P. 237 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 23601-23700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mopntopon 23601	The set of open sets of a metric space 𝑋 is a topology on 𝑋. Remark in [Kreyszig] p. 19. This theorem connects the two concepts and makes available the theorems for topologies for use with metric spaces. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
Theorem	mopntop 23602	The set of open sets of a metric space is a topology. (Contributed by NM, 28-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
Theorem	mopnuni 23603	The union of all open sets in a metric space is its underlying set. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
Theorem	elmopn 23604*	The defining property of an open set of a metric space. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∈ 𝐽 ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))))
Theorem	mopnfss 23605	The family of open sets of a metric space is a collection of subsets of the base set. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ⊆ 𝒫 𝑋)
Theorem	mopnm 23606	The base set of a metric space is open. Part of Theorem T1 of [Kreyszig] p. 19. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ 𝐽)
Theorem	elmopn2 23607*	A defining property of an open set of a metric space. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∈ 𝐽 ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ℝ+ (𝑥(ball‘𝐷)𝑦) ⊆ 𝐴)))
Theorem	mopnss 23608	An open set of a metric space is a subspace of its base set. (Contributed by NM, 3-Sep-2006.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋)
Theorem	isxms 23609	Express the predicate "⟨𝑋, 𝐷⟩ is an extended metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐽 = (TopOpen‘𝐾)    &   ⊢ 𝑋 = (Base‘𝐾)    &   ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)))
Theorem	isxms2 23610	Express the predicate "⟨𝑋, 𝐷⟩ is an extended metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐽 = (TopOpen‘𝐾)    &   ⊢ 𝑋 = (Base‘𝐾)    &   ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐽 = (MetOpen‘𝐷)))
Theorem	isms 23611	Express the predicate "⟨𝑋, 𝐷⟩ is a metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
⊢ 𝐽 = (TopOpen‘𝐾)    &   ⊢ 𝑋 = (Base‘𝐾)    &   ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ 𝐷 ∈ (Met‘𝑋)))
Theorem	isms2 23612	Express the predicate "⟨𝑋, 𝐷⟩ is a metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
⊢ 𝐽 = (TopOpen‘𝐾)    &   ⊢ 𝑋 = (Base‘𝐾)    &   ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝐾 ∈ MetSp ↔ (𝐷 ∈ (Met‘𝑋) ∧ 𝐽 = (MetOpen‘𝐷)))
Theorem	xmstopn 23613	The topology component of an extended metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ 𝐽 = (TopOpen‘𝐾)    &   ⊢ 𝑋 = (Base‘𝐾)    &   ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))
Theorem	mstopn 23614	The topology component of a metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ 𝐽 = (TopOpen‘𝐾)    &   ⊢ 𝑋 = (Base‘𝐾)    &   ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝐾 ∈ MetSp → 𝐽 = (MetOpen‘𝐷))
Theorem	xmstps 23615	An extended metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp)
Theorem	msxms 23616	A metric space is an extended metric space. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)
Theorem	mstps 23617	A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ (𝑀 ∈ MetSp → 𝑀 ∈ TopSp)
Theorem	xmsxmet 23618	The distance function, suitably truncated, is an extended metric on 𝑋. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝑀 ∈ ∞MetSp → 𝐷 ∈ (∞Met‘𝑋))
Theorem	msmet 23619	The distance function, suitably truncated, is a metric on 𝑋. (Contributed by Mario Carneiro, 12-Nov-2013.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝑀 ∈ MetSp → 𝐷 ∈ (Met‘𝑋))
Theorem	msf 23620	The distance function of a metric space is a function into the real numbers. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝑀 ∈ MetSp → 𝐷:(𝑋 × 𝑋)⟶ℝ)
Theorem	xmsxmet2 23621	The distance function, suitably truncated, is an extended metric on 𝑋. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ (𝑀 ∈ ∞MetSp → (𝐷 ↾ (𝑋 × 𝑋)) ∈ (∞Met‘𝑋))
Theorem	msmet2 23622	The distance function, suitably truncated, is a metric on 𝑋. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ (𝑀 ∈ MetSp → (𝐷 ↾ (𝑋 × 𝑋)) ∈ (Met‘𝑋))
Theorem	mscl 23623	Closure of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ)
Theorem	xmscl 23624	Closure of the distance function of an extended metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*)
Theorem	xmsge0 23625	The distance function in an extended metric space is nonnegative. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵))
Theorem	xmseq0 23626	The distance between two points in an extended metric space is zero iff the two points are identical. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵))
Theorem	xmssym 23627	The distance function in an extended metric space is symmetric. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))
Theorem	xmstri2 23628	Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ∞MetSp ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))
Theorem	mstri2 23629	Triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ MetSp ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) + (𝐶𝐷𝐵)))
Theorem	xmstri 23630	Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ∞MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐶𝐷𝐵)))
Theorem	mstri 23631	Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) + (𝐶𝐷𝐵)))
Theorem	xmstri3 23632	Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ∞MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐵𝐷𝐶)))
Theorem	mstri3 23633	Triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) + (𝐵𝐷𝐶)))
Theorem	msrtri 23634	Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (abs‘((𝐴𝐷𝐶) − (𝐵𝐷𝐶))) ≤ (𝐴𝐷𝐵))
Theorem	xmspropd 23635	Property deduction for an extended metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))    &   ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))    ⇒   ⊢ (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐿 ∈ ∞MetSp))
Theorem	mspropd 23636	Property deduction for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))    &   ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))    ⇒   ⊢ (𝜑 → (𝐾 ∈ MetSp ↔ 𝐿 ∈ MetSp))
Theorem	setsmsbas 23637	The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Proof shortened by AV, 12-Nov-2024.)
