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Theorem List for Metamath Proof Explorer - 22301-22400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisms 22301 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)))

Theoremisms2 22302 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)))

Theoremxmstopn 22303 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))

Theoremmstopn 22304 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))

Theoremxmstps 22305 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp)

Theoremmsxms 22306 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)

Theoremmstps 22307 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝑀 ∈ MetSp → 𝑀 ∈ TopSp)

Theoremxmsxmet 22308 The distance function, suitably truncated, is a metric on 𝑋. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝑋 = (Base‘𝑀)    &   𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))       (𝑀 ∈ ∞MetSp → 𝐷 ∈ (∞Met‘𝑋))

Theoremmsmet 22309 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))

Theoremmsf 22310 Mapping of the distance function of a metric space. (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)

Theoremxmsxmet2 22311 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))

Theoremmsmet2 22312 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))

Theoremmscl 22313 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 ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) ∈ ℝ)

Theoremxmscl 22314 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 ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) ∈ ℝ*)

Theoremxmsge0 22315 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 ∧ 𝐴𝑋𝐵𝑋) → 0 ≤ (𝐴𝐷𝐵))

Theoremxmseq0 22316 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 ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵))

Theoremxmssym 22317 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 ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))

Theoremxmstri2 22318 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 ∧ (𝐶𝑋𝐴𝑋𝐵𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))

Theoremmstri2 22319 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 ∧ (𝐶𝑋𝐴𝑋𝐵𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) + (𝐶𝐷𝐵)))

Theoremxmstri 22320 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 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐶𝐷𝐵)))

Theoremmstri 22321 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 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) + (𝐶𝐷𝐵)))

Theoremxmstri3 22322 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 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐵𝐷𝐶)))

Theoremmstri3 22323 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 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) + (𝐵𝐷𝐶)))

Theoremmsrtri 22324 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 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (abs‘((𝐴𝐷𝐶) − (𝐵𝐷𝐶))) ≤ (𝐴𝐷𝐵))

Theoremxmspropd 22325 Property deduction for an extended metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))    &   (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))       (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐿 ∈ ∞MetSp))

Theoremmspropd 22326 Property deduction for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))    &   (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))       (𝜑 → (𝐾 ∈ MetSp ↔ 𝐿 ∈ MetSp))

Theoremsetsmsbas 22327 The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
(𝜑𝑋 = (Base‘𝑀))    &   (𝜑𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   (𝜑𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))       (𝜑𝑋 = (Base‘𝐾))

Theoremsetsmsds 22328 The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
(𝜑𝑋 = (Base‘𝑀))    &   (𝜑𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   (𝜑𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))       (𝜑 → (dist‘𝑀) = (dist‘𝐾))

Theoremsetsmstset 22329 The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
(𝜑𝑋 = (Base‘𝑀))    &   (𝜑𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   (𝜑𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    &   (𝜑𝑀𝑉)       (𝜑 → (MetOpen‘𝐷) = (TopSet‘𝐾))

Theoremsetsmstopn 22330 The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
(𝜑𝑋 = (Base‘𝑀))    &   (𝜑𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   (𝜑𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    &   (𝜑𝑀𝑉)       (𝜑 → (MetOpen‘𝐷) = (TopOpen‘𝐾))

Theoremsetsxms 22331 The constructed metric space is a metric space iff the provided distance function is a metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
(𝜑𝑋 = (Base‘𝑀))    &   (𝜑𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   (𝜑𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    &   (𝜑𝑀𝑉)       (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐷 ∈ (∞Met‘𝑋)))

Theoremsetsms 22332 The constructed metric space is a metric space iff the provided distance function is a metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
(𝜑𝑋 = (Base‘𝑀))    &   (𝜑𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   (𝜑𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    &   (𝜑𝑀𝑉)       (𝜑 → (𝐾 ∈ MetSp ↔ 𝐷 ∈ (Met‘𝑋)))

Theoremtmsval 22333 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)⟩))

