P. 233 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 23201-23300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	tsms0 23201*	The sum of zero is zero. (Contributed by Mario Carneiro, 18-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 0 ∈ (𝐺 tsums (𝑥 ∈ 𝐴 ↦ 0 )))
Theorem	tsmssubm 23202	Evaluate an infinite group sum in a submonoid. (Contributed by Mario Carneiro, 18-Sep-2015.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopSp)    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺))    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ 𝐻 = (𝐺 ↾s 𝑆)    ⇒   ⊢ (𝜑 → (𝐻 tsums 𝐹) = ((𝐺 tsums 𝐹) ∩ 𝑆))
Theorem	tsmsres 23203	Extend an infinite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 18-Sep-2015.) (Revised by AV, 25-Jul-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝑊)    ⇒   ⊢ (𝜑 → (𝐺 tsums (𝐹 ↾ 𝑊)) = (𝐺 tsums 𝐹))
Theorem	tsmsf1o 23204	Re-index an infinite group sum using a bijection. (Contributed by Mario Carneiro, 18-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐻:𝐶–1-1-onto→𝐴)    ⇒   ⊢ (𝜑 → (𝐺 tsums 𝐹) = (𝐺 tsums (𝐹 ∘ 𝐻)))
Theorem	tsmsmhm 23205	Apply a continuous group homomorphism to an infinite group sum. (Contributed by Mario Carneiro, 18-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 𝐾 = (TopOpen‘𝐻)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopSp)    &   ⊢ (𝜑 → 𝐻 ∈ CMnd)    &   ⊢ (𝜑 → 𝐻 ∈ TopSp)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐺 MndHom 𝐻))    &   ⊢ (𝜑 → 𝐶 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹))    ⇒   ⊢ (𝜑 → (𝐶‘𝑋) ∈ (𝐻 tsums (𝐶 ∘ 𝐹)))
Theorem	tsmsadd 23206	The sum of two infinite group sums. (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 → 𝐻:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐺 tsums 𝐻))    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝐺 tsums (𝐹 ∘f + 𝐻)))
Theorem	tsmsinv 23207	Inverse of an infinite group sum. (Contributed by Mario Carneiro, 20-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopGrp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹))    ⇒   ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝐺 tsums (𝐼 ∘ 𝐹)))
Theorem	tsmssub 23208	The difference of two infinite group sums. (Contributed by Mario Carneiro, 20-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopGrp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐻:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐺 tsums 𝐻))    ⇒   ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝐺 tsums (𝐹 ∘f − 𝐻)))
Theorem	tgptsmscls 23209	A sum in a topological group is uniquely determined up to a coset of cls({0}), which is a normal subgroup by clsnsg 23169, 0nsg 18712. (Contributed by Mario Carneiro, 22-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopGrp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹))    ⇒   ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{𝑋}))
Theorem	tgptsmscld 23210	The set of limit points to an infinite sum in a topological group is closed. (Contributed by Mario Carneiro, 22-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopGrp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽))
Theorem	tsmssplit 23211	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 23212*	Lemma for tsmsxp 23214. (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 23213*	Lemma for tsmsxp 23214. (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 23214*	Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 19492 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 23212 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 23215	The class of all topological division rings.
class TopRing
Syntax	ctdrg 23216	The class of all topological division rings.
class TopDRing
Syntax	ctlm 23217	The class of all topological modules.
class TopMod
Syntax	ctvc 23218	The class of all topological vector spaces.
class TopVec
Definition	df-trg 23219	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 23220	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 23221	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 23222	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 23223	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 23224	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 23225	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 23226	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 23227	A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)
Theorem	trgtmd2 23228	A topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopMnd)
Theorem	trgtps 23229	A topological ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopSp)
Theorem	trgring 23230	A topological ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Ring)
Theorem	trggrp 23231	A topological ring is a group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Grp)
Theorem	tdrgtrg 23232	A topological division ring is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopRing)
Theorem	tdrgdrng 23233	A topological division ring is a division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ DivRing)
Theorem	tdrgring 23234	A topological division ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ Ring)
Theorem	tdrgtmd 23235	A topological division ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopMnd)
Theorem	tdrgtps 23236	A topological division ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopSp)
Theorem	istdrg2 23237	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 23238	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 23239	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 23240	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 23241*	Continuity of ring multiplication; analogue of cnmpt12f 22725 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 23242*	Continuity of ring multiplication; analogue of cnmpt22f 22734 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 23243	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 23244	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 23245	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 23246	A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopMnd)
Theorem	tlmtps 23247	A topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopSp)
Theorem	tlmlmod 23248	A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ LMod)
Theorem	tlmtrg 23249	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 23250	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 23251	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 23252	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 23253*	Continuity of scalar multiplication; analogue of cnmpt12f 22725 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 23254*	Continuity of scalar multiplication; analogue of cnmpt22f 22734 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 23255	A topological vector space is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopGrp)
Theorem	tvctlm 23256	A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopVec → 𝑊 ∈ TopMod)
Theorem	tvclmod 23257	A topological vector space is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopVec → 𝑊 ∈ LMod)
Theorem	tvclvec 23258	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 23259	Extend class notation with the class function of uniform structures.
class UnifOn
Definition	df-ust 23260*	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 23261	The defined uniform structure as a function. (Contributed by Thierry Arnoux, 15-Nov-2017.)
⊢ UnifOn Fn V
Theorem	ustval 23262*	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 23263*	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 23264	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 23265	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 23266	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 23267	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 23268	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 23269	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 23270*	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 23271	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 23272	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 23273	A uniform structure cannot be empty. (Contributed by Thierry Arnoux, 16-Nov-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)
Theorem	ustssco 23274	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 23275*	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 23276*	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 23277*	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 23278	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 23279	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 23280	The empty set is not an uniform structure. (Contributed by Thierry Arnoux, 3-Dec-2017.)
⊢ ¬ ∅ ∈ ∪ ran UnifOn
Theorem	ustund 23281	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 23282	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 23283	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 23284	First direction for ustbas 23287. (Contributed by Thierry Arnoux, 16-Nov-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn)
Theorem	ustbas2 23285	Second direction for ustbas 23287. (Contributed by Thierry Arnoux, 16-Nov-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
Theorem	ustuni 23286	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 23287	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 23288	Lemma for ustuqtop 23306. (Contributed by Thierry Arnoux, 5-Dec-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → (𝑉 “ {𝑃}) ⊆ 𝑋)
Theorem	trust 23289	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 23290	Extend class notation with the function inducing a topology from a uniform structure.
class unifTop
Definition	df-utop 23291*	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 23292*	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 23293*	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 23294	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 23295	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 23296	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 23297	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 23298	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 23299*	Lemma for ustuqtop 23306. (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 23300*	Lemma for ustuqtop 23306. (Contributed by Thierry Arnoux, 11-Jan-2018.)
⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))    ⇒   ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)