P. 176 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 17501-17600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	plendxnn 17501	The index value of the order slot is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 30-Oct-2024.)
⊢ (le‘ndx) ∈ ℕ
Theorem	basendxltplendx 17502	The index value of the Base slot is less than the index value of the le slot. (Contributed by AV, 30-Oct-2024.)
⊢ (Base‘ndx) < (le‘ndx)
Theorem	plendxnbasendx 17503	The slot for the order is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 30-Oct-2024.)
⊢ (le‘ndx) ≠ (Base‘ndx)
Theorem	plendxnplusgndx 17504	The slot for the "less than or equal to" ordering is not the slot for the group operation in an extensible structure. Formerly part of proof for oppgle 19543. (Contributed by AV, 18-Oct-2024.)
⊢ (le‘ndx) ≠ (+g‘ndx)
Theorem	plendxnmulrndx 17505	The slot for the "less than or equal to" ordering is not the slot for the ring multiplication operation in an extensible structure. Formerly part of proof for opsrmulr 22323. (Contributed by AV, 1-Nov-2024.)
⊢ (le‘ndx) ≠ (.r‘ndx)
Theorem	plendxnscandx 17506	The slot for the "less than or equal to" ordering is not the slot for the scalar in an extensible structure. Formerly part of proof for opsrsca 22325. (Contributed by AV, 1-Nov-2024.)
⊢ (le‘ndx) ≠ (Scalar‘ndx)
Theorem	plendxnvscandx 17507	The slot for the "less than or equal to" ordering is not the slot for the scalar product in an extensible structure. Formerly part of proof for opsrvsca 22324. (Contributed by AV, 1-Nov-2024.)
⊢ (le‘ndx) ≠ ( ·𝑠 ‘ndx)
Theorem	slotsdifplendx 17508	The index of the slot for the distance is not the index of other slots. Formerly part of proof for cnfldfunALT 21655. (Contributed by AV, 11-Nov-2024.)
⊢ ((*𝑟‘ndx) ≠ (le‘ndx) ∧ (TopSet‘ndx) ≠ (le‘ndx))
Theorem	otpsstr 17509	Functionality of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.) (Revised by AV, 9-Sep-2021.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩}    ⇒   ⊢ 𝐾 Struct ⟨1, ;10⟩
Theorem	otpsbas 17510	The base set of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.) (Revised by AV, 9-Sep-2021.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩}    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐾))
Theorem	otpstset 17511	The open sets of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.) (Revised by AV, 9-Sep-2021.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩}    ⇒   ⊢ (𝐽 ∈ 𝑉 → 𝐽 = (TopSet‘𝐾))
Theorem	otpsle 17512	The order of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.) (Revised by AV, 9-Sep-2021.)
⊢ 𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩}    ⇒   ⊢ ( ≤ ∈ 𝑉 → ≤ = (le‘𝐾))
Theorem	ressle 17513	le is unaffected by restriction. (Contributed by Mario Carneiro, 3-Nov-2015.)
⊢ 𝑊 = (𝐾 ↾s 𝐴)    &   ⊢ ≤ = (le‘𝐾)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ≤ = (le‘𝑊))
Theorem	ocndx 17514	Index value of the df-ocomp 17411 slot. (Contributed by Mario Carneiro, 25-Oct-2015.) (New usage is discouraged.)
⊢ (oc‘ndx) = ;11
Theorem	ocid 17515	Utility theorem: index-independent form of df-ocomp 17411. (Contributed by Mario Carneiro, 25-Oct-2015.)
⊢ oc = Slot (oc‘ndx)
Theorem	basendxnocndx 17516	The slot for the orthocomplementation is not the slot for the base set in an extensible structure. Formerly part of proof for thlbas 21964. (Contributed by AV, 11-Nov-2024.)
⊢ (Base‘ndx) ≠ (oc‘ndx)
Theorem	plendxnocndx 17517	The slot for the orthocomplementation is not the slot for the order in an extensible structure. Formerly part of proof for thlle 21965. (Contributed by AV, 11-Nov-2024.)
⊢ (le‘ndx) ≠ (oc‘ndx)
Theorem	dsndx 17518	Index value of the df-ds 17412 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (New usage is discouraged.)
