P. 244 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 24301-24400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	xmettri3 24301	Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐵𝐷𝐶)))
Theorem	mettri3 24302	Triangle inequality for the distance function of a metric space. (Contributed by NM, 13-Mar-2007.)
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) + (𝐵𝐷𝐶)))
Theorem	xmetrtri 24303	One half of the reverse triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 4-Sep-2015.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐶) +𝑒 -𝑒(𝐵𝐷𝐶)) ≤ (𝐴𝐷𝐵))
Theorem	xmetrtri2 24304	The reverse triangle inequality for the distance function of an extended metric. In order to express the "extended absolute value function", we use the distance function xrsdsval 21369 defined on the extended real structure. (Contributed by Mario Carneiro, 4-Sep-2015.)
⊢ 𝐾 = (dist‘ℝ*𝑠)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐶)𝐾(𝐵𝐷𝐶)) ≤ (𝐴𝐷𝐵))
Theorem	metrtri 24305	Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 5-May-2014.)
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (abs‘((𝐴𝐷𝐶) − (𝐵𝐷𝐶))) ≤ (𝐴𝐷𝐵))
Theorem	xmetgt0 24306	The distance function of an extended metric space is positive for unequal points. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ≠ 𝐵 ↔ 0 < (𝐴𝐷𝐵)))
Theorem	metgt0 24307	The distance function of a metric space is positive for unequal points. Definition 14-1.1(b) of [Gleason] p. 223 and its converse. (Contributed by NM, 27-Aug-2006.)
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ≠ 𝐵 ↔ 0 < (𝐴𝐷𝐵)))
Theorem	metn0 24308	A metric space is nonempty iff its base set is nonempty. (Contributed by NM, 4-Oct-2007.) (Revised by Mario Carneiro, 14-Aug-2015.)
⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 ≠ ∅ ↔ 𝑋 ≠ ∅))
Theorem	xmetres2 24309	Restriction of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (∞Met‘𝑅))
Theorem	metreslem 24310	Lemma for metres 24313. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ (dom 𝐷 = (𝑋 × 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) = (𝐷 ↾ ((𝑋 ∩ 𝑅) × (𝑋 ∩ 𝑅))))
Theorem	metres2 24311	Lemma for metres 24313. (Contributed by FL, 12-Oct-2006.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (Met‘𝑅))
Theorem	xmetres 24312	A restriction of an extended metric is an extended metric. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (∞Met‘(𝑋 ∩ 𝑅)))
Theorem	metres 24313	A restriction of a metric is a metric. (Contributed by NM, 26-Aug-2007.) (Revised by Mario Carneiro, 14-Aug-2015.)
⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (Met‘(𝑋 ∩ 𝑅)))
Theorem	0met 24314	The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
⊢ ∅ ∈ (Met‘∅)
Theorem	prdsdsf 24315*	The product metric is a function into the nonnegative extended reals. In general this means that it is not a metric but rather an *extended* metric (even when all the factors are metrics), but it will be a metric when restricted to regions where it does not take infinite values. (Contributed by Mario Carneiro, 20-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 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ 𝑍)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))    ⇒   ⊢ (𝜑 → 𝐷:(𝐵 × 𝐵)⟶(0[,]+∞))
Theorem	prdsxmetlem 24316*	The product metric is an extended metric. (Contributed by Mario Carneiro, 20-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 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ 𝑍)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))    ⇒   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))
Theorem	prdsxmet 24317*	The product metric is an extended metric. Eliminate disjoint variable conditions from prdsxmetlem 24316. (Contributed by Mario Carneiro, 26-Sep-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 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ 𝑍)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))    ⇒   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))
Theorem	prdsmet 24318*	The product metric is a metric when the index set is finite. (Contributed by Mario Carneiro, 20-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 → 𝐷 ∈ (Met‘𝐵))
Theorem	ressprdsds 24319*	Restriction of a product metric. (Contributed by Mario Carneiro, 16-Sep-2015.)
