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Theorem List for Metamath Proof Explorer - 16101-16200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremotpsle 16101 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))
 
TheoremotpsstrOLD 16102 Obsolete version of otpsstr 16098 as of 9-Sep-2021. (Contributed by Mario Carneiro, 12-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ⟩}       𝐾 Struct ⟨1, 10⟩
 
TheoremotpsbasOLD 16103 Obsolete version of otpsbas 16099 as of 9-Sep-2021. (Contributed by Mario Carneiro, 12-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ⟩}       (𝐵𝑉𝐵 = (Base‘𝐾))
 
TheoremotpstsetOLD 16104 Obsolete version of otpstset 16100 as of 9-Sep-2021. (Contributed by Mario Carneiro, 12-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ⟩}       (𝐽𝑉𝐽 = (TopSet‘𝐾))
 
TheoremotpsleOLD 16105 Obsolete version of otpsle 16101 as of 9-Sep-2021. (Contributed by Mario Carneiro, 12-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ⟩}       ( 𝑉 = (le‘𝐾))
 
Theoremressle 16106 le is unaffected by restriction. (Contributed by Mario Carneiro, 3-Nov-2015.)
𝑊 = (𝐾s 𝐴)    &    = (le‘𝐾)       (𝐴𝑉 = (le‘𝑊))
 
Theoremocndx 16107 Index value of the df-ocomp 16010 slot. (Contributed by Mario Carneiro, 25-Oct-2015.)
(oc‘ndx) = 11
 
Theoremocid 16108 Utility theorem: index-independent form of df-ocomp 16010. (Contributed by Mario Carneiro, 25-Oct-2015.)
oc = Slot (oc‘ndx)
 
Theoremdsndx 16109 Index value of the df-ds 16011 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
(dist‘ndx) = 12
 
Theoremdsid 16110 Utility theorem: index-independent form of df-ds 16011. (Contributed by Mario Carneiro, 23-Dec-2013.)
dist = Slot (dist‘ndx)
 
Theoremunifndx 16111 Index value of the df-unif 16012 slot. (Contributed by Thierry Arnoux, 17-Dec-2017.)
(UnifSet‘ndx) = 13
 
Theoremunifid 16112 Utility theorem: index-independent form of df-unif 16012. (Contributed by Thierry Arnoux, 17-Dec-2017.)
UnifSet = Slot (UnifSet‘ndx)
 
Theoremodrngstr 16113 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 16114 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 16115 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 16116 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 16117 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 16118 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 16119 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 16120 dist is unaffected by restriction. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝐻 = (𝐺s 𝐴)    &   𝐷 = (dist‘𝐺)       (𝐴𝑉𝐷 = (dist‘𝐻))
 
Theoremhomndx 16121 Index value of the df-hom 16013 slot. (Contributed by Mario Carneiro, 7-Jan-2017.)
(Hom ‘ndx) = 14
 
Theoremhomid 16122 Utility theorem: index-independent form of df-hom 16013. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hom = Slot (Hom ‘ndx)
 
Theoremccondx 16123 Index value of the df-cco 16014 slot. (Contributed by Mario Carneiro, 7-Jan-2017.)
(comp‘ndx) = 15
 
Theoremccoid 16124 Utility theorem: index-independent form of df-cco 16014. (Contributed by Mario Carneiro, 7-Jan-2017.)
comp = Slot (comp‘ndx)
 
Theoremresshom 16125 Hom is unaffected by restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
𝐷 = (𝐶s 𝐴)    &   𝐻 = (Hom ‘𝐶)       (𝐴𝑉𝐻 = (Hom ‘𝐷))
 
Theoremressco 16126 comp is unaffected by restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
𝐷 = (𝐶s 𝐴)    &    · = (comp‘𝐶)       (𝐴𝑉· = (comp‘𝐷))
 
Theoremslotsbhcdif 16127 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 16128 Extend class notation with the function returning a subspace topology.
class t
 
Syntaxctopn 16129 Extend class notation with the topology extractor function.
class TopOpen
 
Definitiondf-rest 16130* 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 16131 Define the topology extractor function. This differs from df-tset 16007 when a structure has been restricted using df-ress 15912; 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 16132 The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.)
t Fn (V × V)
 
Theoremtopnfn 16133 The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOpen Fn V
 
