P. 136 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13501-13600   *Has distinct variable group(s)

Type	Label	Description
Statement
Syntax	chom 13501	Extend class notation with the hom-set structure.
class Hom
Syntax	cco 13502	Extend class notation with the composition operation.
class comp
Definition	df-plusg 13503	Define group operation. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
|- +g = Slot 2
Definition	df-mulr 13504	Define ring multiplication. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
|- .r = Slot 3
Definition	df-starv 13505	Define the involution function of a *-ring. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
|- *r = Slot 4
Definition	df-sca 13506	Define scalar field component of a vector space v. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
|- Scalar = Slot 5
Definition	df-vsca 13507	Define scalar product. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
|- .s = Slot 6
Definition	df-ip 13508	Define Hermitian form (inner product). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
|- .i = Slot 8
Definition	df-tset 13509	Define the topology component of a topological space (structure). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
|- TopSet = Slot 9
Definition	df-ple 13510	Define "less than or equal to" ordering extractor for posets and related structures. We use ; 1 0 for the index to avoid conflict with 1 through 9 used for other purposes. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
|- le = Slot ; 1 0
Definition	df-ocomp 13511	Define the orthocomplementation extractor for posets and related structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
|- oc = Slot ; 1 1
Definition	df-ds 13512	Define the distance function component of a metric space (structure). (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
|- dist = Slot ; 1 2
Definition	df-unif 13513	Define the uniform structure component of a uniform space. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- UnifSet = Slot ; 1 3
Definition	df-hom 13514	Define the hom-set component of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- Hom = Slot ; 1 4
Definition	df-cco 13515	Define the composition operation of a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- comp = Slot ; 1 5
Theorem	strleund 13516	Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
|- ( ph -> F Struct <. A , B >. )   &    |- ( ph -> G Struct <. C , D >. )   &    |- ( ph -> B < C )   =>    |- ( ph -> ( F u. G ) Struct <. A , D >. )
Theorem	strleun 13517	Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.)
|- F Struct <. A , B >.   &    |- G Struct <. C , D >.   &    |- B < C   =>    |- ( F u. G ) Struct <. A , D >.
Theorem	strext 13518	Extending the upper range of a structure. This works because when we say that a structure has components in A ... C we are not saying that every slot in that range is present, just that all the slots that are present are within that range. (Contributed by Jim Kingdon, 26-Feb-2025.)
|- ( ph -> F Struct <. A , B >. )   &    |- ( ph -> C e. ( ZZ>= ` B ) )   =>    |- ( ph -> F Struct <. A , C >. )
Theorem	strle1g 13519	Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
|- I e. NN   &    |- A = I   =>    |- ( X e. V -> { <. A , X >. } Struct <. I , I >. )
Theorem	strle2g 13520	Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
|- I e. NN   &    |- A = I   &    |- I < J   &    |- J e. NN   &    |- B = J   =>    |- ( ( X e. V /\ Y e. W ) -> { <. A , X >. , <. B , Y >. } Struct <. I , J >. )
Theorem	strle3g 13521	Make a structure from a triple. (Contributed by Mario Carneiro, 29-Aug-2015.)
|- I e. NN   &    |- A = I   &    |- I < J   &    |- J e. NN   &    |- B = J   &    |- J < K   &    |- K e. NN   &    |- C = K   =>    |- ( ( X e. V /\ Y e. W /\ Z e. P ) -> { <. A , X >. , <. B , Y >. , <. C , Z >. } Struct <. I , K >. )
Theorem	plusgndx 13522	Index value of the df-plusg 13503 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( +g ` ndx ) = 2
Theorem	plusgid 13523	Utility theorem: index-independent form of df-plusg 13503. (Contributed by NM, 20-Oct-2012.)
|- +g = Slot ( +g ` ndx )
Theorem	plusgndxnn 13524	The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 17-Oct-2024.)
|- ( +g ` ndx ) e. NN
Theorem	plusgslid 13525	Slot property of +g. (Contributed by Jim Kingdon, 3-Feb-2023.)
|- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
Theorem	basendxltplusgndx 13526	The index of the slot for the base set is less then the index of the slot for the group operation in an extensible structure. (Contributed by AV, 17-Oct-2024.)
