P. 339 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 33801-33900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	locfinref 33801*	A locally finite refinement of an open cover induces a locally finite open cover with the original index set. This is fact 2 of http://at.yorku.ca/p/a/c/a/02.pdf, it is expressed by exposing a function 𝑓 from the original cover 𝑈, which is taken as the index set. (Contributed by Thierry Arnoux, 31-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝑈)    &   ⊢ (𝜑 → 𝑉 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑉Ref𝑈)    &   ⊢ (𝜑 → 𝑉 ∈ (LocFin‘𝐽))    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
21.3.13.4  Open cover refinement property
Syntax	ccref 33802	The "every open cover has an 𝐴 refinement" predicate.
class CovHasRef𝐴
Definition	df-cref 33803*	Define a statement "every open cover has an 𝐴 refinement" , where 𝐴 is a property for refinements like "finite", "countable", "point finite" or "locally finite". (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)}
Theorem	iscref 33804*	The property that every open cover has an 𝐴 refinement for the topological space 𝐽. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)))
Theorem	crefeq 33805	Equality theorem for the "every open cover has an A refinement" predicate. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ (𝐴 = 𝐵 → CovHasRef𝐴 = CovHasRef𝐵)
Theorem	creftop 33806	A space where every open cover has an 𝐴 refinement is a topological space. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ (𝐽 ∈ CovHasRef𝐴 → 𝐽 ∈ Top)
Theorem	crefi 33807*	The property that every open cover has an 𝐴 refinement for the topological space 𝐽. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶)
Theorem	crefdf 33808*	A formulation of crefi 33807 easier to use for definitions. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐵 = CovHasRef𝐴    &   ⊢ (𝑧 ∈ 𝐴 → 𝜑)    ⇒   ⊢ ((𝐽 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ 𝒫 𝐽(𝜑 ∧ 𝑧Ref𝐶))
Theorem	crefss 33809	The "every open cover has an 𝐴 refinement" predicate respects inclusion. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ (𝐴 ⊆ 𝐵 → CovHasRef𝐴 ⊆ CovHasRef𝐵)
Theorem	cmpcref 33810	Equivalent definition of compact space in terms of open cover refinements. Compact spaces are topologies with finite open cover refinements. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ Comp = CovHasRefFin
Theorem	cmpfiref 33811*	Every open cover of a Compact space has a finite refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Comp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ Fin ∧ 𝑣Ref𝑈))
21.3.13.5  Lindelöf spaces
Syntax	cldlf 33812	Extend class notation with the class of all Lindelöf spaces.
class Ldlf
Definition	df-ldlf 33813	Definition of a Lindelöf space. A Lindelöf space is a topological space in which every open cover has a countable subcover. Definition 1 of [BourbakiTop2] p. 195. (Contributed by Thierry Arnoux, 30-Jan-2020.)
⊢ Ldlf = CovHasRef{𝑥 ∣ 𝑥 ≼ ω}
Theorem	ldlfcntref 33814*	Every open cover of a Lindelöf space has a countable refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Ldlf ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈))
21.3.13.6  Paracompact spaces
Syntax	cpcmp 33815	Extend class notation with the class of all paracompact topologies.
class Paracomp
Definition	df-pcmp 33816	Definition of a paracompact topology. A topology is said to be paracompact iff every open cover has an open refinement that is locally finite. The definition 6 of [BourbakiTop1] p. I.69. also requires the topology to be Hausdorff, but this is dropped here. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ Paracomp = {𝑗 ∣ 𝑗 ∈ CovHasRef(LocFin‘𝑗)}
Theorem	ispcmp 33817	The predicate "is a paracompact topology". (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
Theorem	cmppcmp 33818	Every compact space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ (𝐽 ∈ Comp → 𝐽 ∈ Paracomp)
Theorem	dispcmp 33819	Every discrete space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)
⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Paracomp)
Theorem	pcmplfin 33820*	Given a paracompact topology 𝐽 and an open cover 𝑈, there exists an open refinement 𝑣 that is locally finite. (Contributed by Thierry Arnoux, 31-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))
Theorem	pcmplfinf 33821*	Given a paracompact topology 𝐽 and an open cover 𝑈, there exists an open refinement ran 𝑓 that is locally finite, using the same index as the original cover 𝑈. (Contributed by Thierry Arnoux, 31-Jan-2020.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
21.3.13.7  Spectrum of a ring The prime ideals of a ring 𝑅 can be endowed with the Zariski topology. This is done by defining a function 𝑉 which maps ideals of 𝑅 to closed sets (see for example zarcls0 33828 for the definition of 𝑉). The closed sets of the topology are in the range of 𝑉 (see zartopon 33837). The correspondence with the open sets is made in zarcls 33834. As proved in zart0 33839, the Zariski topology is T0 , but generally not T1 .