Theoremrestval 16134* 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 16135* 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 16136 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 16137 Value of the structure restriction when the topology input is empty. (Contributed by Mario Carneiro, 13-Aug-2015.)
(∅ ↾t 𝐴) = ∅
 
Theoremrestid2 16138 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 16139 The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽t 𝐴) ⊆ 𝒫 𝐴
 
Theoremfirest 16140 The finite intersections operator commutes with restriction. (Contributed by Mario Carneiro, 30-Aug-2015.)
(fi‘(𝐽t 𝐴)) = ((fi‘𝐽) ↾t 𝐴)
 
Theoremrestid 16141 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 16142 Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopSet‘𝑊)       (𝐽t 𝐵) = (TopOpen‘𝑊)
 
Theoremtopnid 16143 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 16144 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 16145 Extend class notation with a function that converts a basis to its corresponding topology.
class topGen
 
Syntaxcpt 16146 Extend class notation with a function whose value is a product topology.
class t
 
Syntaxc0g 16147 Extend class notation with group identity element.
class 0g
 
Syntaxcgsu 16148 Extend class notation to include finitely supported group sums.
class Σg
 
Definitiondf-0g 16149* Define group identity element. Remark: this definition is required here because the symbol 0g is already used in df-gsum 16150. The related theorems are provided later, see grpidval 17307. (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 16150* 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 21977. Remark: this definition is required here because the symbol Σg is already used in df-prds 16155 and df-imas 16215. The related theorems are provided later, see gsumvalx 17317. (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 16151* 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 20808). The first use of this definition is tgval 20807 but the token is used in df-pt 16152. See tgval3 20815 for an alternate expression for the value. (Contributed by NM, 16-Jul-2006.)
topGen = (𝑥 ∈ V ↦ {𝑦𝑦 (𝑥 ∩ 𝒫 𝑦)})
 
Definitiondf-pt 16152* 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 16153 The function constructing structure products.
class Xs
 
Syntaxcpws 16154 The function constructing structure powers.
class s
 
Definitiondf-prds 16155* 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 16156 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 16157* 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 16158* Lemma for structure products. (Contributed by Mario Carneiro, 3-Jan-2015.)
𝐵 = X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))       𝐵 ∈ V
 
Theoremimasvalstr 16159 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 16160 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 16161 Lemma for prdsbas 16164 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 16162* 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 16163 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 16164* 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 16165* 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 16166* 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 16167* 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 16168* 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 16169* 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 16170 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 16171* 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 16172 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 16173 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 16174* 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 16175* 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 16176* 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 16177* 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 16178 Points in the structure product are functions; use this with dffn5 6280 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 16179 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 16180* 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 16181 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 16182* 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 16183 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 16184* 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)))
 
Theoremprdsdsval 16185* Value of the metric in a structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   𝐷 = (dist‘𝑌)       (𝜑 → (𝐹𝐷𝐺) = sup((ran (𝑥𝐼 ↦ ((𝐹𝑥)(dist‘(𝑅𝑥))(𝐺𝑥))) ∪ {0}), ℝ*, < ))
 
Theoremprdsvscaval 16186* Scalar multiplication in a structure product is pointwise. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &    · = ( ·𝑠𝑌)    &   𝐾 = (Base‘𝑆)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)    &   (𝜑𝐹𝐾)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐹 · 𝐺) = (𝑥𝐼 ↦ (𝐹( ·𝑠 ‘(𝑅𝑥))(𝐺𝑥))))
 
Theoremprdsvscafval 16187 Scalar multiplication of a single coordinate in a structure product. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &    · = ( ·𝑠𝑌)    &   𝐾 = (Base‘𝑆)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)    &   (𝜑𝐹𝐾)    &   (𝜑𝐺𝐵)    &   (𝜑𝐽𝐼)       (𝜑 → ((𝐹 · 𝐺)‘𝐽) = (𝐹( ·𝑠 ‘(𝑅𝐽))(𝐺𝐽)))
 
Theoremprdsbas3 16188* The base set of an indexed structure product. (Contributed by Mario Carneiro, 13-Sep-2015.)
𝑌 = (𝑆Xs(𝑥𝐼𝑅))    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑 → ∀𝑥𝐼 𝑅𝑋)    &   𝐾 = (Base‘𝑅)       (𝜑𝐵 = X𝑥𝐼 𝐾)
 