⊢ (𝜑 → 𝑋 = (Base‘𝑀))    &   ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    ⇒   ⊢ (𝜑 → 𝑋 = (Base‘𝐾))
Theorem	setsmsbasOLD 23638	Obsolete proof of setsmsbas 23637 as of 12-Nov-2024. The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝑋 = (Base‘𝑀))    &   ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    ⇒   ⊢ (𝜑 → 𝑋 = (Base‘𝐾))
Theorem	setsmsds 23639	The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Proof shortened by AV, 11-Nov-2024.)
⊢ (𝜑 → 𝑋 = (Base‘𝑀))    &   ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    ⇒   ⊢ (𝜑 → (dist‘𝑀) = (dist‘𝐾))
Theorem	setsmsdsOLD 23640	Obsolete proof of setsmsds 23639 as of 11-Nov-2024. The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝑋 = (Base‘𝑀))    &   ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    ⇒   ⊢ (𝜑 → (dist‘𝑀) = (dist‘𝐾))
Theorem	setsmstset 23641	The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ (𝜑 → 𝑋 = (Base‘𝑀))    &   ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    &   ⊢ (𝜑 → 𝑀 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (MetOpen‘𝐷) = (TopSet‘𝐾))
Theorem	setsmstopn 23642	The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ (𝜑 → 𝑋 = (Base‘𝑀))    &   ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    &   ⊢ (𝜑 → 𝑀 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (MetOpen‘𝐷) = (TopOpen‘𝐾))
Theorem	setsxms 23643	The constructed metric space is a metric space iff the provided distance function is a metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ (𝜑 → 𝑋 = (Base‘𝑀))    &   ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    &   ⊢ (𝜑 → 𝑀 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐷 ∈ (∞Met‘𝑋)))
Theorem	setsms 23644	The constructed metric space is a metric space iff the provided distance function is a metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ (𝜑 → 𝑋 = (Base‘𝑀))    &   ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    &   ⊢ (𝜑 → 𝑀 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐾 ∈ MetSp ↔ 𝐷 ∈ (Met‘𝑋)))
Theorem	tmsval 23645	For any metric there is an associated metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝑀 = {⟨(Base‘ndx), 𝑋⟩, ⟨(dist‘ndx), 𝐷⟩}    &   ⊢ 𝐾 = (toMetSp‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))
Theorem	tmslem 23646	Lemma for tmsbas 23648, tmsds 23649, and tmstopn 23650. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝑀 = {⟨(Base‘ndx), 𝑋⟩, ⟨(dist‘ndx), 𝐷⟩}    &   ⊢ 𝐾 = (toMetSp‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝐷 = (dist‘𝐾) ∧ (MetOpen‘𝐷) = (TopOpen‘𝐾)))
Theorem	tmslemOLD 23647	Obsolete version of tmslem 23646 as of 28-Oct-2024. Lemma for tmsbas 23648, tmsds 23649, and tmstopn 23650. (Contributed by Mario Carneiro, 2-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑀 = {⟨(Base‘ndx), 𝑋⟩, ⟨(dist‘ndx), 𝐷⟩}    &   ⊢ 𝐾 = (toMetSp‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝐷 = (dist‘𝐾) ∧ (MetOpen‘𝐷) = (TopOpen‘𝐾)))
Theorem	tmsbas 23648	The base set of a constructed metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐾 = (toMetSp‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝐾))
Theorem	tmsds 23649	The metric of a constructed metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐾 = (toMetSp‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝐾))
Theorem	tmstopn 23650	The topology of a constructed metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐾 = (toMetSp‘𝐷)    &   ⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 = (TopOpen‘𝐾))
Theorem	tmsxms 23651	The constructed metric space is an extended metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐾 = (toMetSp‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 ∈ ∞MetSp)