Syntax	crspec 33822	Extend class notation with the spectrum of a ring.
class Spec
Definition	df-rspec 33823	Define the spectrum of a ring. (Contributed by Thierry Arnoux, 21-Jan-2024.)
⊢ Spec = (𝑟 ∈ Ring ↦ ((IDLsrg‘𝑟) ↾s (PrmIdeal‘𝑟)))
Theorem	rspecval 33824	Value of the spectrum of the ring 𝑅. Notation 1.1.1 of [EGA] p. 80. (Contributed by Thierry Arnoux, 2-Jun-2024.)
⊢ (𝑅 ∈ Ring → (Spec‘𝑅) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅)))
Theorem	rspecbas 33825	The prime ideals form the base of the spectrum of a ring. (Contributed by Thierry Arnoux, 2-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) = (Base‘𝑆))
Theorem	rspectset 33826*	Topology component of the spectrum of a ring. (Contributed by Thierry Arnoux, 2-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ 𝐽 = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})    ⇒   ⊢ (𝑅 ∈ Ring → 𝐽 = (TopSet‘𝑆))
Theorem	rspectopn 33827*	The topology component of the spectrum of a ring. (Contributed by Thierry Arnoux, 4-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ 𝑃 = (PrmIdeal‘𝑅)    &   ⊢ 𝐽 = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗})    ⇒   ⊢ (𝑅 ∈ Ring → 𝐽 = (TopOpen‘𝑆))
Theorem	zarcls0 33828*	The closure of the identity ideal in the Zariski topology. Proposition 1.1.2(i) of [EGA] p. 80. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})    &   ⊢ 𝑃 = (PrmIdeal‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝑉‘{ 0 }) = 𝑃)
Theorem	zarcls1 33829*	The unit ideal 𝐵 is the only ideal whose closure in the Zariski topology is the empty set. Stronger form of the Proposition 1.1.2(i) of [EGA] p. 80. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ((𝑉‘𝐼) = ∅ ↔ 𝐼 = 𝐵))
Theorem	zarclsun 33830*	The union of two closed sets of the Zariski topology is closed. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉 ∧ 𝑌 ∈ ran 𝑉) → (𝑋 ∪ 𝑌) ∈ ran 𝑉)
Theorem	zarclsiin 33831*	In a Zariski topology, the intersection of the closures of a family of ideals is the closure of the span of their union. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})    &   ⊢ 𝐾 = (RSpan‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) → ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙) = (𝑉‘(𝐾‘∪ 𝑇)))
Theorem	zarclsint 33832*	The intersection of a family of closed sets is closed in the Zariski topology. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ ran 𝑉)
Theorem	zarclssn 33833*	The closed points of Zariski topology are the maximal ideals. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})    &   ⊢ 𝐵 = (LIdeal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ({𝑀} = (𝑉‘𝑀) ↔ 𝑀 ∈ (MaxIdeal‘𝑅)))
Theorem	zarcls 33834*	The open sets of the Zariski topology are the complements of the closed sets. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    &   ⊢ 𝑃 = (PrmIdeal‘𝑅)    &   ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗})    ⇒   ⊢ (𝑅 ∈ Ring → 𝐽 = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉})
Theorem	zartopn 33835*	The Zariski topology is a topology, and its closed sets are images by 𝑉 of the ideals of 𝑅. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    &   ⊢ 𝑃 = (PrmIdeal‘𝑅)    &   ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗})    ⇒   ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘𝑃) ∧ ran 𝑉 = (Clsd‘𝐽)))
Theorem	zartop 33836	The Zariski topology is a topology. Proposition 1.1.2 of [EGA] p. 80. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Top)
Theorem	zartopon 33837	The points of the Zariski topology are the prime ideals. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    &   ⊢ 𝑃 = (PrmIdeal‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐽 ∈ (TopOn‘𝑃))
Theorem	zar0ring 33838	The Zariski Topology of the trivial ring. (Contributed by Thierry Arnoux, 1-Jul-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 𝐽 = {∅})
Theorem	zart0 33839	The Zariski topology is T0 . Corollary 1.1.8 of [EGA] p. 81. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Kol2)
Theorem	zarmxt1 33840	The Zariski topology restricted to maximal ideals is T1 . (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    &   ⊢ 𝑀 = (MaxIdeal‘𝑅)    &   ⊢ 𝑇 = (𝐽 ↾t 𝑀)    ⇒   ⊢ (𝑅 ∈ CRing → 𝑇 ∈ Fre)
Theorem	zarcmplem 33841*	Lemma for zarcmp 33842. (Contributed by Thierry Arnoux, 2-Jul-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    &   ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})    ⇒   ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Comp)
Theorem	zarcmp 33842	The Zariski topology is compact. Proposition 1.1.10(ii) of [EGA], p. 82. (Contributed by Thierry Arnoux, 2-Jul-2024.)