Theoremtmslem 22334 Lemma for tmsbas 22335, tmsds 22336, and tmstopn 22337. (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)))

Theoremtmsbas 22335 The base set of a constructed metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐾 = (toMetSp‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝐾))

Theoremtmsds 22336 The metric of a constructed metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐾 = (toMetSp‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝐾))

Theoremtmstopn 22337 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))

Theoremtmsxms 22338 The constructed metric space is an extended metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐾 = (toMetSp‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → 𝐾 ∈ ∞MetSp)

Theoremtmsms 22339 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)

Theoremimasf1obl 22340 The image of a metric space ball. (Contributed by Mario Carneiro, 28-Aug-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉1-1-onto𝐵)    &   (𝜑𝑅𝑍)    &   𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))    &   𝐷 = (dist‘𝑈)    &   (𝜑𝐸 ∈ (∞Met‘𝑉))    &   (𝜑𝑃𝑉)    &   (𝜑𝑆 ∈ ℝ*)       (𝜑 → ((𝐹𝑃)(ball‘𝐷)𝑆) = (𝐹 “ (𝑃(ball‘𝐸)𝑆)))

Theoremimasf1oxms 22341 The image of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉1-1-onto𝐵)    &   (𝜑𝑅 ∈ ∞MetSp)       (𝜑𝑈 ∈ ∞MetSp)

Theoremimasf1oms 22342 The image of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉1-1-onto𝐵)    &   (𝜑𝑅 ∈ MetSp)       (𝜑𝑈 ∈ MetSp)

Theoremprdsbl 22343* 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 22357) - 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(𝑥𝐼𝑅))    &   𝐵 = (Base‘𝑌)    &   𝑉 = (Base‘𝑅)    &   𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))    &   𝐷 = (dist‘𝑌)    &   (𝜑𝑆𝑊)    &   (𝜑𝐼 ∈ Fin)    &   ((𝜑𝑥𝐼) → 𝑅𝑍)    &   ((𝜑𝑥𝐼) → 𝐸 ∈ (∞Met‘𝑉))    &   (𝜑𝑃𝐵)    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑 → 0 < 𝐴)       (𝜑 → (𝑃(ball‘𝐷)𝐴) = X𝑥𝐼 ((𝑃𝑥)(ball‘𝐸)𝐴))

Theoremmopni 22344* 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) ∧ 𝐴𝐽𝑃𝐴) → ∃𝑥 ∈ ran (ball‘𝐷)(𝑃𝑥𝑥𝐴))

Theoremmopni2 22345* 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) ∧ 𝐴𝐽𝑃𝐴) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴)

Theoremmopni3 22346* 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) ∧ 𝐴𝐽𝑃𝐴) ∧ 𝑅 ∈ ℝ+) → ∃𝑥 ∈ ℝ+ (𝑥 < 𝑅 ∧ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴))

Theoremblssopn 22347 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)

Theoremunimopn 22348 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) ∧ 𝐴𝐽) → 𝐴𝐽)

Theoremmopnin 22349 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) ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)

Theoremmopn0 22350 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)

Theoremrnblopn 22351 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)) → 𝐵𝐽)

Theoremblopn 22352 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) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ 𝐽)

Theoremneibl 22353* 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) ∧ 𝑃𝑋) → (𝑁 ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑁𝑋 ∧ ∃𝑟 ∈ ℝ+ (𝑃(ball‘𝐷)𝑟) ⊆ 𝑁)))

Theoremblnei 22354 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) ∧ 𝑃𝑋𝑅 ∈ ℝ+) → (𝑃(ball‘𝐷)𝑅) ∈ ((nei‘𝐽)‘{𝑃}))

Theoremlpbl 22355* Every ball around a limit point 𝑃 of a subset 𝑆 includes a member of 𝑆 (even if 𝑃𝑆). (Contributed by NM, 9-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)
𝐽 = (MetOpen‘𝐷)       (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋𝑃 ∈ ((limPt‘𝐽)‘𝑆)) ∧ 𝑅 ∈ ℝ+) → ∃𝑥𝑆 𝑥 ∈ (𝑃(ball‘𝐷)𝑅))