⊢ (dist‘ndx) = ;12
Theorem	dsid 17519	Utility theorem: index-independent form of df-ds 17412. (Contributed by Mario Carneiro, 23-Dec-2013.)
⊢ dist = Slot (dist‘ndx)
Theorem	dsndxnn 17520	The index of the slot for the distance in an extensible structure is a positive integer. Formerly part of proof for tmslem 24763. (Contributed by AV, 28-Oct-2024.)
⊢ (dist‘ndx) ∈ ℕ
Theorem	basendxltdsndx 17521	The index of the slot for the base set is less than the index of the slot for the distance in an extensible structure. Formerly part of proof for tmslem 24763. (Contributed by AV, 28-Oct-2024.)
⊢ (Base‘ndx) < (dist‘ndx)
Theorem	dsndxnbasendx 17522	The slot for the distance is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 28-Oct-2024.)
⊢ (dist‘ndx) ≠ (Base‘ndx)
Theorem	dsndxnplusgndx 17523	The slot for the distance function is not the slot for the group operation in an extensible structure. Formerly part of proof for mgpds 20331. (Contributed by AV, 18-Oct-2024.)
⊢ (dist‘ndx) ≠ (+g‘ndx)
Theorem	dsndxnmulrndx 17524	The slot for the distance function is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.)
⊢ (dist‘ndx) ≠ (.r‘ndx)
Theorem	slotsdnscsi 17525	The slots Scalar, ·𝑠 and ·𝑖 are different from the slot dist. Formerly part of sralem 21413 and proofs using it. (Contributed by AV, 29-Oct-2024.)
⊢ ((dist‘ndx) ≠ (Scalar‘ndx) ∧ (dist‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (dist‘ndx) ≠ (·𝑖‘ndx))
Theorem	dsndxntsetndx 17526	The slot for the distance function is not the slot for the topology in an extensible structure. Formerly part of proof for tngds 24929. (Contributed by AV, 29-Oct-2024.)
⊢ (dist‘ndx) ≠ (TopSet‘ndx)
Theorem	slotsdifdsndx 17527	The index of the slot for the distance is not the index of other slots. Formerly part of proof for cnfldfunALT 21655. (Contributed by AV, 11-Nov-2024.)
⊢ ((*𝑟‘ndx) ≠ (dist‘ndx) ∧ (le‘ndx) ≠ (dist‘ndx))
Theorem	unifndx 17528	Index value of the df-unif 17413 slot. (Contributed by Thierry Arnoux, 17-Dec-2017.) (New usage is discouraged.)
⊢ (UnifSet‘ndx) = ;13
Theorem	unifid 17529	Utility theorem: index-independent form of df-unif 17413. (Contributed by Thierry Arnoux, 17-Dec-2017.)
⊢ UnifSet = Slot (UnifSet‘ndx)
Theorem	unifndxnn 17530	The index of the slot for the uniform set in an extensible structure is a positive integer. Formerly part of proof for tuslem 24547. (Contributed by AV, 28-Oct-2024.)
⊢ (UnifSet‘ndx) ∈ ℕ
Theorem	basendxltunifndx 17531	The index of the slot for the base set is less than the index of the slot for the uniform set in an extensible structure. Formerly part of proof for tuslem 24547. (Contributed by AV, 28-Oct-2024.)
⊢ (Base‘ndx) < (UnifSet‘ndx)
Theorem	unifndxnbasendx 17532	The slot for the uniform set is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
⊢ (UnifSet‘ndx) ≠ (Base‘ndx)
Theorem	unifndxntsetndx 17533	The slot for the uniform set is not the slot for the topology in an extensible structure. Formerly part of proof for tuslem 24547. (Contributed by AV, 28-Oct-2024.)
⊢ (UnifSet‘ndx) ≠ (TopSet‘ndx)
Theorem	slotsdifunifndx 17534	The index of the slot for the uniform set is not the index of other slots. Formerly part of proof for cnfldfunALT 21655. (Contributed by AV, 10-Nov-2024.)