⊢ (𝜑 → 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅)))    &   ⊢ (𝜑 → 𝐻 = (𝑇Xs(𝑥 ∈ 𝐼 ↦ (𝑅 ↾s 𝐴))))    &   ⊢ 𝐵 = (Base‘𝐻)    &   ⊢ 𝐷 = (dist‘𝑌)    &   ⊢ 𝐸 = (dist‘𝐻)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ 𝑍)    ⇒   ⊢ (𝜑 → 𝐸 = (𝐷 ↾ (𝐵 × 𝐵)))
Theorem	resspwsds 24320	Restriction of a power metric. (Contributed by Mario Carneiro, 16-Sep-2015.)
⊢ (𝜑 → 𝑌 = (𝑅 ↑s 𝐼))    &   ⊢ (𝜑 → 𝐻 = ((𝑅 ↾s 𝐴) ↑s 𝐼))    &   ⊢ 𝐵 = (Base‘𝐻)    &   ⊢ 𝐷 = (dist‘𝑌)    &   ⊢ 𝐸 = (dist‘𝐻)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    ⇒   ⊢ (𝜑 → 𝐸 = (𝐷 ↾ (𝐵 × 𝐵)))
Theorem	imasdsf1olem 24321*	Lemma for imasdsf1o 24322. (Contributed by Mario Carneiro, 21-Aug-2015.) (Proof shortened by AV, 6-Oct-2020.)
⊢ (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 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ 𝑊 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))    &   ⊢ 𝑆 = {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = (𝐹‘𝑋) ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = (𝐹‘𝑌) ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))}    &   ⊢ 𝑇 = ∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔)))    ⇒   ⊢ (𝜑 → ((𝐹‘𝑋)𝐷(𝐹‘𝑌)) = (𝑋𝐸𝑌))
Theorem	imasdsf1o 24322	The distance function is transferred across an image structure under a bijection. (Contributed by Mario Carneiro, 20-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 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝐹‘𝑋)𝐷(𝐹‘𝑌)) = (𝑋𝐸𝑌))
Theorem	imasf1oxmet 24323	The image of an extended metric is an extended metric. (Contributed by Mario Carneiro, 20-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 → 𝐷 ∈ (∞Met‘𝐵))
Theorem	imasf1omet 24324	The image of a metric is a metric. (Contributed by Mario Carneiro, 21-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 → 𝐷 ∈ (Met‘𝐵))
Theorem	xpsdsfn 24325	Closure of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝑌 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ 𝑃 = (dist‘𝑇)    ⇒   ⊢ (𝜑 → 𝑃 Fn ((𝑋 × 𝑌) × (𝑋 × 𝑌)))
Theorem	xpsdsfn2 24326	Closure of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝑌 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ 𝑃 = (dist‘𝑇)    ⇒   ⊢ (𝜑 → 𝑃 Fn ((Base‘𝑇) × (Base‘𝑇)))
Theorem	xpsxmetlem 24327*	Lemma for xpsxmet 24328. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝑌 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ 𝑃 = (dist‘𝑇)    &   ⊢ 𝑀 = ((dist‘𝑅) ↾ (𝑋 × 𝑋))    &   ⊢ 𝑁 = ((dist‘𝑆) ↾ (𝑌 × 𝑌))    &   ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌))    ⇒   ⊢ (𝜑 → (dist‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})) ∈ (∞Met‘ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})))
Theorem	xpsxmet 24328	A product metric of extended metrics is an extended metric. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝑌 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ 𝑃 = (dist‘𝑇)    &   ⊢ 𝑀 = ((dist‘𝑅) ↾ (𝑋 × 𝑋))    &   ⊢ 𝑁 = ((dist‘𝑆) ↾ (𝑌 × 𝑌))    &   ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌))    ⇒   ⊢ (𝜑 → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
Theorem	xpsdsval 24329	Value of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝑌 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ 𝑃 = (dist‘𝑇)    &   ⊢ 𝑀 = ((dist‘𝑅) ↾ (𝑋 × 𝑋))    &   ⊢ 𝑁 = ((dist‘𝑆) ↾ (𝑌 × 𝑌))    &   ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐵⟩𝑃⟨𝐶, 𝐷⟩) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ))
Theorem	xpsmet 24330	The direct product of two metric spaces. Definition 14-1.5 of [Gleason] p. 225. (Contributed by NM, 20-Jun-2007.) (Revised by Mario Carneiro, 20-Aug-2015.)