Theorem	tmsms 23652	The constructed metric space is a metric space given a metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐾 = (toMetSp‘𝐷)    ⇒   ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐾 ∈ MetSp)
Theorem	imasf1obl 23653	The image of a metric space ball. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))    &   ⊢ 𝐷 = (dist‘𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉))    &   ⊢ (𝜑 → 𝑃 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ ℝ*)    ⇒   ⊢ (𝜑 → ((𝐹‘𝑃)(ball‘𝐷)𝑆) = (𝐹 “ (𝑃(ball‘𝐸)𝑆)))
Theorem	imasf1oxms 23654	The image of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ ∞MetSp)    ⇒   ⊢ (𝜑 → 𝑈 ∈ ∞MetSp)
Theorem	imasf1oms 23655	The image of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ MetSp)    ⇒   ⊢ (𝜑 → 𝑈 ∈ MetSp)
Theorem	prdsbl 23656*	A ball in the product metric for finite index set is the Cartesian product of balls in all coordinates. For infinite index set this is no longer true; instead the correct statement is that a *closed ball* is the product of closed balls in each coordinate (where closed ball means a set of the form in blcld 23670) - for a counterexample the point 𝑝 in ℝ↑ℕ whose 𝑛-th coordinate is 1 − 1 / 𝑛 is in X𝑛 ∈ ℕball(0, 1) but is not in the 1-ball of the product (since 𝑑(0, 𝑝) = 1). The last assumption, 0 < 𝐴, is needed only in the case 𝐼 = ∅, when the right side evaluates to {∅} and the left evaluates to ∅ if 𝐴 ≤ 0 and {∅} if 0 < 𝐴. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅))    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝑉 = (Base‘𝑅)    &   ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))    &   ⊢ 𝐷 = (dist‘𝑌)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ 𝑍)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 0 < 𝐴)    ⇒   ⊢ (𝜑 → (𝑃(ball‘𝐷)𝐴) = X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴))
Theorem	mopni 23657*	An open set of a metric space includes a ball around each of its points. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴) → ∃𝑥 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴))
Theorem	mopni2 23658*	An open set of a metric space includes a ball around each of its points. (Contributed by NM, 2-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴)
Theorem	mopni3 23659*	An open set of a metric space includes an arbitrarily small ball around each of its points. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴) ∧ 𝑅 ∈ ℝ+) → ∃𝑥 ∈ ℝ+ (𝑥 < 𝑅 ∧ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴))
Theorem	blssopn 23660	The balls of a metric space are open sets. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → ran (ball‘𝐷) ⊆ 𝐽)
Theorem	unimopn 23661	The union of a collection of open sets of a metric space is open. Theorem T2 of [Kreyszig] p. 19. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ⊆ 𝐽) → ∪ 𝐴 ∈ 𝐽)
Theorem	mopnin 23662	The intersection of two open sets of a metric space is open. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽)
Theorem	mopn0 23663	The empty set is an open set of a metric space. Part of Theorem T1 of [Kreyszig] p. 19. (Contributed by NM, 4-Sep-2006.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∅ ∈ 𝐽)
Theorem	rnblopn 23664	A ball of a metric space is an open set. (Contributed by NM, 12-Sep-2006.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ ran (ball‘𝐷)) → 𝐵 ∈ 𝐽)
Theorem	blopn 23665	A ball of a metric space is an open set. (Contributed by NM, 9-Mar-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ 𝐽)