|- ( Base ` ndx ) < ( +g ` ndx )
Theorem	opelstrsl 13527	The slot of a structure which contains an ordered pair for that slot. (Contributed by Jim Kingdon, 5-Feb-2023.)
|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )   &    |- ( ph -> S Struct X )   &    |- ( ph -> V e. Y )   &    |- ( ph -> <. ( E ` ndx ) , V >. e. S )   =>    |- ( ph -> V = ( E ` S ) )
Theorem	opelstrbas 13528	The base set of a structure with a base set. (Contributed by AV, 10-Nov-2021.)
|- ( ph -> S Struct X )   &    |- ( ph -> V e. Y )   &    |- ( ph -> <. ( Base ` ndx ) , V >. e. S )   =>    |- ( ph -> V = ( Base ` S ) )
Theorem	1strstrg 13529	A constructed one-slot structure. (Contributed by AV, 27-Mar-2020.) (Revised by Jim Kingdon, 28-Jan-2023.)
|- G = { <. ( Base ` ndx ) , B >. }   =>    |- ( B e. V -> G Struct <. 1 , 1 >. )
Theorem	1strbas 13530	The base set of a constructed one-slot structure. (Contributed by AV, 27-Mar-2020.)
|- G = { <. ( Base ` ndx ) , B >. }   =>    |- ( B e. V -> B = ( Base ` G ) )
Theorem	2strstrndx 13531	A constructed two-slot structure not depending on the hard-coded index value of the base set. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 14-Dec-2025.)
|- G = { <. ( Base ` ndx ) , B >. , <. N , .+ >. }   &    |- ( Base ` ndx ) < N   &    |- N e. NN   =>    |- ( ( B e. V /\ .+ e. W ) -> G Struct <. ( Base ` ndx ) , N >. )
Theorem	2strstrg 13532	A constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.) Use 2strstrndx 13531 instead. (New usage is discouraged.)
|- G = { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. }   &    |- E = Slot N   &    |- 1 < N   &    |- N e. NN   =>    |- ( ( B e. V /\ .+ e. W ) -> G Struct <. 1 , N >. )
Theorem	2strbasg 13533	The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
|- G = { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. }   &    |- E = Slot N   &    |- 1 < N   &    |- N e. NN   =>    |- ( ( B e. V /\ .+ e. W ) -> B = ( Base ` G ) )
Theorem	2stropg 13534	The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
|- G = { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. }   &    |- E = Slot N   &    |- 1 < N   &    |- N e. NN   =>    |- ( ( B e. V /\ .+ e. W ) -> .+ = ( E ` G ) )
Theorem	2strstr1g 13535	A constructed two-slot structure. Version of 2strstrg 13532 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
|- G = { <. ( Base ` ndx ) , B >. , <. N , .+ >. }   &    |- ( Base ` ndx ) < N   &    |- N e. NN   =>    |- ( ( B e. V /\ .+ e. W ) -> G Struct <. ( Base ` ndx ) , N >. )
Theorem	2strbas1g 13536	The base set of a constructed two-slot structure. Version of 2strbasg 13533 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
|- G = { <. ( Base ` ndx ) , B >. , <. N , .+ >. }   &    |- ( Base ` ndx ) < N   &    |- N e. NN   =>    |- ( ( B e. V /\ .+ e. W ) -> B = ( Base ` G ) )
Theorem	2strop1g 13537	The other slot of a constructed two-slot structure. Version of 2stropg 13534 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
|- G = { <. ( Base ` ndx ) , B >. , <. N , .+ >. }   &    |- ( Base ` ndx ) < N   &    |- N e. NN   &    |- E = Slot N   =>    |- ( ( B e. V /\ .+ e. W ) -> .+ = ( E ` G ) )
Theorem	basendxnplusgndx 13538	The slot for the base set is not the slot for the group operation in an extensible structure. (Contributed by AV, 14-Nov-2021.)