⊢ 𝑆 = (Spec‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑆)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Comp)
Theorem	rspectps 33843	The spectrum of a ring 𝑅 is a topological space. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑆 = (Spec‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → 𝑆 ∈ TopSp)
Theorem	rhmpreimacnlem 33844*	Lemma for rhmpreimacn 33845. (Contributed by Thierry Arnoux, 7-Jul-2024.)
⊢ 𝑇 = (Spec‘𝑅)    &   ⊢ 𝑈 = (Spec‘𝑆)    &   ⊢ 𝐴 = (PrmIdeal‘𝑅)    &   ⊢ 𝐵 = (PrmIdeal‘𝑆)    &   ⊢ 𝐽 = (TopOpen‘𝑇)    &   ⊢ 𝐾 = (TopOpen‘𝑈)    &   ⊢ 𝐺 = (𝑖 ∈ 𝐵 ↦ (◡𝐹 “ 𝑖))    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))    &   ⊢ (𝜑 → ran 𝐹 = (Base‘𝑆))    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   ⊢ 𝑉 = (𝑗 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑗 ⊆ 𝑘})    &   ⊢ 𝑊 = (𝑗 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑗 ⊆ 𝑘})    ⇒   ⊢ (𝜑 → (𝑊‘(𝐹 “ 𝐼)) = (◡𝐺 “ (𝑉‘𝐼)))
Theorem	rhmpreimacn 33845*	The function mapping a prime ideal to its preimage by a surjective ring homomorphism is continuous, when considering the Zariski topology. Corollary 1.2.3 of [EGA], p. 83. Notice that the direction of the continuous map 𝐺 is reverse: the original ring homomorphism 𝐹 goes from 𝑅 to 𝑆, but the continuous map 𝐺 goes from 𝐵 to 𝐴. This mapping is also called "induced map on prime spectra" or "pullback on primes". (Contributed by Thierry Arnoux, 8-Jul-2024.)
⊢ 𝑇 = (Spec‘𝑅)    &   ⊢ 𝑈 = (Spec‘𝑆)    &   ⊢ 𝐴 = (PrmIdeal‘𝑅)    &   ⊢ 𝐵 = (PrmIdeal‘𝑆)    &   ⊢ 𝐽 = (TopOpen‘𝑇)    &   ⊢ 𝐾 = (TopOpen‘𝑈)    &   ⊢ 𝐺 = (𝑖 ∈ 𝐵 ↦ (◡𝐹 “ 𝑖))    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))    &   ⊢ (𝜑 → ran 𝐹 = (Base‘𝑆))    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))
21.3.13.8  Pseudometrics
Syntax	cmetid 33846	Extend class notation with the class of metric identifications.
class ~Met
Syntax	cpstm 33847	Extend class notation with the metric induced by a pseudometric.