Theorem	neibl 23666*	The neighborhoods around a point 𝑃 of a metric space are those subsets containing a ball around 𝑃. Definition of neighborhood in [Kreyszig] p. 19. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑟 ∈ ℝ+ (𝑃(ball‘𝐷)𝑟) ⊆ 𝑁)))
Theorem	blnei 23667	A ball around a point is a neighborhood of the point. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → (𝑃(ball‘𝐷)𝑅) ∈ ((nei‘𝐽)‘{𝑃}))
Theorem	lpbl 23668*	Every ball around a limit point 𝑃 of a subset 𝑆 includes a member of 𝑆 (even if 𝑃 ∉ 𝑆). (Contributed by NM, 9-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → ∃𝑥 ∈ 𝑆 𝑥 ∈ (𝑃(ball‘𝐷)𝑅))
Theorem	blsscls2 23669*	A smaller closed ball is contained in a larger open ball. (Contributed by Mario Carneiro, 10-Jan-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝑆 = {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅}    ⇒   ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ* ∧ 𝑇 ∈ ℝ* ∧ 𝑅 < 𝑇)) → 𝑆 ⊆ (𝑃(ball‘𝐷)𝑇))
Theorem	blcld 23670*	A "closed ball" in a metric space is actually closed. (Contributed by Mario Carneiro, 31-Dec-2013.) (Revised by Mario Carneiro, 24-Aug-2015.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝑆 = {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅}    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → 𝑆 ∈ (Clsd‘𝐽))
Theorem	blcls 23671*	The closure of an open ball in a metric space is contained in the corresponding closed ball. (Equality need not hold; for example, with the discrete metric, the closed ball of radius 1 is the whole space, but the open ball of radius 1 is just a point, whose closure is also a point.) (Contributed by Mario Carneiro, 31-Dec-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝑆 = {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅}    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → ((cls‘𝐽)‘(𝑃(ball‘𝐷)𝑅)) ⊆ 𝑆)
Theorem	blsscls 23672	If two concentric balls have different radii, the closure of the smaller one is contained in the larger one. (Contributed by Mario Carneiro, 5-Jan-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ* ∧ 𝑆 ∈ ℝ* ∧ 𝑅 < 𝑆)) → ((cls‘𝐽)‘(𝑃(ball‘𝐷)𝑅)) ⊆ (𝑃(ball‘𝐷)𝑆))
Theorem	metss 23673*	Two ways of saying that metric 𝐷 generates a finer topology than metric 𝐶. (Contributed by Mario Carneiro, 12-Nov-2013.) (Revised by Mario Carneiro, 24-Aug-2015.)
⊢ 𝐽 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (𝐽 ⊆ 𝐾 ↔ ∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟)))
Theorem	metequiv 23674*	Two ways of saying that two metrics generate the same topology. Two metrics satisfying the right-hand side are said to be (topologically) equivalent. (Contributed by Jeff Hankins, 21-Jun-2009.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (𝐽 = 𝐾 ↔ ∀𝑥 ∈ 𝑋 (∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ ∀𝑎 ∈ ℝ+ ∃𝑏 ∈ ℝ+ (𝑥(ball‘𝐶)𝑏) ⊆ (𝑥(ball‘𝐷)𝑎))))
Theorem	metequiv2 23675*	If there is a sequence of radii approaching zero for which the balls of both metrics coincide, then the generated topologies are equivalent. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ 𝐽 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)) → 𝐽 = 𝐾))
Theorem	metss2lem 23676*	Lemma for metss2 23677. (Contributed by Mario Carneiro, 14-Sep-2015.)
⊢ 𝐽 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))    ⇒   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑆 ∈ ℝ+)) → (𝑥(ball‘𝐷)(𝑆 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑆))
Theorem	metss2 23677*	If the metric 𝐷 is "strongly finer" than 𝐶 (meaning that there is a positive real constant 𝑅 such that 𝐶(𝑥, 𝑦) ≤ 𝑅 · 𝐷(𝑥, 𝑦)), then 𝐷 generates a finer topology. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they generate the same topology.) (Contributed by Mario Carneiro, 14-Sep-2015.)