Theoremblsscls2 22356* A smaller closed ball is contained in a larger open ball. (Contributed by Mario Carneiro, 10-Jan-2014.)
𝐽 = (MetOpen‘𝐷)    &   𝑆 = {𝑧𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅}       (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑅 ∈ ℝ*𝑇 ∈ ℝ*𝑅 < 𝑇)) → 𝑆 ⊆ (𝑃(ball‘𝐷)𝑇))

Theoremblcld 22357* 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 = {𝑧𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅}       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝑆 ∈ (Clsd‘𝐽))

Theoremblcls 22358* 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 = {𝑧𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅}       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → ((cls‘𝐽)‘(𝑃(ball‘𝐷)𝑅)) ⊆ 𝑆)

Theoremblsscls 22359 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) ∧ 𝑃𝑋) ∧ (𝑅 ∈ ℝ*𝑆 ∈ ℝ*𝑅 < 𝑆)) → ((cls‘𝐽)‘(𝑃(ball‘𝐷)𝑅)) ⊆ (𝑃(ball‘𝐷)𝑆))

Theoremmetss 22360* 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)) → (𝐽𝐾 ↔ ∀𝑥𝑋𝑟 ∈ ℝ+𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟)))

Theoremmetequiv 22361* 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 ↔ ∀𝑥𝑋 (∀𝑟 ∈ ℝ+𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ ∀𝑎 ∈ ℝ+𝑏 ∈ ℝ+ (𝑥(ball‘𝐶)𝑏) ⊆ (𝑥(ball‘𝐷)𝑎))))

Theoremmetequiv2 22362* 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)) → (∀𝑥𝑋𝑟 ∈ ℝ+𝑠 ∈ ℝ+ (𝑠𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)) → 𝐽 = 𝐾))

Theoremmetss2lem 22363* Lemma for metss2 22364. (Contributed by Mario Carneiro, 14-Sep-2015.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)    &   (𝜑𝐶 ∈ (Met‘𝑋))    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝑅 ∈ ℝ+)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))       ((𝜑 ∧ (𝑥𝑋𝑆 ∈ ℝ+)) → (𝑥(ball‘𝐷)(𝑆 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑆))

Theoremmetss2 22364* 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)    &   (𝜑𝐶 ∈ (Met‘𝑋))    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝑅 ∈ ℝ+)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))       (𝜑𝐽𝐾)

Theoremcomet 22365* The composition of an extended metric with a monotonic subadditive function is an extended metric. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝐹:(0[,]+∞)⟶ℝ*)    &   ((𝜑𝑥 ∈ (0[,]+∞)) → ((𝐹𝑥) = 0 ↔ 𝑥 = 0))    &   ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))    &   ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝐹‘(𝑥 +𝑒 𝑦)) ≤ ((𝐹𝑥) +𝑒 (𝐹𝑦)))       (𝜑 → (𝐹𝐷) ∈ (∞Met‘𝑋))

Theoremstdbdmetval 22366* Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))       ((𝑅𝑉𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅))

Theoremstdbdxmet 22367* 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 = (𝑥𝑋, 𝑦𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋))

Theoremstdbdmet 22368* 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 = (𝑥𝑋, 𝑦𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ+) → 𝐷 ∈ (Met‘𝑋))

Theoremstdbdbl 22369* 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 = (𝑥𝑋, 𝑦𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))       (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) → (𝑃(ball‘𝐷)𝑆) = (𝑃(ball‘𝐶)𝑆))

Theoremstdbdmopn 22370* The standard bounded metric corresponding to 𝐶 generates the same topology as 𝐶. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))    &   𝐽 = (MetOpen‘𝐶)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → 𝐽 = (MetOpen‘𝐷))