Theoremprdsbasmpt2 16189* A constructed tuple is a point in a structure product iff each coordinate is in the proper base set. (Contributed by Mario Carneiro, 3-Jul-2015.) (Revised by Mario Carneiro, 13-Sep-2015.)
𝑌 = (𝑆Xs(𝑥𝐼𝑅))    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑 → ∀𝑥𝐼 𝑅𝑋)    &   𝐾 = (Base‘𝑅)       (𝜑 → ((𝑥𝐼𝑈) ∈ 𝐵 ↔ ∀𝑥𝐼 𝑈𝐾))
 
Theoremprdsbascl 16190* An element of the base has projections closed in the factors. (Contributed by Mario Carneiro, 27-Aug-2015.)
𝑌 = (𝑆Xs(𝑥𝐼𝑅))    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑 → ∀𝑥𝐼 𝑅𝑋)    &   𝐾 = (Base‘𝑅)    &   (𝜑𝐹𝐵)       (𝜑 → ∀𝑥𝐼 (𝐹𝑥) ∈ 𝐾)
 
Theoremprdsdsval2 16191* Value of the metric in a structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝑌 = (𝑆Xs(𝑥𝐼𝑅))    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑 → ∀𝑥𝐼 𝑅𝑋)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   𝐸 = (dist‘𝑅)    &   𝐷 = (dist‘𝑌)       (𝜑 → (𝐹𝐷𝐺) = sup((ran (𝑥𝐼 ↦ ((𝐹𝑥)𝐸(𝐺𝑥))) ∪ {0}), ℝ*, < ))
 
Theoremprdsdsval3 16192* Value of the metric in a structure product. (Contributed by Mario Carneiro, 27-Aug-2015.)
𝑌 = (𝑆Xs(𝑥𝐼𝑅))    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑 → ∀𝑥𝐼 𝑅𝑋)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   𝐾 = (Base‘𝑅)    &   𝐸 = ((dist‘𝑅) ↾ (𝐾 × 𝐾))    &   𝐷 = (dist‘𝑌)       (𝜑 → (𝐹𝐷𝐺) = sup((ran (𝑥𝐼 ↦ ((𝐹𝑥)𝐸(𝐺𝑥))) ∪ {0}), ℝ*, < ))
 
Theorempwsval 16193 Value of a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐹 = (Scalar‘𝑅)       ((𝑅𝑉𝐼𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅})))
 
Theorempwsbas 16194 Base set of a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑅)       ((𝑅𝑉𝐼𝑊) → (𝐵𝑚 𝐼) = (Base‘𝑌))
 
Theorempwselbasb 16195 Membership in the base set of a structure product. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑅)    &   𝑉 = (Base‘𝑌)       ((𝑅𝑊𝐼𝑍) → (𝑋𝑉𝑋:𝐼𝐵))
 
Theorempwselbas 16196 An element of a structure power is a function from the index set to the base set of the structure. (Contributed by Mario Carneiro, 11-Jan-2015.) (Revised by Mario Carneiro, 5-Jun-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑅)    &   𝑉 = (Base‘𝑌)    &   (𝜑𝑅𝑊)    &   (𝜑𝐼𝑍)    &   (𝜑𝑋𝑉)       (𝜑𝑋:𝐼𝐵)
 
Theorempwsplusgval 16197 Value of addition in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑅𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &    + = (+g𝑅)    &    = (+g𝑌)       (𝜑 → (𝐹 𝐺) = (𝐹𝑓 + 𝐺))
 
Theorempwsmulrval 16198 Value of multiplication in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑅𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &    · = (.r𝑅)    &    = (.r𝑌)       (𝜑 → (𝐹 𝐺) = (𝐹𝑓 · 𝐺))
 
Theorempwsle 16199 Ordering in a structure power. (Contributed by Mario Carneiro, 16-Aug-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑌)    &   𝑂 = (le‘𝑅)    &    = (le‘𝑌)       ((𝑅𝑉𝐼𝑊) → = ( ∘𝑟 𝑂 ∩ (𝐵 × 𝐵)))
 
Theorempwsleval 16200* Ordering in a structure power. (Contributed by Mario Carneiro, 16-Aug-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑌)    &   𝑂 = (le‘𝑅)    &    = (le‘𝑌)    &   (𝜑𝑅𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐹 𝐺 ↔ ∀𝑥𝐼 (𝐹𝑥)𝑂(𝐺𝑥)))
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