⊢ (((+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx) ∧ (*𝑟‘ndx) ≠ (UnifSet‘ndx)) ∧ ((le‘ndx) ≠ (UnifSet‘ndx) ∧ (dist‘ndx) ≠ (UnifSet‘ndx)))
Theorem	ressunif 17535	UnifSet is unaffected by restriction. (Contributed by Thierry Arnoux, 7-Dec-2017.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ 𝑈 = (UnifSet‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝑈 = (UnifSet‘𝐻))
Theorem	odrngstr 17536	Functionality of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.) (Proof shortened by AV, 15-Sep-2021.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩})    ⇒   ⊢ 𝑊 Struct ⟨1, ;12⟩
Theorem	odrngbas 17537	The base set of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩})    ⇒   ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝑊))
Theorem	odrngplusg 17538	The addition operation of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩})    ⇒   ⊢ ( + ∈ 𝑉 → + = (+g‘𝑊))
Theorem	odrngmulr 17539	The multiplication operation of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩})    ⇒   ⊢ ( · ∈ 𝑉 → · = (.r‘𝑊))
Theorem	odrngtset 17540	The open sets of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩})    ⇒   ⊢ (𝐽 ∈ 𝑉 → 𝐽 = (TopSet‘𝑊))
Theorem	odrngle 17541	The order of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩})    ⇒   ⊢ ( ≤ ∈ 𝑉 → ≤ = (le‘𝑊))
Theorem	odrngds 17542	The metric of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩})    ⇒   ⊢ (𝐷 ∈ 𝑉 → 𝐷 = (dist‘𝑊))
Theorem	ressds 17543	dist is unaffected by restriction. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ 𝐷 = (dist‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐷 = (dist‘𝐻))
Theorem	homndx 17544	Index value of the df-hom 17414 slot. (Contributed by Mario Carneiro, 7-Jan-2017.) (New usage is discouraged.)
⊢ (Hom ‘ndx) = ;14
Theorem	homid 17545	Utility theorem: index-independent form of df-hom 17414. (Contributed by Mario Carneiro, 7-Jan-2017.)
⊢ Hom = Slot (Hom ‘ndx)
Theorem	ccondx 17546	Index value of the df-cco 17415 slot. (Contributed by Mario Carneiro, 7-Jan-2017.) (New usage is discouraged.)
⊢ (comp‘ndx) = ;15
Theorem	ccoid 17547	Utility theorem: index-independent form of df-cco 17415. (Contributed by Mario Carneiro, 7-Jan-2017.)
⊢ comp = Slot (comp‘ndx)
Theorem	slotsbhcdif 17548	The slots Base, Hom and comp are different. (Contributed by AV, 5-Mar-2020.) (Proof shortened by AV, 28-Oct-2024.)
⊢ ((Base‘ndx) ≠ (Hom ‘ndx) ∧ (Base‘ndx) ≠ (comp‘ndx) ∧ (Hom ‘ndx) ≠ (comp‘ndx))
Theorem	slotsdifplendx2 17549	The index of the slot for the "less than or equal to" ordering is not the index of other slots. Formerly part of proof for prstcleval 50585. (Contributed by AV, 12-Nov-2024.)
⊢ ((le‘ndx) ≠ (comp‘ndx) ∧ (le‘ndx) ≠ (Hom ‘ndx))
Theorem	slotsdifocndx 17550	The index of the slot for the orthocomplementation is not the index of other slots. Formerly part of proof for prstcocval 50587. (Contributed by AV, 12-Nov-2024.)
⊢ ((oc‘ndx) ≠ (comp‘ndx) ∧ (oc‘ndx) ≠ (Hom ‘ndx))
Theorem	resshom 17551	Hom is unaffected by restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
⊢ 𝐷 = (𝐶 ↾s 𝐴)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐻 = (Hom ‘𝐷))
Theorem	ressco 17552	comp is unaffected by restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
⊢ 𝐷 = (𝐶 ↾s 𝐴)    &   ⊢ · = (comp‘𝐶)    ⇒   ⊢ (𝐴 ∈ 𝑉 → · = (comp‘𝐷))
7.1.3  Definition of the structure product
Syntax	crest 17553	Extend class notation with the function returning a subspace topology.
class ↾t
Syntax	ctopn 17554	Extend class notation with the topology extractor function.
class TopOpen
Definition	df-rest 17555*	Function returning the subspace topology induced by the topology 𝑦 and the set 𝑥. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.)
