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Theorem List for Metamath Proof Explorer - 16501-16600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremressvsca 16501 ·𝑠 is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
𝐻 = (𝐺s 𝐴)    &    · = ( ·𝑠𝐺)       (𝐴𝑉· = ( ·𝑠𝐻))

Theoremressip 16502 The inner product is unaffected by restriction. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝐻 = (𝐺s 𝐴)    &    , = (·𝑖𝐺)       (𝐴𝑉, = (·𝑖𝐻))

Theoremphlstr 16503 A constructed pre-Hilbert space is a structure. Starting from lmodstr 16486 (which has 4 members), we chain strleun 16441 once more, adding an ordered pair to the function, to get all 5 members. (Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
𝐻 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩})       𝐻 Struct ⟨1, 8⟩

Theoremphlbase 16504 The base set of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
𝐻 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩})       (𝐵𝑋𝐵 = (Base‘𝐻))

Theoremphlplusg 16505 The additive operation of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
𝐻 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩})       ( +𝑋+ = (+g𝐻))

Theoremphlsca 16506 The ring of scalars of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
𝐻 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩})       (𝑇𝑋𝑇 = (Scalar‘𝐻))

Theoremphlvsca 16507 The scalar product operation of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
𝐻 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩})       ( ·𝑋· = ( ·𝑠𝐻))

Theoremphlip 16508 The inner product (Hermitian form) operation of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.)
𝐻 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑇⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩})       ( ,𝑋, = (·𝑖𝐻))

Theoremtsetndx 16509 Index value of the df-tset 16434 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(TopSet‘ndx) = 9

Theoremtsetid 16510 Utility theorem: index-independent form of df-tset 16434. (Contributed by NM, 20-Oct-2012.)
TopSet = Slot (TopSet‘ndx)

Theoremtopgrpstr 16511 A constructed topological group is a structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}       𝑊 Struct ⟨1, 9⟩

Theoremtopgrpbas 16512 The base set of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}       (𝐵𝑋𝐵 = (Base‘𝑊))

Theoremtopgrpplusg 16513 The additive operation of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}       ( +𝑋+ = (+g𝑊))

Theoremtopgrptset 16514 The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝑊 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}       (𝐽𝑋𝐽 = (TopSet‘𝑊))

Theoremresstset 16515 TopSet is unaffected by restriction. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐻 = (𝐺s 𝐴)    &   𝐽 = (TopSet‘𝐺)       (𝐴𝑉𝐽 = (TopSet‘𝐻))

Theoremplendx 16516 Index value of the df-ple 16435 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
(le‘ndx) = 10

Theorempleid 16517 Utility theorem: self-referencing, index-independent form of df-ple 16435. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.)
le = Slot (le‘ndx)

Theoremotpsstr 16518 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⟩

Theoremotpsbas 16519 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), ⟩}       (𝐵𝑉𝐵 = (Base‘𝐾))

Theoremotpstset 16520 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), ⟩}       (𝐽𝑉𝐽 = (TopSet‘𝐾))

Theoremotpsle 16521 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))

Theoremressle 16522 le is unaffected by restriction. (Contributed by Mario Carneiro, 3-Nov-2015.)
𝑊 = (𝐾s 𝐴)    &    = (le‘𝐾)       (𝐴𝑉 = (le‘𝑊))

Theoremocndx 16523 Index value of the df-ocomp 16436 slot. (Contributed by Mario Carneiro, 25-Oct-2015.)
(oc‘ndx) = 11

Theoremocid 16524 Utility theorem: index-independent form of df-ocomp 16436. (Contributed by Mario Carneiro, 25-Oct-2015.)
oc = Slot (oc‘ndx)

Theoremdsndx 16525 Index value of the df-ds 16437 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(dist‘ndx) = 12

Theoremdsid 16526 Utility theorem: index-independent form of df-ds 16437. (Contributed by Mario Carneiro, 23-Dec-2013.)
dist = Slot (dist‘ndx)

Theoremunifndx 16527 Index value of the df-unif 16438 slot. (Contributed by Thierry Arnoux, 17-Dec-2017.)
(UnifSet‘ndx) = 13

Theoremunifid 16528 Utility theorem: index-independent form of df-unif 16438. (Contributed by Thierry Arnoux, 17-Dec-2017.)
UnifSet = Slot (UnifSet‘ndx)

Theoremodrngstr 16529 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⟩

Theoremodrngbas 16530 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⟩})       (𝐵𝑉𝐵 = (Base‘𝑊))

Theoremodrngplusg 16531 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+ = (+g𝑊))

Theoremodrngmulr 16532 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· = (.r𝑊))

Theoremodrngtset 16533 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⟩})       (𝐽𝑉𝐽 = (TopSet‘𝑊))

Theoremodrngle 16534 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))