|- ( Base ` ndx ) =/= ( +g ` ndx )
Theorem	grpstrg 13539	A constructed group is a structure on 1 ... 2. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
|- G = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. }   =>    |- ( ( B e. V /\ .+ e. W ) -> G Struct <. 1 , 2 >. )
Theorem	grpbaseg 13540	The base set of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
|- G = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. }   =>    |- ( ( B e. V /\ .+ e. W ) -> B = ( Base ` G ) )
Theorem	grpplusgg 13541	The operation of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
|- G = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. }   =>    |- ( ( B e. V /\ .+ e. W ) -> .+ = ( +g ` G ) )
Theorem	ressplusgd 13542	+g is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
|- ( ph -> H = ( G ↾s A ) )   &    |- ( ph -> .+ = ( +g ` G ) )   &    |- ( ph -> A e. V )   &    |- ( ph -> G e. W )   =>    |- ( ph -> .+ = ( +g ` H ) )
Theorem	mulrndx 13543	Index value of the df-mulr 13504 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( .r ` ndx ) = 3
Theorem	mulridx 13544	Utility theorem: index-independent form of df-mulr 13504. (Contributed by Mario Carneiro, 8-Jun-2013.)
|- .r = Slot ( .r ` ndx )
Theorem	mulrslid 13545	Slot property of .r. (Contributed by Jim Kingdon, 3-Feb-2023.)
|- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
Theorem	plusgndxnmulrndx 13546	The slot for the group (addition) operation is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020.)
|- ( +g ` ndx ) =/= ( .r ` ndx )
Theorem	basendxnmulrndx 13547	The slot for the base set is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020.)
|- ( Base ` ndx ) =/= ( .r ` ndx )
Theorem	rngstrg 13548	A constructed ring is a structure. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Jim Kingdon, 3-Feb-2023.)
|- R = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. }   =>    |- ( ( B e. V /\ .+ e. W /\ .x. e. X ) -> R Struct <. 1 , 3 >. )
Theorem	rngbaseg 13549	The base set of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 3-Feb-2023.)
|- R = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. }   =>    |- ( ( B e. V /\ .+ e. W /\ .x. e. X ) -> B = ( Base ` R ) )
Theorem	rngplusgg 13550	The additive operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
|- R = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. }   =>    |- ( ( B e. V /\ .+ e. W /\ .x. e. X ) -> .+ = ( +g ` R ) )
Theorem	rngmulrg 13551	The multiplicative operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
|- R = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. }   =>    |- ( ( B e. V /\ .+ e. W /\ .x. e. X ) -> .x. = ( .r ` R ) )
Theorem	starvndx 13552	Index value of the df-starv 13505 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( *r ` ndx ) = 4
Theorem	starvid 13553	Utility theorem: index-independent form of df-starv 13505. (Contributed by Mario Carneiro, 6-Oct-2013.)
|- *r = Slot ( *r ` ndx )
Theorem	starvslid 13554	Slot property of *r. (Contributed by Jim Kingdon, 4-Feb-2023.)
|- ( *r = Slot ( *r ` ndx ) /\ ( *r ` ndx ) e. NN )
Theorem	starvndxnbasendx 13555	The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
|- ( *r ` ndx ) =/= ( Base ` ndx )
Theorem	starvndxnplusgndx 13556	The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
|- ( *r ` ndx ) =/= ( +g ` ndx )
Theorem	starvndxnmulrndx 13557	The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
|- ( *r ` ndx ) =/= ( .r ` ndx )
Theorem	ressmulrg 13558	.r is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
|- S = ( R ↾s A )   &    |- .x. = ( .r ` R )   =>    |- ( ( A e. V /\ R e. W ) -> .x. = ( .r ` S ) )
Theorem	srngstrd 13559	A constructed star ring is a structure. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
|- R = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. } u. { <. ( *r ` ndx ) , .* >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .x. e. X )   &    |- ( ph -> .* e. Y )   =>    |- ( ph -> R Struct <. 1 , 4 >. )
Theorem	srngbased 13560	The base set of a constructed star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
|- R = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. } u. { <. ( *r ` ndx ) , .* >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .x. e. X )   &    |- ( ph -> .* e. Y )   =>    |- ( ph -> B = ( Base ` R ) )
Theorem	srngplusgd 13561	The addition operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.) (Revised by Jim Kingdon, 5-Feb-2023.)
|- R = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. } u. { <. ( *r ` ndx ) , .* >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .x. e. X )   &    |- ( ph -> .* e. Y )   =>    |- ( ph -> .+ = ( +g ` R ) )
Theorem	srngmulrd 13562	The multiplication operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.)
