid ax-mulf 11261. (Revised by GG, 30-Apr-2025.)
⊢ ℂfld ∈ DivRing
Theorem	cnflddiv 21688	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 11261. (Revised by GG, 30-Apr-2025.)
⊢ / = (/r‘ℂfld)
Theorem	cnfldinv 21689	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 21690	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 21691	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 21692	The complex numbers form a *-ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ ℂfld ∈ *-Ring
Theorem	xrsmgm 21693	The "additive group" of the extended reals is a magma. (Contributed by AV, 30-Jan-2020.)
⊢ ℝ*𝑠 ∈ Mgm
Theorem	xrsnsgrp 21694	The "additive group" of the extended reals is not a semigroup. (Contributed by AV, 30-Jan-2020.)
⊢ ℝ*𝑠 ∉ Smgrp
Theorem	xrsmgmdifsgrp 21695	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 21681. (Contributed by AV, 30-Jan-2020.)
⊢ ℝ*𝑠 ∈ (Mgm ∖ Smgrp)
Theorem	xrsds 21696*	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 21697	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 21698	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 21699	Lemma for xrsdsreclb 21700. (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 21700	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)))