Theoremmopnex 22371* 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))

Theoremmethaus 22372 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)

Theoremmet1stc 22373 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)

Theoremmet2ndci 22374 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) ∧ (𝐴𝑋𝐴 ≼ ω ∧ ((cls‘𝐽)‘𝐴) = 𝑋)) → 𝐽 ∈ 2nd𝜔)

Theoremmet2ndc 22375* 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)))

Theoremmetrest 22376 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) ∧ 𝑌𝑋) → (𝐽t 𝑌) = 𝐾)

Theoremressxms 22377 The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (𝐾s 𝐴) ∈ ∞MetSp)

Theoremressms 22378 The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
((𝐾 ∈ MetSp ∧ 𝐴𝑉) → (𝐾s 𝐴) ∈ MetSp)

Theoremprdsmslem1 22379 Lemma for prdsms 22383. The distance function of a product structure is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑊)    &   (𝜑𝐼 ∈ Fin)    &   𝐷 = (dist‘𝑌)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑅:𝐼⟶MetSp)       (𝜑𝐷 ∈ (Met‘𝐵))

Theoremprdsxmslem1 22380 Lemma for prdsms 22383. The distance function of a product structure is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑊)    &   (𝜑𝐼 ∈ Fin)    &   𝐷 = (dist‘𝑌)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑅:𝐼⟶∞MetSp)       (𝜑𝐷 ∈ (∞Met‘𝐵))

Theoremprdsxmslem2 22381* Lemma for prdsxms 22382. 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)    &   (𝜑𝑆𝑊)    &   (𝜑𝐼 ∈ Fin)    &   𝐷 = (dist‘𝑌)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑅:𝐼⟶∞MetSp)    &   𝐽 = (TopOpen‘𝑌)    &   𝑉 = (Base‘(𝑅𝑘))    &   𝐸 = ((dist‘(𝑅𝑘)) ↾ (𝑉 × 𝑉))    &   𝐾 = (TopOpen‘(𝑅𝑘))    &   𝐶 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘𝐼 (𝑔𝑘))}       (𝜑𝐽 = (MetOpen‘𝐷))

Theoremprdsxms 22382 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)       ((𝑆𝑊𝐼 ∈ Fin ∧ 𝑅:𝐼⟶∞MetSp) → 𝑌 ∈ ∞MetSp)

Theoremprdsms 22383 The indexed product structure is a metric space when the index set is finite. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝑌 = (𝑆Xs𝑅)       ((𝑆𝑊𝐼 ∈ Fin ∧ 𝑅:𝐼⟶MetSp) → 𝑌 ∈ MetSp)

Theorempwsxms 22384 The product of a finite family of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ ∞MetSp ∧ 𝐼 ∈ Fin) → 𝑌 ∈ ∞MetSp)

Theorempwsms 22385 The product of a finite family of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → 𝑌 ∈ MetSp)

Theoremxpsxms 22386 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)

Theoremxpsms 22387 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)

Theoremtmsxps 22388 Express the product of two metrics as another metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝑃 = (dist‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))    &   (𝜑𝑀 ∈ (∞Met‘𝑋))    &   (𝜑𝑁 ∈ (∞Met‘𝑌))       (𝜑𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))

Theoremtmsxpsmopn 22389 Express the product of two metrics as another metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝑃 = (dist‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))    &   (𝜑𝑀 ∈ (∞Met‘𝑋))    &   (𝜑𝑁 ∈ (∞Met‘𝑌))    &   𝐽 = (MetOpen‘𝑀)    &   𝐾 = (MetOpen‘𝑁)    &   𝐿 = (MetOpen‘𝑃)       (𝜑𝐿 = (𝐽 ×t 𝐾))

Theoremtmsxpsval 22390 Value of the product of two metrics. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝑃 = (dist‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))    &   (𝜑𝑀 ∈ (∞Met‘𝑋))    &   (𝜑𝑁 ∈ (∞Met‘𝑌))    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑌)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑌)       (𝜑 → (⟨𝐴, 𝐵𝑃𝐶, 𝐷⟩) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))