⊢ ↾t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥)))
Definition	df-topn 17556	Define the topology extractor function. This differs from df-tset 17409 when a structure has been restricted using df-ress 17371; in this case the TopSet component will still have a topology over the larger set, and this function fixes this by restricting the topology as well. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤)))
Theorem	restfn 17557	The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.)
⊢ ↾t Fn (V × V)
Theorem	topnfn 17558	The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ TopOpen Fn V
Theorem	restval 17559*	The subspace topology induced by the topology 𝐽 on the set 𝐴. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.)
⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
Theorem	elrest 17560*	The predicate "is an open set of a subspace topology". (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
⊢ ((𝐽 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐽 𝐴 = (𝑥 ∩ 𝐵)))
Theorem	elrestr 17561	Sufficient condition for being an open set in a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
⊢ ((𝐽 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ∧ 𝐴 ∈ 𝐽) → (𝐴 ∩ 𝑆) ∈ (𝐽 ↾t 𝑆))
Theorem	0rest 17562	Value of the structure restriction when the topology input is empty. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ (∅ ↾t 𝐴) = ∅
Theorem	restid2 17563	The subspace topology over a subset of the base set is the original topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ⊆ 𝒫 𝐴) → (𝐽 ↾t 𝐴) = 𝐽)
Theorem	restsspw 17564	The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴
Theorem	firest 17565	The finite intersections operator commutes with restriction. (Contributed by Mario Carneiro, 30-Aug-2015.)
⊢ (fi‘(𝐽 ↾t 𝐴)) = ((fi‘𝐽) ↾t 𝐴)
Theorem	restid 17566	The subspace topology of the base set is the original topology. (Contributed by Jeff Hankins, 9-Jul-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ 𝑉 → (𝐽 ↾t 𝑋) = 𝐽)
Theorem	topnval 17567	Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopSet‘𝑊)    ⇒   ⊢ (𝐽 ↾t 𝐵) = (TopOpen‘𝑊)
Theorem	topnid 17568	Value of the topology extractor function when the topology is defined over the same set as the base. (Contributed by Mario Carneiro, 13-Aug-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopSet‘𝑊)    ⇒   ⊢ (𝐽 ⊆ 𝒫 𝐵 → 𝐽 = (TopOpen‘𝑊))
Theorem	topnpropd 17569	The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.)
⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   ⊢ (𝜑 → (TopSet‘𝐾) = (TopSet‘𝐿))    ⇒   ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
Syntax	ctg 17570	Extend class notation with a function that converts a basis to its corresponding topology.
class topGen
Syntax	cpt 17571	Extend class notation with a function whose value is a product topology.
class ∏t
Syntax	c0g 17572	Extend class notation with group identity element.
class 0g
Syntax	cgsu 17573	Extend class notation to include finitely supported group sums.
class Σg
Definition	df-0g 17574*	Define group identity element. Remark: this definition is required here because the symbol 0g is already used in df-gsum 17575. The related theorems are provided later, see grpidval 18802. (Contributed by NM, 20-Aug-2011.)
⊢ 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥))))
Definition	df-gsum 17575*	Define a finite group sum (also called "iterated sum") of a structure. Given 𝐺 Σg 𝐹 where 𝐹:𝐴⟶(Base‘𝐺), the set of indices is 𝐴 and the values are given by 𝐹 at each index. A group sum over a multiplicative group may be viewed as a product. The definition is meaningful in different contexts, depending on the size of the index set 𝐴 and each demanding different properties of 𝐺. 1. If 𝐴 = ∅ and 𝐺 has an identity element, then the sum equals this identity. See gsum0 18835. 2. If 𝐴 = (𝑀...𝑁) and 𝐺 is any magma, then the sum is the sum of the elements, evaluated left-to-right, i.e., ((𝐹‘1) + (𝐹‘2)) + (𝐹‘3), etc. See gsumval2 18837 and gsumnunsn 35108. 3. If 𝐴 is a finite set (or is nonzero for finitely many indices) and 𝐺 is a commutative monoid, then the sum adds up these elements in some order, which is then uniquely defined. See gsumval3 20083. 4. If 𝐴 is an infinite set and 𝐺 is a Hausdorff topological group, then there is a meaningful sum, but Σg cannot handle this case. See df-tsms 24408. Remark: this definition is required here because the symbol Σg is already used in df-prds 17580 and df-imas 17642. The related theorems are provided later, see gsumvalx 18827. (Contributed by FL, 5-Sep-2010.) (Revised by FL, 17-Oct-2011.) (Revised by Mario Carneiro, 7-Dec-2014.)
