P. 234 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 23301-23400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	tsmssplit 23301	Split a topological group sum into two parts. (Contributed by Mario Carneiro, 19-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums (𝐹 ↾ 𝐶)))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐺 tsums (𝐹 ↾ 𝐷)))    &   ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅)    &   ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷))    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝐺 tsums 𝐹))
Theorem	tsmsxplem1 23302*	Lemma for tsmsxp 23304. (Contributed by Mario Carneiro, 21-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopGrp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵)    &   ⊢ (𝜑 → 𝐻:𝐴⟶𝐵)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐻‘𝑗) ∈ (𝐺 tsums (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ (𝜑 → 𝐿 ∈ 𝐽)    &   ⊢ (𝜑 → 0 ∈ 𝐿)    &   ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐴 ∩ Fin))    &   ⊢ (𝜑 → dom 𝐷 ⊆ 𝐾)    &   ⊢ (𝜑 → 𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ (𝒫 𝐶 ∩ Fin)(ran 𝐷 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝐾 ((𝐻‘𝑥) − (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝐿))
Theorem	tsmsxplem2 23303*	Lemma for tsmsxp 23304. (Contributed by Mario Carneiro, 21-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopGrp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵)    &   ⊢ (𝜑 → 𝐻:𝐴⟶𝐵)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐻‘𝑗) ∈ (𝐺 tsums (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ (𝜑 → 𝐿 ∈ 𝐽)    &   ⊢ (𝜑 → 0 ∈ 𝐿)    &   ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐴 ∩ Fin))    &   ⊢ (𝜑 → ∀𝑐 ∈ 𝑆 ∀𝑑 ∈ 𝑇 (𝑐 + 𝑑) ∈ 𝑈)    &   ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝐶 ∩ Fin))    &   ⊢ (𝜑 → 𝐷 ⊆ (𝐾 × 𝑁))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐾 ((𝐻‘𝑥) − (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑁)))) ∈ 𝐿)    &   ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐾 × 𝑁))) ∈ 𝑆)    &   ⊢ (𝜑 → ∀𝑔 ∈ (𝐿 ↑m 𝐾)(𝐺 Σg 𝑔) ∈ 𝑇)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝐻 ↾ 𝐾)) ∈ 𝑈)
Theorem	tsmsxp 23304*	Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 19575 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 23302 for the main proof; this part mostly sets up the local assumptions. (Contributed by Mario Carneiro, 21-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopGrp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵)    &   ⊢ (𝜑 → 𝐻:𝐴⟶𝐵)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐻‘𝑗) ∈ (𝐺 tsums (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))    ⇒   ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ (𝐺 tsums 𝐻))
12.2.8  Topological rings, fields, vector spaces
Syntax	ctrg 23305	The class of all topological division rings.
class TopRing
Syntax	ctdrg 23306	The class of all topological division rings.
class TopDRing
Syntax	ctlm 23307	The class of all topological modules.
class TopMod
Syntax	ctvc 23308	The class of all topological vector spaces.
class TopVec
Definition	df-trg 23309	Define a topological ring, which is a ring such that the addition is a topological group operation and the multiplication is continuous. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd}
Definition	df-tdrg 23310	Define a topological division ring (which differs from a topological field only in being potentially noncommutative), which is a division ring and topological ring such that the unit group of the division ring (which is the set of nonzero elements) is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ TopDRing = {𝑟 ∈ (TopRing ∩ DivRing) ∣ ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp}
Definition	df-tlm 23311	Define a topological left module, which is just what its name suggests: instead of a group over a ring with a scalar product connecting them, it is a topological group over a topological ring with a continuous scalar product. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ TopMod = {𝑤 ∈ (TopMnd ∩ LMod) ∣ ((Scalar‘𝑤) ∈ TopRing ∧ ( ·sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)))}
Definition	df-tvc 23312	Define a topological left vector space, which is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ TopVec = {𝑤 ∈ TopMod ∣ (Scalar‘𝑤) ∈ TopDRing}
Theorem	istrg 23313	Express the predicate "𝑅 is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd))
Theorem	trgtmd 23314	The multiplicative monoid of a topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ (𝑅 ∈ TopRing → 𝑀 ∈ TopMnd)
Theorem	istdrg 23315	Express the predicate "𝑅 is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s 𝑈) ∈ TopGrp))
Theorem	tdrgunit 23316	The unit group of a topological division ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ (𝑅 ∈ TopDRing → (𝑀 ↾s 𝑈) ∈ TopGrp)
Theorem	trgtgp 23317	A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)
Theorem	trgtmd2 23318	A topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopMnd)
Theorem	trgtps 23319	A topological ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopSp)
Theorem	trgring 23320	A topological ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Ring)
Theorem	trggrp 23321	A topological ring is a group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Grp)