class pstoMet
Definition	df-metid 33848*	Define the metric identification relation for a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ ~Met = (𝑑 ∈ ∪ ran PsMet ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ dom dom 𝑑 ∧ 𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0)})
Definition	df-pstm 33849*	Define the metric induced by a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ pstoMet = (𝑑 ∈ ∪ ran PsMet ↦ (𝑎 ∈ (dom dom 𝑑 / (~Met‘𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met‘𝑑)) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)}))
Theorem	metidval 33850*	Value of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑥𝐷𝑦) = 0)})
Theorem	metidss 33851	As a relation, the metric identification is a subset of a Cartesian product. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) ⊆ (𝑋 × 𝑋))
Theorem	metidv 33852	𝐴 and 𝐵 identify by the metric 𝐷 if their distance is zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴(~Met‘𝐷)𝐵 ↔ (𝐴𝐷𝐵) = 0))
Theorem	metideq 33853	Basic property of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐸) = (𝐵𝐷𝐹))
Theorem	metider 33854	The metric identification is an equivalence relation. (Contributed by Thierry Arnoux, 11-Feb-2018.)
⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) Er 𝑋)
Theorem	pstmval 33855*	Value of the metric induced by a pseudometric 𝐷. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ ∼ = (~Met‘𝐷)    ⇒   ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑎 ∈ (𝑋 / ∼ ), 𝑏 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝐷𝑦)}))
Theorem	pstmfval 33856	Function value of the metric induced by a pseudometric 𝐷 (Contributed by Thierry Arnoux, 11-Feb-2018.)
⊢ ∼ = (~Met‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ([𝐴] ∼ (pstoMet‘𝐷)[𝐵] ∼ ) = (𝐴𝐷𝐵))
Theorem	pstmxmet 33857	The metric induced by a pseudometric is a full-fledged metric on the equivalence classes of the metric identification. (Contributed by Thierry Arnoux, 11-Feb-2018.)
⊢ ∼ = (~Met‘𝐷)    ⇒   ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) ∈ (∞Met‘(𝑋 / ∼ )))
21.3.13.9  Continuity - misc additions
Theorem	hauseqcn 33858	In a Hausdorff topology, two continuous functions which agree on a dense set agree everywhere. (Contributed by Thierry Arnoux, 28-Dec-2017.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝐾 ∈ Haus)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑋)    &   ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)    ⇒   ⊢ (𝜑 → 𝐹 = 𝐺)
21.3.13.10  Topology of the closed unit interval
Theorem	elunitge0 33859	An element of the closed unit interval is positive. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 20-Dec-2016.)
⊢ (𝐴 ∈ (0[,]1) → 0 ≤ 𝐴)
Theorem	unitssxrge0 33860	The closed unit interval is a subset of the set of the extended nonnegative reals. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 12-Dec-2016.)
⊢ (0[,]1) ⊆ (0[,]+∞)
Theorem	unitdivcld 33861	Necessary conditions for a quotient to be in the closed unit interval. (somewhat too strong, it would be sufficient that A and B are in RR+) (Contributed by Thierry Arnoux, 20-Dec-2016.)
⊢ ((𝐴 ∈ (0[,]1) ∧ 𝐵 ∈ (0[,]1) ∧ 𝐵 ≠ 0) → (𝐴 ≤ 𝐵 ↔ (𝐴 / 𝐵) ∈ (0[,]1)))
Theorem	iistmd 33862	The closed unit interval forms a topological monoid under multiplication. (Contributed by Thierry Arnoux, 25-Mar-2017.)
⊢ 𝐼 = ((mulGrp‘ℂfld) ↾s (0[,]1))    ⇒   ⊢ 𝐼 ∈ TopMnd
21.3.13.11  Topology of ` ( RR X. RR ) `
Theorem	unicls 33863	The union of the closed set is the underlying set of the topology. (Contributed by Thierry Arnoux, 21-Sep-2017.)
⊢ 𝐽 ∈ Top    &   ⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ∪ (Clsd‘𝐽) = 𝑋
Theorem	tpr2tp 33864	The usual topology on (ℝ × ℝ) is the product topology of the usual topology on ℝ. (Contributed by Thierry Arnoux, 21-Sep-2017.)
⊢ 𝐽 = (topGen‘ran (,))    ⇒   ⊢ (𝐽 ×t 𝐽) ∈ (TopOn‘(ℝ × ℝ))
Theorem	tpr2uni 33865	The usual topology on (ℝ × ℝ) is the product topology of the usual topology on ℝ. (Contributed by Thierry Arnoux, 21-Sep-2017.)
⊢ 𝐽 = (topGen‘ran (,))    ⇒   ⊢ ∪ (𝐽 ×t 𝐽) = (ℝ × ℝ)
Theorem	xpinpreima 33866	Rewrite the cartesian product of two sets as the intersection of their preimage by 1st and 2nd, the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.)