⊢ Σg = (𝑤 ∈ V, 𝑓 ∈ V ↦ ⦋{𝑥 ∈ (Base‘𝑤) ∣ ∀𝑦 ∈ (Base‘𝑤)((𝑥(+g‘𝑤)𝑦) = 𝑦 ∧ (𝑦(+g‘𝑤)𝑥) = 𝑦)} / 𝑜⦌if(ran 𝑓 ⊆ 𝑜, (0g‘𝑤), if(dom 𝑓 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))), (℩𝑥∃𝑔[(◡𝑓 “ (V ∖ 𝑜)) / 𝑦](𝑔:(1...(♯‘𝑦))–1-1-onto→𝑦 ∧ 𝑥 = (seq1((+g‘𝑤), (𝑓 ∘ 𝑔))‘(♯‘𝑦)))))))
Definition	df-topgen 17576*	Define a function that converts a basis to its corresponding topology. Equivalent to the definition of a topology generated by a basis in [Munkres] p. 78 (see tgval2 23236). The first use of this definition is tgval 23235 but the token is used in df-pt 17577. See tgval3 23243 for an alternate expression for the value. (Contributed by NM, 16-Jul-2006.)
⊢ topGen = (𝑥 ∈ V ↦ {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)})
Definition	df-pt 17577*	Define the product topology on a collection of topologies. For convenience, it is defined on arbitrary collections of sets, expressed as a function from some index set to the subbases of each factor space. (Contributed by Mario Carneiro, 3-Feb-2015.)
⊢ ∏t = (𝑓 ∈ V ↦ (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔‘𝑦) ∈ (𝑓‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝑓‘𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(𝑔‘𝑦))}))
Syntax	cprds 17578	The function constructing structure products.
class Xs
Syntax	cpws 17579	The function constructing structure powers.
class ↑s
Definition	df-prds 17580*	Define a structure product. This can be a product of groups, rings, modules, or ordered topological fields; any unused components will have garbage in them but this is usually not relevant for the purpose of inheriting the structures present in the factors. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
⊢ Xs = (𝑠 ∈ V, 𝑟 ∈ V ↦ ⦋X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) / 𝑣⦌⦋(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) / ℎ⦌(({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ ((2nd ‘𝑎)ℎ𝑐), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})))
Theorem	reldmprds 17581	The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
⊢ Rel dom Xs
Definition	df-pws 17582*	Define a structure power, which is just a structure product where all the factors are the same. (Contributed by Mario Carneiro, 11-Jan-2015.)
⊢ ↑s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)Xs(𝑖 × {𝑟})))
Theorem	prdsbasex 17583*	Lemma for structure products. (Contributed by Mario Carneiro, 3-Jan-2015.)
⊢ 𝐵 = X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))    ⇒   ⊢ 𝐵 ∈ V
Theorem	imasvalstr 17584	An image structure value is a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
⊢ 𝑈 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ {⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), 𝐿⟩, ⟨(dist‘ndx), 𝐷⟩})    ⇒   ⊢ 𝑈 Struct ⟨1, ;12⟩
Theorem	prdsvalstr 17585	Structure product value is a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
⊢ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), 𝐿⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ∙ ⟩})) Struct ⟨1, ;15⟩
Theorem	prdsbaslem 17586	Lemma for prdsbas 17590 and similar theorems. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 12-Jul-2024.)