Theorem	tdrgtrg 23322	A topological division ring is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopRing)
Theorem	tdrgdrng 23323	A topological division ring is a division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ DivRing)
Theorem	tdrgring 23324	A topological division ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ Ring)
Theorem	tdrgtmd 23325	A topological division ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopMnd)
Theorem	tdrgtps 23326	A topological division ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopSp)
Theorem	istdrg2 23327	A topological-ring division ring is a topological division ring iff the group of nonzero elements is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp))
Theorem	mulrcn 23328	The functionalization of the ring multiplication operation is a continuous function in a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐽 = (TopOpen‘𝑅)    &   ⊢ 𝑇 = (+𝑓‘(mulGrp‘𝑅))    ⇒   ⊢ (𝑅 ∈ TopRing → 𝑇 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
Theorem	invrcn2 23329	The multiplicative inverse function is a continuous function from the unit group (that is, the nonzero numbers) to itself. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐽 = (TopOpen‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ (𝑅 ∈ TopDRing → 𝐼 ∈ ((𝐽 ↾t 𝑈) Cn (𝐽 ↾t 𝑈)))
Theorem	invrcn 23330	The multiplicative inverse function is a continuous function from the unit group (that is, the nonzero numbers) to the field. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐽 = (TopOpen‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ (𝑅 ∈ TopDRing → 𝐼 ∈ ((𝐽 ↾t 𝑈) Cn 𝐽))
Theorem	cnmpt1mulr 23331*	Continuity of ring multiplication; analogue of cnmpt12f 22815 which cannot be used directly because .r is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐽 = (TopOpen‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ TopRing)    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝐾 Cn 𝐽))
Theorem	cnmpt2mulr 23332*	Continuity of ring multiplication; analogue of cnmpt22f 22824 which cannot be used directly because .r is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐽 = (TopOpen‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ TopRing)    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴 · 𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
Theorem	dvrcn 23333	The division function is continuous in a topological field. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐽 = (TopOpen‘𝑅)    &   ⊢ / = (/r‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ (𝑅 ∈ TopDRing → / ∈ ((𝐽 ×t (𝐽 ↾t 𝑈)) Cn 𝐽))
Theorem	istlm 23334	The predicate "𝑊 is a topological left module". (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ · = ( ·sf ‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (TopOpen‘𝐹)    ⇒   ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
Theorem	vscacn 23335	The scalar multiplication is continuous in a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ · = ( ·sf ‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (TopOpen‘𝐹)    ⇒   ⊢ (𝑊 ∈ TopMod → · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
Theorem	tlmtmd 23336	A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopMnd)
Theorem	tlmtps 23337	A topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopSp)
Theorem	tlmlmod 23338	A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ LMod)
Theorem	tlmtrg 23339	The scalar ring of a topological module is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopRing)
Theorem	tlmscatps 23340	The scalar ring of a topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)
Theorem	istvc 23341	A topological vector space is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ 𝐹 ∈ TopDRing))
Theorem	tvctdrg 23342	The scalar field of a topological vector space is a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ TopVec → 𝐹 ∈ TopDRing)
Theorem	cnmpt1vsca 23343*	Continuity of scalar multiplication; analogue of cnmpt12f 22815 which cannot be used directly because ·𝑠 is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝐾 = (TopOpen‘𝐹)    &   ⊢ (𝜑 → 𝑊 ∈ TopMod)    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐿 Cn 𝐾))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐿 Cn 𝐽))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝐿 Cn 𝐽))
Theorem	cnmpt2vsca 23344*	Continuity of scalar multiplication; analogue of cnmpt22f 22824 which cannot be used directly because ·𝑠 is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝐾 = (TopOpen‘𝐹)    &   ⊢ (𝜑 → 𝑊 ∈ TopMod)    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐿 ×t 𝑀) Cn 𝐾))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴 · 𝐵)) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))
Theorem	tlmtgp 23345	A topological vector space is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopGrp)
Theorem	tvctlm 23346	A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopVec → 𝑊 ∈ TopMod)
Theorem	tvclmod 23347	A topological vector space is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopVec → 𝑊 ∈ LMod)
Theorem	tvclvec 23348	A topological vector space is a vector space. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopVec → 𝑊 ∈ LVec)
12.3  Uniform Structures and Spaces
12.3.1  Uniform structures
Syntax	cust 23349	Extend class notation with the class function of uniform structures.