⊢ (𝐴 × 𝐵) = ((◡(1st ↾ (V × V)) “ 𝐴) ∩ (◡(2nd ↾ (V × V)) “ 𝐵))
Theorem	xpinpreima2 33867	Rewrite the cartesian product of two sets as the intersection of their preimage by 1st and 2nd, the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.)
⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → (𝐴 × 𝐵) = ((◡(1st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ (◡(2nd ↾ (𝐸 × 𝐹)) “ 𝐵)))
Theorem	sqsscirc1 33868	The complex square of side 𝐷 is a subset of the complex circle of radius 𝐷. (Contributed by Thierry Arnoux, 25-Sep-2017.)
⊢ ((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ+) → ((𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2)) → (√‘((𝑋↑2) + (𝑌↑2))) < 𝐷))
Theorem	sqsscirc2 33869	The complex square of side 𝐷 is a subset of the complex disc of radius 𝐷. (Contributed by Thierry Arnoux, 25-Sep-2017.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐷 ∈ ℝ+) → (((abs‘(ℜ‘(𝐵 − 𝐴))) < (𝐷 / 2) ∧ (abs‘(ℑ‘(𝐵 − 𝐴))) < (𝐷 / 2)) → (abs‘(𝐵 − 𝐴)) < 𝐷))
Theorem	cnre2csqlem 33870*	Lemma for cnre2csqima 33871. (Contributed by Thierry Arnoux, 27-Sep-2017.)
⊢ (𝐺 ↾ (ℝ × ℝ)) = (𝐻 ∘ 𝐹)    &   ⊢ 𝐹 Fn (ℝ × ℝ)    &   ⊢ 𝐺 Fn V    &   ⊢ (𝑥 ∈ (ℝ × ℝ) → (𝐺‘𝑥) ∈ ℝ)    &   ⊢ ((𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐹) → (𝐻‘(𝑥 − 𝑦)) = ((𝐻‘𝑥) − (𝐻‘𝑦)))    ⇒   ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ (◡(𝐺 ↾ (ℝ × ℝ)) “ (((𝐺‘𝑋) − 𝐷)(,)((𝐺‘𝑋) + 𝐷))) → (abs‘(𝐻‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷))
Theorem	cnre2csqima 33871*	Image of a centered square by the canonical bijection from (ℝ × ℝ) to ℂ. (Contributed by Thierry Arnoux, 27-Sep-2017.)
⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))    ⇒   ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ ((((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷)) × (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷))) → ((abs‘(ℜ‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷 ∧ (abs‘(ℑ‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷)))
Theorem	tpr2rico 33872*	For any point of an open set of the usual topology on (ℝ × ℝ) there is an open square which contains that point and is entirely in the open set. This is square is actually a ball by the (𝑙↑+∞) norm 𝑋. (Contributed by Thierry Arnoux, 21-Sep-2017.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐺 = (𝑢 ∈ ℝ, 𝑣 ∈ ℝ ↦ (𝑢 + (i · 𝑣)))    &   ⊢ 𝐵 = ran (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦))    ⇒   ⊢ ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑟 ∈ 𝐵 (𝑋 ∈ 𝑟 ∧ 𝑟 ⊆ 𝐴))
21.3.13.12  Order topology - misc. additions
Theorem	cnvordtrestixx 33873*	The restriction of the 'greater than' order to an interval gives the same topology as the subspace topology. (Contributed by Thierry Arnoux, 1-Apr-2017.)