Theoremodrngds 16535 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⟩})       (𝐷𝑉𝐷 = (dist‘𝑊))

Theoremressds 16536 dist is unaffected by restriction. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝐻 = (𝐺s 𝐴)    &   𝐷 = (dist‘𝐺)       (𝐴𝑉𝐷 = (dist‘𝐻))

Theoremhomndx 16537 Index value of the df-hom 16439 slot. (Contributed by Mario Carneiro, 7-Jan-2017.)
(Hom ‘ndx) = 14

Theoremhomid 16538 Utility theorem: index-independent form of df-hom 16439. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hom = Slot (Hom ‘ndx)

Theoremccondx 16539 Index value of the df-cco 16440 slot. (Contributed by Mario Carneiro, 7-Jan-2017.)
(comp‘ndx) = 15

Theoremccoid 16540 Utility theorem: index-independent form of df-cco 16440. (Contributed by Mario Carneiro, 7-Jan-2017.)
comp = Slot (comp‘ndx)

Theoremresshom 16541 Hom is unaffected by restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
𝐷 = (𝐶s 𝐴)    &   𝐻 = (Hom ‘𝐶)       (𝐴𝑉𝐻 = (Hom ‘𝐷))

Theoremressco 16542 comp is unaffected by restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
𝐷 = (𝐶s 𝐴)    &    · = (comp‘𝐶)       (𝐴𝑉· = (comp‘𝐷))

Theoremslotsbhcdif 16543 The slots Base, Hom and comp are different. (Contributed by AV, 5-Mar-2020.)
((Base‘ndx) ≠ (Hom ‘ndx) ∧ (Base‘ndx) ≠ (comp‘ndx) ∧ (Hom ‘ndx) ≠ (comp‘ndx))

7.1.3  Definition of the structure product

Syntaxcrest 16544 Extend class notation with the function returning a subspace topology.
class t

Syntaxctopn 16545 Extend class notation with the topology extractor function.
class TopOpen

Definitiondf-rest 16546* 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 (𝑦𝑗 ↦ (𝑦𝑥)))

Definitiondf-topn 16547 Define the topology extractor function. This differs from df-tset 16434 when a structure has been restricted using df-ress 16341; 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)))

Theoremrestfn 16548 The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.)
t Fn (V × V)

Theoremtopnfn 16549 The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOpen Fn V

Theoremrestval 16550* 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.)
((𝐽𝑉𝐴𝑊) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))

Theoremelrest 16551* The predicate "is an open set of a subspace topology". (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
((𝐽𝑉𝐵𝑊) → (𝐴 ∈ (𝐽t 𝐵) ↔ ∃𝑥𝐽 𝐴 = (𝑥𝐵)))

Theoremelrestr 16552 Sufficient condition for being an open set in a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
((𝐽𝑉𝑆𝑊𝐴𝐽) → (𝐴𝑆) ∈ (𝐽t 𝑆))

Theorem0rest 16553 Value of the structure restriction when the topology input is empty. (Contributed by Mario Carneiro, 13-Aug-2015.)
(∅ ↾t 𝐴) = ∅

Theoremrestid2 16554 The subspace topology over a subset of the base set is the original topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → (𝐽t 𝐴) = 𝐽)

Theoremrestsspw 16555 The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽t 𝐴) ⊆ 𝒫 𝐴

Theoremfirest 16556 The finite intersections operator commutes with restriction. (Contributed by Mario Carneiro, 30-Aug-2015.)
(fi‘(𝐽t 𝐴)) = ((fi‘𝐽) ↾t 𝐴)

Theoremrestid 16557 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       (𝐽𝑉 → (𝐽t 𝑋) = 𝐽)

Theoremtopnval 16558 Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopSet‘𝑊)       (𝐽t 𝐵) = (TopOpen‘𝑊)

Theoremtopnid 16559 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 𝐵𝐽 = (TopOpen‘𝑊))

Theoremtopnpropd 16560 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))

Syntaxctg 16561 Extend class notation with a function that converts a basis to its corresponding topology.
class topGen

Syntaxcpt 16562 Extend class notation with a function whose value is a product topology.
class t

Syntaxc0g 16563 Extend class notation with group identity element.
class 0g

Syntaxcgsu 16564 Extend class notation to include finitely supported group sums.
class Σg

Definitiondf-0g 16565* Define group identity element. Remark: this definition is required here because the symbol 0g is already used in df-gsum 16566. The related theorems are provided later, see grpidval 17722. (Contributed by NM, 20-Aug-2011.)
0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g𝑔)𝑥) = 𝑥 ∧ (𝑥(+g𝑔)𝑒) = 𝑥))))

Definitiondf-gsum 16566* Define the group sum for the structure 𝐺 of a finite sequence of elements whose values are defined by the expression 𝐵 and whose set of indices is 𝐴. It may be viewed as a product (if 𝐺 is a multiplication), a sum (if 𝐺 is an addition) or any other operation. The variable 𝑘 is normally a free variable in 𝐵 (i.e., 𝐵 can be thought of as 𝐵(𝑘)). 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.