|- R = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. } u. { <. ( *r ` ndx ) , .* >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .x. e. X )   &    |- ( ph -> .* e. Y )   =>    |- ( ph -> .x. = ( .r ` R ) )
Theorem	srnginvld 13563	The involution function of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.)
|- R = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. } u. { <. ( *r ` ndx ) , .* >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .x. e. X )   &    |- ( ph -> .* e. Y )   =>    |- ( ph -> .* = ( *r ` R ) )
Theorem	scandx 13564	Index value of the df-sca 13506 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- (Scalar ` ndx ) = 5
Theorem	scaid 13565	Utility theorem: index-independent form of scalar df-sca 13506. (Contributed by Mario Carneiro, 19-Jun-2014.)
|- Scalar = Slot (Scalar ` ndx )
Theorem	scaslid 13566	Slot property of Scalar. (Contributed by Jim Kingdon, 5-Feb-2023.)
|- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
Theorem	scandxnbasendx 13567	The slot for the scalar is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
|- (Scalar ` ndx ) =/= ( Base ` ndx )
Theorem	scandxnplusgndx 13568	The slot for the scalar field is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
|- (Scalar ` ndx ) =/= ( +g ` ndx )
Theorem	scandxnmulrndx 13569	The slot for the scalar field is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
|- (Scalar ` ndx ) =/= ( .r ` ndx )
Theorem	vscandx 13570	Index value of the df-vsca 13507 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( .s ` ndx ) = 6
Theorem	vscaid 13571	Utility theorem: index-independent form of scalar product df-vsca 13507. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- .s = Slot ( .s ` ndx )
Theorem	vscandxnbasendx 13572	The slot for the scalar product is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
|- ( .s ` ndx ) =/= ( Base ` ndx )
Theorem	vscandxnplusgndx 13573	The slot for the scalar product is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
|- ( .s ` ndx ) =/= ( +g ` ndx )
Theorem	vscandxnmulrndx 13574	The slot for the scalar product is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
|- ( .s ` ndx ) =/= ( .r ` ndx )
Theorem	vscandxnscandx 13575	The slot for the scalar product is not the slot for the scalar field in an extensible structure. (Contributed by AV, 18-Oct-2024.)
|- ( .s ` ndx ) =/= (Scalar ` ndx )
Theorem	vscaslid 13576	Slot property of .s. (Contributed by Jim Kingdon, 5-Feb-2023.)
|- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
Theorem	lmodstrd 13577	A constructed left module or left vector space is a structure. (Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
|- W = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , F >. } u. { <. ( .s ` ndx ) , .x. >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. X )   &    |- ( ph -> F e. Y )   &    |- ( ph -> .x. e. Z )   =>    |- ( ph -> W Struct <. 1 , 6 >. )
Theorem	lmodbased 13578	The base set of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
|- W = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , F >. } u. { <. ( .s ` ndx ) , .x. >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. X )   &    |- ( ph -> F e. Y )   &    |- ( ph -> .x. e. Z )   =>    |- ( ph -> B = ( Base ` W ) )
Theorem	lmodplusgd 13579	The additive operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
|- W = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , F >. } u. { <. ( .s ` ndx ) , .x. >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. X )   &    |- ( ph -> F e. Y )   &    |- ( ph -> .x. e. Z )   =>    |- ( ph -> .+ = ( +g ` W ) )
Theorem	lmodscad 13580	The set of scalars of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
|- W = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , F >. } u. { <. ( .s ` ndx ) , .x. >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. X )   &    |- ( ph -> F e. Y )   &    |- ( ph -> .x. e. Z )   =>    |- ( ph -> F = (Scalar ` W ) )
Theorem	lmodvscad 13581	The scalar product operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 7-Feb-2023.)
|- W = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , F >. } u. { <. ( .s ` ndx ) , .x. >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. X )   &    |- ( ph -> F e. Y )   &    |- ( ph -> .x. e. Z )   =>    |- ( ph -> .x. = ( .s ` W ) )
Theorem	ipndx 13582	Index value of the df-ip 13508 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( .i ` ndx ) = 8
Theorem	ipid 13583	Utility theorem: index-independent form of df-ip 13508. (Contributed by Mario Carneiro, 6-Oct-2013.)