⊢ 𝐴 ⊆ ℝ*    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥[,]𝑦) ⊆ 𝐴)    ⇒   ⊢ ((ordTop‘ ≤ ) ↾t 𝐴) = (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴)))
Theorem	prsdm 33874	Domain of the relation of a proset. (Contributed by Thierry Arnoux, 11-Sep-2015.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    ⇒   ⊢ (𝐾 ∈ Proset → dom ≤ = 𝐵)
Theorem	prsrn 33875	Range of the relation of a proset. (Contributed by Thierry Arnoux, 11-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    ⇒   ⊢ (𝐾 ∈ Proset → ran ≤ = 𝐵)
Theorem	prsss 33876	Relation of a subproset. (Contributed by Thierry Arnoux, 13-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    ⇒   ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
Theorem	prsssdm 33877	Domain of a subproset relation. (Contributed by Thierry Arnoux, 12-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    ⇒   ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) = 𝐴)
Theorem	ordtprsval 33878*	Value of the order topology for a proset. (Contributed by Thierry Arnoux, 11-Sep-2015.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    &   ⊢ 𝐸 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥})    &   ⊢ 𝐹 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})    ⇒   ⊢ (𝐾 ∈ Proset → (ordTop‘ ≤ ) = (topGen‘(fi‘({𝐵} ∪ (𝐸 ∪ 𝐹)))))
Theorem	ordtprsuni 33879*	Value of the order topology. (Contributed by Thierry Arnoux, 13-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    &   ⊢ 𝐸 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥})    &   ⊢ 𝐹 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})    ⇒   ⊢ (𝐾 ∈ Proset → 𝐵 = ∪ ({𝐵} ∪ (𝐸 ∪ 𝐹)))
Theorem	ordtcnvNEW 33880	The order dual generates the same topology as the original order. (Contributed by Mario Carneiro, 3-Sep-2015.) (Revised by Thierry Arnoux, 13-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    ⇒   ⊢ (𝐾 ∈ Proset → (ordTop‘◡ ≤ ) = (ordTop‘ ≤ ))
Theorem	ordtrestNEW 33881	The subspace topology of an order topology is in general finer than the topology generated by the restricted order, but we do have inclusion in one direction. (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    ⇒   ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
Theorem	ordtrest2NEWlem 33882*	Lemma for ordtrest2NEW 33883. (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    &   ⊢ (𝜑 → 𝐾 ∈ Toset)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝐵 ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴)    ⇒   ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
Theorem	ordtrest2NEW 33883*	An interval-closed set 𝐴 in a total order has the same subspace topology as the restricted order topology. (An interval-closed set is the same thing as an open or half-open or closed interval in ℝ, but in other sets like ℚ there are interval-closed sets like (π, +∞) ∩ ℚ that are not intervals.) (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    &   ⊢ (𝜑 → 𝐾 ∈ Toset)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝐵 ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴)    ⇒   ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) = ((ordTop‘ ≤ ) ↾t 𝐴))
Theorem	ordtconnlem1 33884*	Connectedness in the order topology of a toset. This is the "easy" direction of ordtconn 33885. See also reconnlem1 24861. (Contributed by Thierry Arnoux, 14-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    &   ⊢ 𝐽 = (ordTop‘ ≤ )    ⇒   ⊢ ((𝐾 ∈ Toset ∧ 𝐴 ⊆ 𝐵) → ((𝐽 ↾t 𝐴) ∈ Conn → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑟 ∈ 𝐵 ((𝑥 ≤ 𝑟 ∧ 𝑟 ≤ 𝑦) → 𝑟 ∈ 𝐴)))
Theorem	ordtconn 33885	Connectedness in the order topology of a complete uniform totally ordered space. (Contributed by Thierry Arnoux, 15-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    &   ⊢ 𝐽 = (ordTop‘ ≤ )    ⇒   ⊢ ⊤
21.3.13.13  Continuity in topological spaces - misc. additions
Theorem	mndpluscn 33886*	A mapping that is both a homeomorphism and a monoid homomorphism preserves the "continuousness" of the operation. (Contributed by Thierry Arnoux, 25-Mar-2017.)
⊢ 𝐹 ∈ (𝐽Homeo𝐾)    &   ⊢ + :(𝐵 × 𝐵)⟶𝐵    &   ⊢ ∗ :(𝐶 × 𝐶)⟶𝐶    &   ⊢ 𝐽 ∈ (TopOn‘𝐵)    &   ⊢ 𝐾 ∈ (TopOn‘𝐶)    &   ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))    &   ⊢ + ∈ ((𝐽 ×t 𝐽) Cn 𝐽)    ⇒   ⊢ ∗ ∈ ((𝐾 ×t 𝐾) Cn 𝐾)
Theorem	mhmhmeotmd 33887	Deduce a Topological Monoid using mapping that is both a homeomorphism and a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.)