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. (𝐵(1) + 𝐵(2)) + 𝐵(3) etc.

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.

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 22432. Remark: this definition is required here because the symbol Σg is already used in df-prds 16571 and df-imas 16631. The related theorems are provided later, see gsumvalx 17732. (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(+g𝑤)𝑦) = 𝑦 ∧ (𝑦(+g𝑤)𝑥) = 𝑦)} / 𝑜if(ran 𝑓𝑜, (0g𝑤), if(dom 𝑓 ∈ ran ..., (℩𝑥𝑚𝑛 ∈ (ℤ𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝑤), 𝑓)‘𝑛))), (℩𝑥𝑔[(𝑓 “ (V ∖ 𝑜)) / 𝑦](𝑔:(1...(♯‘𝑦))–1-1-onto𝑦𝑥 = (seq1((+g𝑤), (𝑓𝑔))‘(♯‘𝑦)))))))

Definitiondf-topgen 16567* 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 21262). The first use of this definition is tgval 21261 but the token is used in df-pt 16568. See tgval3 21269 for an alternate expression for the value. (Contributed by NM, 16-Jul-2006.)
topGen = (𝑥 ∈ V ↦ {𝑦𝑦 (𝑥 ∩ 𝒫 𝑦)})

Definitiondf-pt 16568* 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(𝑔𝑦) ∈ (𝑓𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓𝑧)(𝑔𝑦) = (𝑓𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(𝑔𝑦))}))

Syntaxcprds 16569 The function constructing structure products.
class Xs

Syntaxcpws 16570 The function constructing structure powers.
class s

Definitiondf-prds 16571* 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.)
Xs = (𝑠 ∈ V, 𝑟 ∈ V ↦ X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) / 𝑣(𝑓𝑣, 𝑔𝑣X𝑥 ∈ dom 𝑟((𝑓𝑥)(Hom ‘(𝑟𝑥))(𝑔𝑥))) / (({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(+g‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(.r‘(𝑟𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(·𝑖‘(𝑟𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓𝑥)(le‘(𝑟𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓𝑣, 𝑔𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓𝑥)(dist‘(𝑟𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐𝑣 ↦ (𝑑 ∈ (𝑐(2nd𝑎)), 𝑒 ∈ (𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑟𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩})))

Theoremreldmprds 16572 The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.)
Rel dom Xs

Definitiondf-pws 16573* 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})))

Theoremprdsbasex 16574* Lemma for structure products. (Contributed by Mario Carneiro, 3-Jan-2015.)
𝐵 = X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))       𝐵 ∈ V

Theoremimasvalstr 16575 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.)
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⟩

Theoremprdsvalstr 16576 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⟩

Theoremprdsvallem 16577 Lemma for prdsbas 16580 and similar theorems. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝑈 = (({⟨(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 = (𝐸𝑈)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝜑𝑇 ∈ 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), ⟩}))       (𝜑𝐴 = 𝑇)

Theoremprdsval 16578* 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.)
𝑃 = (𝑆Xs𝑅)    &   𝐾 = (Base‘𝑆)    &   (𝜑 → dom 𝑅 = 𝐼)    &   (𝜑𝐵 = X𝑥𝐼 (Base‘(𝑅𝑥)))    &   (𝜑+ = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))))    &   (𝜑× = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))))    &   (𝜑· = (𝑓𝐾, 𝑔𝐵 ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))))    &   (𝜑, = (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))))    &   (𝜑𝑂 = (∏t‘(TopOpen ∘ 𝑅)))    &   (𝜑 = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))})    &   (𝜑𝐷 = (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )))    &   (𝜑𝐻 = (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))))    &   (𝜑 = (𝑎 ∈ (𝐵 × 𝐵), 𝑐𝐵 ↦ (𝑑 ∈ (𝑐𝐻(2nd𝑎)), 𝑒 ∈ (𝐻𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥))))))    &   (𝜑𝑆𝑊)    &   (𝜑𝑅𝑍)       (𝜑𝑃 = (({⟨(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), ⟩})))

Theoremprdssca 16579 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.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)       (𝜑𝑆 = (Scalar‘𝑃))

Theoremprdsbas 16580* 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.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)       (𝜑𝐵 = X𝑥𝐼 (Base‘(𝑅𝑥)))

Theoremprdsplusg 16581* 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.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &    + = (+g𝑃)       (𝜑+ = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))))

Theoremprdsmulr 16582* 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.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &    · = (.r𝑃)       (𝜑· = (𝑓𝐵, 𝑔𝐵 ↦ (𝑥𝐼 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))))