⊢ 𝐽 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))    ⇒   ⊢ (𝜑 → 𝐽 ⊆ 𝐾)
Theorem	comet 23678*	The composition of an extended metric with a monotonic subadditive function is an extended metric. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝐹:(0[,]+∞)⟶ℝ*)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]+∞)) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = 0))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝐹‘(𝑥 +𝑒 𝑦)) ≤ ((𝐹‘𝑥) +𝑒 (𝐹‘𝑦)))    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐷) ∈ (∞Met‘𝑋))
Theorem	stdbdmetval 23679*	Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅))
Theorem	stdbdxmet 23680*	The standard bounded metric is an extended metric given an extended metric and a positive extended real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))    ⇒   ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋))
Theorem	stdbdmet 23681*	The standard bounded metric is a proper metric given an extended metric and a positive real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))    ⇒   ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ+) → 𝐷 ∈ (Met‘𝑋))
Theorem	stdbdbl 23682*	The standard bounded metric corresponding to 𝐶 generates the same balls as 𝐶 for radii less than 𝑅. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))    ⇒   ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) → (𝑃(ball‘𝐷)𝑆) = (𝑃(ball‘𝐶)𝑆))
Theorem	stdbdmopn 23683*	The standard bounded metric corresponding to 𝐶 generates the same topology as 𝐶. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))    &   ⊢ 𝐽 = (MetOpen‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → 𝐽 = (MetOpen‘𝐷))
Theorem	mopnex 23684*	The topology generated by an extended metric can also be generated by a true metric. Thus, "metrizable topologies" can equivalently be defined in terms of metrics or extended metrics. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∃𝑑 ∈ (Met‘𝑋)𝐽 = (MetOpen‘𝑑))
Theorem	methaus 23685	The topology generated by a metric space is Hausdorff. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Haus)
Theorem	met1stc 23686	The topology generated by a metric space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ 1stω)
Theorem	met2ndci 23687	A separable metric space (a metric space with a countable dense subset) is second-countable. (Contributed by Mario Carneiro, 13-Apr-2015.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐽 ∈ 2ndω)
Theorem	met2ndc 23688*	A metric space is second-countable iff it is separable (has a countable dense subset). (Contributed by Mario Carneiro, 13-Apr-2015.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐽 ∈ 2ndω ↔ ∃𝑥 ∈ 𝒫 𝑋(𝑥 ≼ ω ∧ ((cls‘𝐽)‘𝑥) = 𝑋)))
Theorem	metrest 23689	Two alternate formulations of a subspace topology of a metric space topology. (Contributed by Jeff Hankins, 19-Aug-2009.) (Proof shortened by Mario Carneiro, 5-Jan-2014.)
⊢ 𝐷 = (𝐶 ↾ (𝑌 × 𝑌))    &   ⊢ 𝐽 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) = 𝐾)
Theorem	ressxms 23690	The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (𝐾 ↾s 𝐴) ∈ ∞MetSp)
Theorem	ressms 23691	The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ ((𝐾 ∈ MetSp ∧ 𝐴 ∈ 𝑉) → (𝐾 ↾s 𝐴) ∈ MetSp)
Theorem	prdsmslem1 23692	Lemma for prdsms 23696. The distance function of a product structure is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ 𝐷 = (dist‘𝑌)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝑅:𝐼⟶MetSp)    ⇒   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝐵))
Theorem	prdsxmslem1 23693	Lemma for prdsms 23696. The distance function of a product structure is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ 𝐷 = (dist‘𝑌)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝑅:𝐼⟶∞MetSp)    ⇒   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))
Theorem	prdsxmslem2 23694*	Lemma for prdsxms 23695. The topology generated by the supremum metric is the same as the product topology, when the index set is finite. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ 𝐷 = (dist‘𝑌)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝑅:𝐼⟶∞MetSp)    &   ⊢ 𝐽 = (TopOpen‘𝑌)    &   ⊢ 𝑉 = (Base‘(𝑅‘𝑘))    &   ⊢ 𝐸 = ((dist‘(𝑅‘𝑘)) ↾ (𝑉 × 𝑉))    &   ⊢ 𝐾 = (TopOpen‘(𝑅‘𝑘))    &   ⊢ 𝐶 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘))}    ⇒   ⊢ (𝜑 → 𝐽 = (MetOpen‘𝐷))
Theorem	prdsxms 23695	The indexed product structure is an extended metric space when the index set is finite. (Although the extended metric is still valid when the index set is infinite, it no longer agrees with the product topology, which is not metrizable in any case.) (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    ⇒   ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → 𝑌 ∈ ∞MetSp)
Theorem	prdsms 23696	The indexed product structure is a metric space when the index set is finite. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    ⇒   ⊢ ((𝑆 ∈ 𝑊 ∧ 𝐼 ∈ Fin ∧ 𝑅:𝐼⟶MetSp) → 𝑌 ∈ MetSp)
Theorem	pwsxms 23697	A power of an extended metric space is an extended metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    ⇒   ⊢ ((𝑅 ∈ ∞MetSp ∧ 𝐼 ∈ Fin) → 𝑌 ∈ ∞MetSp)
Theorem	pwsms 23698	A power of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    ⇒   ⊢ ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → 𝑌 ∈ MetSp)
Theorem	xpsxms 23699	A binary product of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    ⇒   ⊢ ((𝑅 ∈ ∞MetSp ∧ 𝑆 ∈ ∞MetSp) → 𝑇 ∈ ∞MetSp)
Theorem	xpsms 23700	A binary product of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    ⇒   ⊢ ((𝑅 ∈ MetSp ∧ 𝑆 ∈ MetSp) → 𝑇 ∈ MetSp)