class UnifOn
Definition	df-ust 23350*	Definition of a uniform structure. Definition 1 of [BourbakiTop1] p. II.1. A uniform structure is used to give a generalization of the idea of Cauchy's sequence. This definition is analogous to TopOn. Elements of an uniform structure are called entourages. (Contributed by FL, 29-May-2014.) (Revised by Thierry Arnoux, 15-Nov-2017.)
⊢ UnifOn = (𝑥 ∈ V ↦ {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣 ∈ 𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑢) ∧ ∀𝑤 ∈ 𝑢 (𝑣 ∩ 𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝑢 (𝑤 ∘ 𝑤) ⊆ 𝑣)))})
Theorem	ustfn 23351	The defined uniform structure as a function. (Contributed by Thierry Arnoux, 15-Nov-2017.)
⊢ UnifOn Fn V
Theorem	ustval 23352*	The class of all uniform structures for a base 𝑋. (Contributed by Thierry Arnoux, 15-Nov-2017.) (Revised by AV, 17-Sep-2021.)
⊢ (𝑋 ∈ 𝑉 → (UnifOn‘𝑋) = {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑢 ∧ ∀𝑣 ∈ 𝑢 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑢) ∧ ∀𝑤 ∈ 𝑢 (𝑣 ∩ 𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝑢 (𝑤 ∘ 𝑤) ⊆ 𝑣)))})
Theorem	isust 23353*	The predicate "𝑈 is a uniform structure with base 𝑋". (Contributed by Thierry Arnoux, 15-Nov-2017.) (Revised by AV, 17-Sep-2021.)
⊢ (𝑋 ∈ 𝑉 → (𝑈 ∈ (UnifOn‘𝑋) ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣 ∈ 𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣)))))
Theorem	ustssxp 23354	Entourages are subsets of the Cartesian product of the base set. (Contributed by Thierry Arnoux, 19-Nov-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑋 × 𝑋))
Theorem	ustssel 23355	A uniform structure is upward closed. Condition FI of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 19-Nov-2017.) (Proof shortened by AV, 17-Sep-2021.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑊 ⊆ (𝑋 × 𝑋)) → (𝑉 ⊆ 𝑊 → 𝑊 ∈ 𝑈))
Theorem	ustbasel 23356	The full set is always an entourage. Condition FIIb of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 19-Nov-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
Theorem	ustincl 23357	A uniform structure is closed under finite intersection. Condition FII of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 30-Nov-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑊 ∈ 𝑈) → (𝑉 ∩ 𝑊) ∈ 𝑈)
Theorem	ustdiag 23358	The diagonal set is included in any entourage, i.e. any point is 𝑉 -close to itself. Condition UI of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ( I ↾ 𝑋) ⊆ 𝑉)
Theorem	ustinvel 23359	If 𝑉 is an entourage, so is its inverse. Condition UII of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ◡𝑉 ∈ 𝑈)
Theorem	ustexhalf 23360*	For each entourage 𝑉 there is an entourage 𝑤 that is "not more than half as large". Condition UIII of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑉)
Theorem	ustrel 23361	The elements of uniform structures, called entourages, are relations. (Contributed by Thierry Arnoux, 15-Nov-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → Rel 𝑉)
Theorem	ustfilxp 23362	A uniform structure on a nonempty base is a filter. Remark 3 of [BourbakiTop1] p. II.2. (Contributed by Thierry Arnoux, 15-Nov-2017.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → 𝑈 ∈ (Fil‘(𝑋 × 𝑋)))
Theorem	ustne0 23363	A uniform structure cannot be empty. (Contributed by Thierry Arnoux, 16-Nov-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)
Theorem	ustssco 23364	In an uniform structure, any entourage 𝑉 is a subset of its composition with itself. (Contributed by Thierry Arnoux, 5-Jan-2018.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑉 ∘ 𝑉))
Theorem	ustexsym 23365*	In an uniform structure, for any entourage 𝑉, there exists a smaller symmetrical entourage. (Contributed by Thierry Arnoux, 4-Jan-2018.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑉))
Theorem	ustex2sym 23366*	In an uniform structure, for any entourage 𝑉, there exists a symmetrical entourage smaller than half 𝑉. (Contributed by Thierry Arnoux, 16-Jan-2018.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑉))
Theorem	ustex3sym 23367*	In an uniform structure, for any entourage 𝑉, there exists a symmetrical entourage smaller than a third of 𝑉. (Contributed by Thierry Arnoux, 16-Jan-2018.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑉))
Theorem	ustref 23368	Any element of the base set is "near" itself, i.e. entourages are reflexive. (Contributed by Thierry Arnoux, 16-Nov-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 𝐴𝑉𝐴)
Theorem	ust0 23369	The unique uniform structure of the empty set is the empty set. Remark 3 of [BourbakiTop1] p. II.2. (Contributed by Thierry Arnoux, 15-Nov-2017.)