⊢ 𝐹 ∈ (𝑆 MndHom 𝑇)    &   ⊢ 𝐹 ∈ ((TopOpen‘𝑆)Homeo(TopOpen‘𝑇))    &   ⊢ 𝑆 ∈ TopMnd    &   ⊢ 𝑇 ∈ TopSp    ⇒   ⊢ 𝑇 ∈ TopMnd
Theorem	rmulccn 33888*	Multiplication by a real constant is a continuous function. (Contributed by Thierry Arnoux, 23-May-2017.) Avoid ax-mulf 11232. (Revised by GG, 16-Mar-2025.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ (𝑥 · 𝐶)) ∈ (𝐽 Cn 𝐽))
Theorem	raddcn 33889*	Addition in the real numbers is a continuous function. (Contributed by Thierry Arnoux, 23-May-2017.)
⊢ 𝐽 = (topGen‘ran (,))    ⇒   ⊢ (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
Theorem	xrmulc1cn 33890*	The operation multiplying an extended real number by a nonnegative constant is continuous. (Contributed by Thierry Arnoux, 5-Jul-2017.)
⊢ 𝐽 = (ordTop‘ ≤ )    &   ⊢ 𝐹 = (𝑥 ∈ ℝ* ↦ (𝑥 ·e 𝐶))    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐽))
Theorem	fmcncfil 33891	The image of a Cauchy filter by a continuous filter map is a Cauchy filter. (Contributed by Thierry Arnoux, 12-Nov-2017.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝐾 = (MetOpen‘𝐸)    ⇒   ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝐸 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝐵 ∈ (CauFil‘𝐷)) → ((𝑌 FilMap 𝐹)‘𝐵) ∈ (CauFil‘𝐸))
21.3.13.14  Topology of the extended nonnegative real numbers ordered monoid
Theorem	xrge0hmph 33892	The extended nonnegative reals are homeomorphic to the closed unit interval. (Contributed by Thierry Arnoux, 24-Mar-2017.)
⊢ II ≃ ((ordTop‘ ≤ ) ↾t (0[,]+∞))
Theorem	xrge0iifcnv 33893*	Define a bijection from [0, 1] onto [0, +∞]. (Contributed by Thierry Arnoux, 29-Mar-2017.)
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))    ⇒   ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦))))
Theorem	xrge0iifcv 33894*	The defined function's value in the real. (Contributed by Thierry Arnoux, 1-Apr-2017.)
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))    ⇒   ⊢ (𝑋 ∈ (0(,]1) → (𝐹‘𝑋) = -(log‘𝑋))
Theorem	xrge0iifiso 33895*	The defined bijection from the closed unit interval onto the extended nonnegative reals is an order isomorphism. (Contributed by Thierry Arnoux, 31-Mar-2017.)
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))    ⇒   ⊢ 𝐹 Isom < , ◡ < ((0[,]1), (0[,]+∞))
Theorem	xrge0iifhmeo 33896*	Expose a homeomorphism from the closed unit interval to the extended nonnegative reals. (Contributed by Thierry Arnoux, 1-Apr-2017.)
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))    &   ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞))    ⇒   ⊢ 𝐹 ∈ (IIHomeo𝐽)
Theorem	xrge0iifhom 33897*	The defined function from the closed unit interval to the extended nonnegative reals is a monoid homomorphism. (Contributed by Thierry Arnoux, 5-Apr-2017.)
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))    &   ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞))    ⇒   ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0[,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) +𝑒 (𝐹‘𝑌)))
Theorem	xrge0iif1 33898*	Condition for the defined function, -(log‘𝑥) to be a monoid homomorphism. (Contributed by Thierry Arnoux, 20-Jun-2017.)
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))    &   ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞))    ⇒   ⊢ (𝐹‘1) = 0
Theorem	xrge0iifmhm 33899*	The defined function from the closed unit interval to the extended nonnegative reals is a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.)
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))    &   ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞))    ⇒   ⊢ 𝐹 ∈ (((mulGrp‘ℂfld) ↾s (0[,]1)) MndHom (ℝ*𝑠 ↾s (0[,]+∞)))
Theorem	xrge0pluscn 33900*	The addition operation of the extended nonnegative real numbers monoid is continuous. (Contributed by Thierry Arnoux, 24-Mar-2017.)
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))    &   ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞))    &   ⊢ + = ( +𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))    ⇒   ⊢ + ∈ ((𝐽 ×t 𝐽) Cn 𝐽)