⊢ (𝜑 → 𝑈 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), 𝐿⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ∙ ⟩})))    &   ⊢ 𝐴 = (𝐸‘𝑈)    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    &   ⊢ {⟨(𝐸‘ndx), 𝑇⟩} ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), 𝐿⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ∙ ⟩}))    ⇒   ⊢ (𝜑 → 𝐴 = 𝑇)
Theorem	prdsvallem 17587*	Lemma for prdsval 17588. (Contributed by Stefan O'Rear, 3-Jan-2015.) Extracted from the former proof of prdsval 17588, dependency on df-hom 17414 removed. (Revised by AV, 13-Oct-2024.)
⊢ (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) ∈ V
Theorem	prdsval 17588*	Value of the structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ (𝜑 → dom 𝑅 = 𝐼)    &   ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))    &   ⊢ (𝜑 → + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))))    &   ⊢ (𝜑 → × = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)))))    &   ⊢ (𝜑 → · = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))))    &   ⊢ (𝜑 → , = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))))    &   ⊢ (𝜑 → 𝑂 = (∏t‘(TopOpen ∘ 𝑅)))    &   ⊢ (𝜑 → ≤ = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))})    &   ⊢ (𝜑 → 𝐷 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < )))    &   ⊢ (𝜑 → 𝐻 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))))    &   ⊢ (𝜑 → ∙ = (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))))    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    ⇒   ⊢ (𝜑 → 𝑃 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ∙ ⟩})))
Theorem	prdssca 17589	Scalar ring of a structure product. (Contributed by Stefan O'Rear, 5-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝑆 = (Scalar‘𝑃))
Theorem	prdsbas 17590*	Base set of a structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → dom 𝑅 = 𝐼)    ⇒   ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
Theorem	prdsplusg 17591*	Addition in a structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → dom 𝑅 = 𝐼)    &   ⊢ + = (+g‘𝑃)    ⇒   ⊢ (𝜑 → + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))))
Theorem	prdsmulr 17592*	Multiplication in a structure product. (Contributed by Mario Carneiro, 11-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → dom 𝑅 = 𝐼)    &   ⊢ · = (.r‘𝑃)    ⇒   ⊢ (𝜑 → · = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)))))
Theorem	prdsvsca 17593*	Scalar multiplication in a structure product. (Contributed by Stefan O'Rear, 5-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → dom 𝑅 = 𝐼)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ · = ( ·𝑠 ‘𝑃)    ⇒   ⊢ (𝜑 → · = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))))
Theorem	prdsip 17594*	Inner product in a structure product. (Contributed by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → dom 𝑅 = 𝐼)    &   ⊢ , = (·𝑖‘𝑃)    ⇒   ⊢ (𝜑 → , = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))))
Theorem	prdsle 17595*	Structure product weak ordering. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → dom 𝑅 = 𝐼)    &   ⊢ ≤ = (le‘𝑃)    ⇒   ⊢ (𝜑 → ≤ = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))})
Theorem	prdsless 17596	Closure of the order relation on a structure product. (Contributed by Mario Carneiro, 16-Aug-2015.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → dom 𝑅 = 𝐼)    &   ⊢ ≤ = (le‘𝑃)    ⇒   ⊢ (𝜑 → ≤ ⊆ (𝐵 × 𝐵))
Theorem	prdsds 17597*	Structure product distance function. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → dom 𝑅 = 𝐼)    &   ⊢ 𝐷 = (dist‘𝑃)    ⇒   ⊢ (𝜑 → 𝐷 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < )))
Theorem	prdsdsfn 17598	Structure product distance function. (Contributed by Mario Carneiro, 15-Sep-2015.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → dom 𝑅 = 𝐼)    &   ⊢ 𝐷 = (dist‘𝑃)    ⇒   ⊢ (𝜑 → 𝐷 Fn (𝐵 × 𝐵))
Theorem	prdstset 17599	Structure product topology. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → dom 𝑅 = 𝐼)    &   ⊢ 𝑂 = (TopSet‘𝑃)    ⇒   ⊢ (𝜑 → 𝑂 = (∏t‘(TopOpen ∘ 𝑅)))
Theorem	prdshom 17600*	Structure product hom-sets. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → dom 𝑅 = 𝐼)    &   ⊢ 𝐻 = (Hom ‘𝑃)    ⇒   ⊢ (𝜑 → 𝐻 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))))