|- .i = Slot ( .i ` ndx )
Theorem	ipslid 13584	Slot property of .i. (Contributed by Jim Kingdon, 7-Feb-2023.)
|- ( .i = Slot ( .i ` ndx ) /\ ( .i ` ndx ) e. NN )
Theorem	ipndxnbasendx 13585	The slot for the inner product is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
|- ( .i ` ndx ) =/= ( Base ` ndx )
Theorem	ipndxnplusgndx 13586	The slot for the inner product is not the slot for the group operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
|- ( .i ` ndx ) =/= ( +g ` ndx )
Theorem	ipndxnmulrndx 13587	The slot for the inner product is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
|- ( .i ` ndx ) =/= ( .r ` ndx )
Theorem	slotsdifipndx 13588	The slot for the scalar is not the index of other slots. (Contributed by AV, 12-Nov-2024.)
|- ( ( .s ` ndx ) =/= ( .i ` ndx ) /\ (Scalar ` ndx ) =/= ( .i ` ndx ) )
Theorem	ipsstrd 13589	A constructed inner product space is a structure. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
|- A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .X. e. X )   &    |- ( ph -> S e. Y )   &    |- ( ph -> .x. e. Q )   &    |- ( ph -> I e. Z )   =>    |- ( ph -> A Struct <. 1 , 8 >. )
Theorem	ipsbased 13590	The base set of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
|- A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .X. e. X )   &    |- ( ph -> S e. Y )   &    |- ( ph -> .x. e. Q )   &    |- ( ph -> I e. Z )   =>    |- ( ph -> B = ( Base ` A ) )
Theorem	ipsaddgd 13591	The additive operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
|- A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .X. e. X )   &    |- ( ph -> S e. Y )   &    |- ( ph -> .x. e. Q )   &    |- ( ph -> I e. Z )   =>    |- ( ph -> .+ = ( +g ` A ) )
Theorem	ipsmulrd 13592	The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
|- A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .X. e. X )   &    |- ( ph -> S e. Y )   &    |- ( ph -> .x. e. Q )   &    |- ( ph -> I e. Z )   =>    |- ( ph -> .X. = ( .r ` A ) )
Theorem	ipsscad 13593	The set of scalars of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.)
|- A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .X. e. X )   &    |- ( ph -> S e. Y )   &    |- ( ph -> .x. e. Q )   &    |- ( ph -> I e. Z )   =>    |- ( ph -> S = (Scalar ` A ) )
Theorem	ipsvscad 13594	The scalar product operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.)
|- A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .X. e. X )   &    |- ( ph -> S e. Y )   &    |- ( ph -> .x. e. Q )   &    |- ( ph -> I e. Z )   =>    |- ( ph -> .x. = ( .s ` A ) )
Theorem	ipsipd 13595	The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.)
|- A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .X. e. X )   &    |- ( ph -> S e. Y )   &    |- ( ph -> .x. e. Q )   &    |- ( ph -> I e. Z )   =>    |- ( ph -> I = ( .i ` A ) )
Theorem	ressscag 13596	Scalar is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
|- H = ( G ↾s A )   &    |- F = (Scalar ` G )   =>    |- ( ( G e. X /\ A e. V ) -> F = (Scalar ` H ) )
Theorem	ressvscag 13597	.s is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
|- H = ( G ↾s A )   &    |- .x. = ( .s ` G )   =>    |- ( ( G e. X /\ A e. V ) -> .x. = ( .s ` H ) )
Theorem	ressipg 13598	The inner product is unaffected by restriction. (Contributed by Thierry Arnoux, 16-Jun-2019.)
|- H = ( G ↾s A )   &    |- ., = ( .i ` G )   =>    |- ( ( G e. X /\ A e. V ) -> ., = ( .i ` H ) )
Theorem	tsetndx 13599	Index value of the df-tset 13509 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- (TopSet ` ndx ) = 9
Theorem	tsetid 13600	Utility theorem: index-independent form of df-tset 13509. (Contributed by NM, 20-Oct-2012.)
|- TopSet = Slot (TopSet ` ndx )