Theoremprdsvsca 16583* 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.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &   𝐾 = (Base‘𝑆)    &    · = ( ·𝑠𝑃)       (𝜑· = (𝑓𝐾, 𝑔𝐵 ↦ (𝑥𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))))

Theoremprdsip 16584* Inner product in a structure product. (Contributed by Thierry Arnoux, 16-Jun-2019.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &    , = (·𝑖𝑃)       (𝜑, = (𝑓𝐵, 𝑔𝐵 ↦ (𝑆 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))))

Theoremprdsle 16585* Structure product weak ordering. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &    = (le‘𝑃)       (𝜑 = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥𝐼 (𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))})

Theoremprdsless 16586 Closure of the order relation on a structure product. (Contributed by Mario Carneiro, 16-Aug-2015.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &    = (le‘𝑃)       (𝜑 ⊆ (𝐵 × 𝐵))

Theoremprdsds 16587* Structure product distance function. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &   𝐷 = (dist‘𝑃)       (𝜑𝐷 = (𝑓𝐵, 𝑔𝐵 ↦ sup((ran (𝑥𝐼 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )))

Theoremprdsdsfn 16588 Structure product distance function. (Contributed by Mario Carneiro, 15-Sep-2015.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &   𝐷 = (dist‘𝑃)       (𝜑𝐷 Fn (𝐵 × 𝐵))

Theoremprdstset 16589 Structure product topology. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &   𝑂 = (TopSet‘𝑃)       (𝜑𝑂 = (∏t‘(TopOpen ∘ 𝑅)))

Theoremprdshom 16590* Structure product hom-sets. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &   𝐻 = (Hom ‘𝑃)       (𝜑𝐻 = (𝑓𝐵, 𝑔𝐵X𝑥𝐼 ((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))))

Theoremprdsco 16591* Structure product composition operation. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &   𝐻 = (Hom ‘𝑃)    &    = (comp‘𝑃)       (𝜑 = (𝑎 ∈ (𝐵 × 𝐵), 𝑐𝐵 ↦ (𝑑 ∈ (𝑐𝐻(2nd𝑎)), 𝑒 ∈ (𝐻𝑎) ↦ (𝑥𝐼 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥))))))

Theoremprdsbas2 16592* The base set of a structure product is an indexed set product. (Contributed by Stefan O'Rear, 10-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)       (𝜑𝐵 = X𝑥𝐼 (Base‘(𝑅𝑥)))

Theoremprdsbasmpt 16593* A constructed tuple is a point in a structure product iff each coordinate is in the proper base set. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)       (𝜑 → ((𝑥𝐼𝑈) ∈ 𝐵 ↔ ∀𝑥𝐼 𝑈 ∈ (Base‘(𝑅𝑥))))

Theoremprdsbasfn 16594 Points in the structure product are functions; use this with dffn5 6552 to establish equalities. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)    &   (𝜑𝑇𝐵)       (𝜑𝑇 Fn 𝐼)

Theoremprdsbasprj 16595 Each point in a structure product restricts on each coordinate to the relevant base set. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)    &   (𝜑𝑇𝐵)    &   (𝜑𝐽𝐼)       (𝜑 → (𝑇𝐽) ∈ (Base‘(𝑅𝐽)))

Theoremprdsplusgval 16596* Value of a componentwise sum in a structure product. (Contributed by Stefan O'Rear, 10-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &    + = (+g𝑌)       (𝜑 → (𝐹 + 𝐺) = (𝑥𝐼 ↦ ((𝐹𝑥)(+g‘(𝑅𝑥))(𝐺𝑥))))

Theoremprdsplusgfval 16597 Value of a structure product sum at a single coordinate. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &    + = (+g𝑌)    &   (𝜑𝐽𝐼)       (𝜑 → ((𝐹 + 𝐺)‘𝐽) = ((𝐹𝐽)(+g‘(𝑅𝐽))(𝐺𝐽)))

Theoremprdsmulrval 16598* Value of a componentwise ring product in a structure product. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &    · = (.r𝑌)       (𝜑 → (𝐹 · 𝐺) = (𝑥𝐼 ↦ ((𝐹𝑥)(.r‘(𝑅𝑥))(𝐺𝑥))))

Theoremprdsmulrfval 16599 Value of a structure product's ring product at a single coordinate. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &    · = (.r𝑌)    &   (𝜑𝐽𝐼)       (𝜑 → ((𝐹 · 𝐺)‘𝐽) = ((𝐹𝐽)(.r‘(𝑅𝐽))(𝐺𝐽)))

Theoremprdsleval 16600* Value of the product ordering in a structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &    = (le‘𝑌)       (𝜑 → (𝐹 𝐺 ↔ ∀𝑥𝐼 (𝐹𝑥)(le‘(𝑅𝑥))(𝐺𝑥)))

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