⊢ (UnifOn‘∅) = {{∅}}
Theorem	ustn0 23370	The empty set is not an uniform structure. (Contributed by Thierry Arnoux, 3-Dec-2017.)
⊢ ¬ ∅ ∈ ∪ ran UnifOn
Theorem	ustund 23371	If two intersecting sets 𝐴 and 𝐵 are both small in 𝑉, their union is small in (𝑉↑2). Proposition 1 of [BourbakiTop1] p. II.12. This proposition actually does not require any axiom of the definition of uniform structures. (Contributed by Thierry Arnoux, 17-Nov-2017.)
⊢ (𝜑 → (𝐴 × 𝐴) ⊆ 𝑉)    &   ⊢ (𝜑 → (𝐵 × 𝐵) ⊆ 𝑉)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) ≠ ∅)    ⇒   ⊢ (𝜑 → ((𝐴 ∪ 𝐵) × (𝐴 ∪ 𝐵)) ⊆ (𝑉 ∘ 𝑉))
Theorem	ustelimasn 23372	Any point 𝐴 is near enough to itself. (Contributed by Thierry Arnoux, 18-Nov-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝑉 “ {𝐴}))
Theorem	ustneism 23373	For a point 𝐴 in 𝑋, (𝑉 “ {𝐴}) is small enough in (𝑉 ∘ ◡𝑉). This proposition actually does not require any axiom of the definition of uniform structures. (Contributed by Thierry Arnoux, 18-Nov-2017.)
⊢ ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴 ∈ 𝑋) → ((𝑉 “ {𝐴}) × (𝑉 “ {𝐴})) ⊆ (𝑉 ∘ ◡𝑉))
Theorem	elrnust 23374	First direction for ustbas 23377. (Contributed by Thierry Arnoux, 16-Nov-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn)
Theorem	ustbas2 23375	Second direction for ustbas 23377. (Contributed by Thierry Arnoux, 16-Nov-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
Theorem	ustuni 23376	The set union of a uniform structure is the Cartesian product of its base. (Contributed by Thierry Arnoux, 5-Dec-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ 𝑈 = (𝑋 × 𝑋))
Theorem	ustbas 23377	Recover the base of an uniform structure 𝑈. ∪ ran UnifOn is to UnifOn what Top is to TopOn. (Contributed by Thierry Arnoux, 16-Nov-2017.)
⊢ 𝑋 = dom ∪ 𝑈    ⇒   ⊢ (𝑈 ∈ ∪ ran UnifOn ↔ 𝑈 ∈ (UnifOn‘𝑋))
Theorem	ustimasn 23378	Lemma for ustuqtop 23396. (Contributed by Thierry Arnoux, 5-Dec-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → (𝑉 “ {𝑃}) ⊆ 𝑋)
Theorem	trust 23379	The trace of a uniform structure 𝑈 on a subset 𝐴 is a uniform structure on 𝐴. Definition 3 of [BourbakiTop1] p. II.9. (Contributed by Thierry Arnoux, 2-Dec-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
12.3.2  The topology induced by an uniform structure
Syntax	cutop 23380	Extend class notation with the function inducing a topology from a uniform structure.
class unifTop
Definition	df-utop 23381*	Definition of a topology induced by a uniform structure. Definition 3 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 17-Nov-2017.)