Theoremtmsxpsval2 22391 Value of the product of two metrics. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝑃 = (dist‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))    &   (𝜑𝑀 ∈ (∞Met‘𝑋))    &   (𝜑𝑁 ∈ (∞Met‘𝑌))    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑌)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑌)       (𝜑 → (⟨𝐴, 𝐵𝑃𝐶, 𝐷⟩) = if((𝐴𝑀𝐶) ≤ (𝐵𝑁𝐷), (𝐵𝑁𝐷), (𝐴𝑀𝐶)))

12.4.5  Continuity in metric spaces

Theoremmetcnp3 22392* Two ways to express that 𝐹 is continuous at 𝑃 for metric spaces. Proposition 14-4.2 of [Gleason] p. 240. (Contributed by NM, 17-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦))))

Theoremmetcnp 22393* Two ways to say a mapping from metric 𝐶 to metric 𝐷 is continuous at point 𝑃. (Contributed by NM, 11-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑋 ((𝑃𝐶𝑤) < 𝑧 → ((𝐹𝑃)𝐷(𝐹𝑤)) < 𝑦))))

Theoremmetcnp2 22394* Two ways to say a mapping from metric 𝐶 to metric 𝐷 is continuous at point 𝑃. The distance arguments are swapped compared to metcnp 22393 (and Munkres' metcn 22395) for compatibility with df-lm 21081. Definition 1.3-3 of [Kreyszig] p. 20. (Contributed by NM, 4-Jun-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑋 ((𝑤𝐶𝑃) < 𝑧 → ((𝐹𝑤)𝐷(𝐹𝑃)) < 𝑦))))

Theoremmetcn 22395* Two ways to say a mapping from metric 𝐶 to metric 𝐷 is continuous. Theorem 10.1 of [Munkres] p. 127. The second biconditional argument says that for every positive "epsilon" 𝑦 there is a positive "delta" 𝑧 such that a distance less than delta in 𝐶 maps to a distance less than epsilon in 𝐷. (Contributed by NM, 15-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝐹𝑥)𝐷(𝐹𝑤)) < 𝑦))))

Theoremmetcnpi 22396* Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 22393. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ+)) → ∃𝑥 ∈ ℝ+𝑦𝑋 ((𝑃𝐶𝑦) < 𝑥 → ((𝐹𝑃)𝐷(𝐹𝑦)) < 𝐴))

Theoremmetcnpi2 22397* Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 22394. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ+)) → ∃𝑥 ∈ ℝ+𝑦𝑋 ((𝑦𝐶𝑃) < 𝑥 → ((𝐹𝑦)𝐷(𝐹𝑃)) < 𝐴))

Theoremmetcnpi3 22398* Epsilon-delta property of a metric space function continuous at 𝑃. A variation of metcnpi2 22397 with non-strict ordering. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ+)) → ∃𝑥 ∈ ℝ+𝑦𝑋 ((𝑦𝐶𝑃) ≤ 𝑥 → ((𝐹𝑦)𝐷(𝐹𝑃)) ≤ 𝐴))

Theoremtxmetcnp 22399* Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)    &   𝐿 = (MetOpen‘𝐸)       (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) → (𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑧 ∈ ℝ+𝑤 ∈ ℝ+𝑢𝑋𝑣𝑌 (((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤) → ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧))))

Theoremtxmetcn 22400* Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)    &   𝐿 = (MetOpen‘𝐸)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) → (𝐹 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑥𝑋𝑦𝑌𝑧 ∈ ℝ+𝑤 ∈ ℝ+𝑢𝑋𝑣𝑌 (((𝑥𝐶𝑢) < 𝑤 ∧ (𝑦𝐷𝑣) < 𝑤) → ((𝑥𝐹𝑦)𝐸(𝑢𝐹𝑣)) < 𝑧))))

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