⊢ 𝑇 = (𝑅 ×s 𝑆)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝑌 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ 𝑃 = (dist‘𝑇)    &   ⊢ 𝑀 = ((dist‘𝑅) ↾ (𝑋 × 𝑋))    &   ⊢ 𝑁 = ((dist‘𝑆) ↾ (𝑌 × 𝑌))    &   ⊢ (𝜑 → 𝑀 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝑁 ∈ (Met‘𝑌))    ⇒   ⊢ (𝜑 → 𝑃 ∈ (Met‘(𝑋 × 𝑌)))
12.4.3  Metric space balls
Theorem	blfvalps 24331*	The value of the ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Feb-2018.)
⊢ (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷) = (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}))
Theorem	blfval 24332*	The value of the ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Proof shortened by Thierry Arnoux, 11-Feb-2018.)
⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) = (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}))
Theorem	blvalps 24333*	The ball around a point 𝑃 is the set of all points whose distance from 𝑃 is less than the ball's radius 𝑅. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) = {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅})
Theorem	blval 24334*	The ball around a point 𝑃 is the set of all points whose distance from 𝑃 is less than the ball's radius 𝑅. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) = {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅})
Theorem	elblps 24335	Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) < 𝑅)))
Theorem	elbl 24336	Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) < 𝑅)))
Theorem	elbl2ps 24337	Membership in a ball. (Contributed by NM, 9-Mar-2007.) (Revised by Thierry Arnoux, 11-Mar-2018.)
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑃𝐷𝐴) < 𝑅))
Theorem	elbl2 24338	Membership in a ball. (Contributed by NM, 9-Mar-2007.)
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑃𝐷𝐴) < 𝑅))
Theorem	elbl3ps 24339	Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.)
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴𝐷𝑃) < 𝑅))
Theorem	elbl3 24340	Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.)
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴𝐷𝑃) < 𝑅))
Theorem	blcomps 24341	Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.) (Revised by Thierry Arnoux, 11-Mar-2018.)
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ 𝑃 ∈ (𝐴(ball‘𝐷)𝑅)))
Theorem	blcom 24342	Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.)
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ 𝑃 ∈ (𝐴(ball‘𝐷)𝑅)))
Theorem	xblpnfps 24343	The infinity ball in an extended metric is the set of all points that are a finite distance from the center. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝐴 ∈ (𝑃(ball‘𝐷)+∞) ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) ∈ ℝ)))
Theorem	xblpnf 24344	The infinity ball in an extended metric is the set of all points that are a finite distance from the center. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝐴 ∈ (𝑃(ball‘𝐷)+∞) ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) ∈ ℝ)))
Theorem	blpnf 24345	The infinity ball in a standard metric is just the whole space. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑃(ball‘𝐷)+∞) = 𝑋)
Theorem	bldisj 24346	Two balls are disjoint if the center-to-center distance is more than the sum of the radii. (Contributed by Mario Carneiro, 30-Dec-2013.)
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ* ∧ 𝑆 ∈ ℝ* ∧ (𝑅 +𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ((𝑃(ball‘𝐷)𝑅) ∩ (𝑄(ball‘𝐷)𝑆)) = ∅)
Theorem	blgt0 24347	A nonempty ball implies that the radius is positive. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ 𝐴 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 < 𝑅)
Theorem	bl2in 24348	Two balls are disjoint if they don't overlap. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑅 ≤ ((𝑃𝐷𝑄) / 2))) → ((𝑃(ball‘𝐷)𝑅) ∩ (𝑄(ball‘𝐷)𝑅)) = ∅)
Theorem	xblss2ps 24349	One ball is contained in another if the center-to-center distance is less than the difference of the radii. In this version of blss2 24352 for extended metrics, we have to assume the balls are a finite distance apart, or else 𝑃 will not even be in the infinity ball around 𝑄. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
⊢ (𝜑 → 𝐷 ∈ (PsMet‘𝑋))    &   ⊢ (𝜑 → 𝑃 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ*)    &   ⊢ (𝜑 → 𝑆 ∈ ℝ*)    &   ⊢ (𝜑 → (𝑃𝐷𝑄) ∈ ℝ)    &   ⊢ (𝜑 → (𝑃𝐷𝑄) ≤ (𝑆 +𝑒 -𝑒𝑅))    ⇒   ⊢ (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆))
Theorem	xblss2 24350	One ball is contained in another if the center-to-center distance is less than the difference of the radii. In this version of blss2 24352 for extended metrics, we have to assume the balls are a finite distance apart, or else 𝑃 will not even be in the infinity ball around 𝑄. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑃 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ*)    &   ⊢ (𝜑 → 𝑆 ∈ ℝ*)    &   ⊢ (𝜑 → (𝑃𝐷𝑄) ∈ ℝ)    &   ⊢ (𝜑 → (𝑃𝐷𝑄) ≤ (𝑆 +𝑒 -𝑒𝑅))    ⇒   ⊢ (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆))
Theorem	blss2ps 24351	One ball is contained in another if the center-to-center distance is less than the difference of the radii. (Contributed by Mario Carneiro, 15-Jan-2014.) (Revised by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ (𝑃𝐷𝑄) ≤ (𝑆 − 𝑅))) → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆))
Theorem	blss2 24352	One ball is contained in another if the center-to-center distance is less than the difference of the radii. (Contributed by Mario Carneiro, 15-Jan-2014.) (Revised by Mario Carneiro, 23-Aug-2015.)
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ (𝑃𝐷𝑄) ≤ (𝑆 − 𝑅))) → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆))
Theorem	blhalf 24353	A ball of radius 𝑅 / 2 is contained in a ball of radius 𝑅 centered at any point inside the smaller ball. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jan-2014.)
⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑌 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑍 ∈ (𝑌(ball‘𝑀)(𝑅 / 2)))) → (𝑌(ball‘𝑀)(𝑅 / 2)) ⊆ (𝑍(ball‘𝑀)𝑅))
Theorem	blfps 24354	Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
⊢ (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋)
Theorem	blf 24355	Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋)
Theorem	blrnps 24356*	Membership in the range of the ball function. Note that ran (ball‘𝐷) is the collection of all balls for metric 𝐷. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐴 ∈ ran (ball‘𝐷) ↔ ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ* 𝐴 = (𝑥(ball‘𝐷)𝑟)))
Theorem	blrn 24357*	Membership in the range of the ball function. Note that ran (ball‘𝐷) is the collection of all balls for metric 𝐷. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∈ ran (ball‘𝐷) ↔ ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ* 𝐴 = (𝑥(ball‘𝐷)𝑟)))
Theorem	xblcntrps 24358	A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ (𝑅 ∈ ℝ* ∧ 0 < 𝑅)) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅))
Theorem	xblcntr 24359	A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ (𝑅 ∈ ℝ* ∧ 0 < 𝑅)) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅))
Theorem	blcntrps 24360	A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅))
Theorem	blcntr 24361	A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅))
Theorem	xbln0 24362	A ball is nonempty iff the radius is positive. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → ((𝑃(ball‘𝐷)𝑅) ≠ ∅ ↔ 0 < 𝑅))
Theorem	bln0 24363	A ball is not empty. (Contributed by NM, 6-Oct-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → (𝑃(ball‘𝐷)𝑅) ≠ ∅)
Theorem	blelrnps 24364	A ball belongs to the set of balls of a metric space. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ ran (ball‘𝐷))
Theorem	blelrn 24365	A ball belongs to the set of balls of a metric space. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ ran (ball‘𝐷))
Theorem	blssm 24366	A ball is a subset of the base set of a metric space. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ 𝑋)
Theorem	unirnblps 24367	The union of the set of balls of a metric space is its base set. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
⊢ (𝐷 ∈ (PsMet‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋)
Theorem	unirnbl 24368	The union of the set of balls of a metric space is its base set. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ (𝐷 ∈ (∞Met‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋)
Theorem	blin 24369	The intersection of two balls with the same center is the smaller of them. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ* ∧ 𝑆 ∈ ℝ*)) → ((𝑃(ball‘𝐷)𝑅) ∩ (𝑃(ball‘𝐷)𝑆)) = (𝑃(ball‘𝐷)if(𝑅 ≤ 𝑆, 𝑅, 𝑆)))
Theorem	ssblps 24370	The size of a ball increases monotonically with its radius. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 24-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ* ∧ 𝑆 ∈ ℝ*) ∧ 𝑅 ≤ 𝑆) → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑃(ball‘𝐷)𝑆))
Theorem	ssbl 24371	The size of a ball increases monotonically with its radius. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ* ∧ 𝑆 ∈ ℝ*) ∧ 𝑅 ≤ 𝑆) → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑃(ball‘𝐷)𝑆))
Theorem	blssps 24372*	Any point 𝑃 in a ball 𝐵 can be centered in another ball that is a subset of 𝐵. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵 ∈ ran (ball‘𝐷) ∧ 𝑃 ∈ 𝐵) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)
Theorem	blss 24373*	Any point 𝑃 in a ball 𝐵 can be centered in another ball that is a subset of 𝐵. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ ran (ball‘𝐷) ∧ 𝑃 ∈ 𝐵) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)
Theorem	blssexps 24374*	Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) → (∃𝑥 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴) ↔ ∃𝑟 ∈ ℝ+ (𝑃(ball‘𝐷)𝑟) ⊆ 𝐴))
Theorem	blssex 24375*	Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (∃𝑥 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴) ↔ ∃𝑟 ∈ ℝ+ (𝑃(ball‘𝐷)𝑟) ⊆ 𝐴))
Theorem	ssblex 24376*	A nested ball exists whose radius is less than any desired amount. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ 𝑆 ∈ ℝ+)) → ∃𝑥 ∈ ℝ+ (𝑥 < 𝑅 ∧ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑃(ball‘𝐷)𝑆)))
Theorem	blin2 24377*	Given any two balls and a point in their intersection, there is a ball contained in the intersection with the given center point. (Contributed by Mario Carneiro, 12-Nov-2013.)
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵 ∩ 𝐶))
Theorem	blbas 24378	The balls of a metric space form a basis for a topology. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 15-Jan-2014.)
⊢ (𝐷 ∈ (∞Met‘𝑋) → ran (ball‘𝐷) ∈ TopBases)
Theorem	blres 24379	A ball in a restricted metric space. (Contributed by Mario Carneiro, 5-Jan-2014.)
⊢ 𝐶 = (𝐷 ↾ (𝑌 × 𝑌))    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝑋 ∩ 𝑌) ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐶)𝑅) = ((𝑃(ball‘𝐷)𝑅) ∩ 𝑌))
Theorem	xmeterval 24380	Value of the "finitely separated" relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ ∼ = (◡𝐷 “ ℝ)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∼ 𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ)))
Theorem	xmeter 24381	The "finitely separated" relation is an equivalence relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ ∼ = (◡𝐷 “ ℝ)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∼ Er 𝑋)
Theorem	xmetec 24382	The equivalence classes under the finite separation equivalence relation are infinity balls. Thus, by erdisj 8695, infinity balls are either identical or disjoint, quite unlike the usual situation with Euclidean balls which admit many kinds of overlap. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ ∼ = (◡𝐷 “ ℝ)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → [𝑃] ∼ = (𝑃(ball‘𝐷)+∞))
Theorem	blssec 24383	A ball centered at 𝑃 is contained in the set of points finitely separated from 𝑃. This is just an application of ssbl 24371 to the infinity ball. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ ∼ = (◡𝐷 “ ℝ)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑆) ⊆ [𝑃] ∼ )
Theorem	blpnfctr 24384	The infinity ball in an extended metric acts like an ultrametric ball in that every point in the ball is also its center. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝐴 ∈ (𝑃(ball‘𝐷)+∞)) → (𝑃(ball‘𝐷)+∞) = (𝐴(ball‘𝐷)+∞))
Theorem	xmetresbl 24385	An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 24382, this shows that any extended metric space can be "factored" into the disjoint union of proper metric spaces, with points in the same region measured by that region's metric, and points in different regions being distance +∞ from each other. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ 𝐵 = (𝑃(ball‘𝐷)𝑅)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (Met‘𝐵))
12.4.4  Open sets of a metric space
Theorem	mopnval 24386	An open set is a subset of a metric space which includes a ball around each of its points. Definition 1.3-2 of [Kreyszig] p. 18. The object (MetOpen‘𝐷) is the family of all open sets in the metric space determined by the metric 𝐷. By mopntop 24388, the open sets of a metric space form a topology 𝐽, whose base set is ∪ 𝐽 by mopnuni 24389. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 = (topGen‘ran (ball‘𝐷)))
Theorem	mopntopon 24387	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 24388	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 24389	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 24390*	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 24391	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 24392	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 24393*	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 24394	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 24395	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 24396	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 24397	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 24398	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 24399	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 24400	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))