⊢ unifTop = (𝑢 ∈ ∪ ran UnifOn ↦ {𝑎 ∈ 𝒫 dom ∪ 𝑢 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎})
Theorem	utopval 23382*	The topology induced by a uniform structure 𝑈. (Contributed by Thierry Arnoux, 30-Nov-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
Theorem	elutop 23383*	Open sets in the topology induced by an uniform structure 𝑈 on 𝑋 (Contributed by Thierry Arnoux, 30-Nov-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴)))
Theorem	utoptop 23384	The topology induced by a uniform structure 𝑈 is a topology. (Contributed by Thierry Arnoux, 30-Nov-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
Theorem	utopbas 23385	The base of the topology induced by a uniform structure 𝑈. (Contributed by Thierry Arnoux, 5-Dec-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ (unifTop‘𝑈))
Theorem	utoptopon 23386	Topology induced by a uniform structure 𝑈 with its base set. (Contributed by Thierry Arnoux, 5-Jan-2018.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ (TopOn‘𝑋))
Theorem	restutop 23387	Restriction of a topology induced by an uniform structure. (Contributed by Thierry Arnoux, 12-Dec-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))))
Theorem	restutopopn 23388	The restriction of the topology induced by an uniform structure to an open set. (Contributed by Thierry Arnoux, 16-Dec-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → ((unifTop‘𝑈) ↾t 𝐴) = (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))))
Theorem	ustuqtoplem 23389*	Lemma for ustuqtop 23396. (Contributed by Thierry Arnoux, 11-Jan-2018.)
⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))    ⇒   ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ (𝑁‘𝑃) ↔ ∃𝑤 ∈ 𝑈 𝐴 = (𝑤 “ {𝑃})))
Theorem	ustuqtop0 23390*	Lemma for ustuqtop 23396. (Contributed by Thierry Arnoux, 11-Jan-2018.)
⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))    ⇒   ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
Theorem	ustuqtop1 23391*	Lemma for ustuqtop 23396, similar to ssnei2 22265. (Contributed by Thierry Arnoux, 11-Jan-2018.)
⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))    ⇒   ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑏 ∈ (𝑁‘𝑝))
Theorem	ustuqtop2 23392*	Lemma for ustuqtop 23396. (Contributed by Thierry Arnoux, 11-Jan-2018.)
⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))    ⇒   ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝))
Theorem	ustuqtop3 23393*	Lemma for ustuqtop 23396, similar to elnei 22260. (Contributed by Thierry Arnoux, 11-Jan-2018.)
⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))    ⇒   ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎)
Theorem	ustuqtop4 23394*	Lemma for ustuqtop 23396. (Contributed by Thierry Arnoux, 11-Jan-2018.)
⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))    ⇒   ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑏 ∈ (𝑁‘𝑝)∀𝑞 ∈ 𝑏 𝑎 ∈ (𝑁‘𝑞))
Theorem	ustuqtop5 23395*	Lemma for ustuqtop 23396. (Contributed by Thierry Arnoux, 11-Jan-2018.)
⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))    ⇒   ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))
Theorem	ustuqtop 23396*	For a given uniform structure 𝑈 on a set 𝑋, there is a unique topology 𝑗 such that the set ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) is the filter of the neighborhoods of 𝑝 for that topology. Proposition 1 of [BourbakiTop1] p. II.3. (Contributed by Thierry Arnoux, 11-Jan-2018.)
⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))    ⇒   ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑝 ∈ 𝑋 (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝}))
Theorem	utopsnneiplem 23397*	The neighborhoods of a point 𝑃 for the topology induced by an uniform space 𝑈. (Contributed by Thierry Arnoux, 11-Jan-2018.)
⊢ 𝐽 = (unifTop‘𝑈)    &   ⊢ 𝐾 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)}    &   ⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))    ⇒   ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
Theorem	utopsnneip 23398*	The neighborhoods of a point 𝑃 for the topology induced by an uniform space 𝑈. (Contributed by Thierry Arnoux, 13-Jan-2018.)
⊢ 𝐽 = (unifTop‘𝑈)    ⇒   ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
Theorem	utopsnnei 23399	Images of singletons by entourages 𝑉 are neighborhoods of those singletons. (Contributed by Thierry Arnoux, 13-Jan-2018.)
⊢ 𝐽 = (unifTop‘𝑈)    ⇒   ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → (𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}))
Theorem	utop2nei 23400	For any symmetrical entourage 𝑉 and any relation 𝑀, build a neighborhood of 𝑀. First part of proposition 2 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 14-Jan-2018.)
⊢ 𝐽 = (unifTop‘𝑈)    ⇒   ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑉 ∘ (𝑀 ∘ 𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘𝑀))