P. 135 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13401-13500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	strleun 13401	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 13402	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 13403	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 13404	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 13405	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 13406	Index value of the df-plusg 13387 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( +g ` ndx ) = 2
Theorem	plusgid 13407	Utility theorem: index-independent form of df-plusg 13387. (Contributed by NM, 20-Oct-2012.)
|- +g = Slot ( +g ` ndx )
Theorem	plusgndxnn 13408	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 13409	Slot property of +g. (Contributed by Jim Kingdon, 3-Feb-2023.)
|- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
Theorem	basendxltplusgndx 13410	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 13411	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 13412	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 13413	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 13414	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 13415	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 13416	A constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.) Use 2strstrndx 13415 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 13417	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 13418	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 13419	A constructed two-slot structure. Version of 2strstrg 13416 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 13420	The base set of a constructed two-slot structure. Version of 2strbasg 13417 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 13421	The other slot of a constructed two-slot structure. Version of 2stropg 13418 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 13422	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 13423	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 13424	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 13425	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 13426	+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 13427	Index value of the df-mulr 13388 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( .r ` ndx ) = 3
Theorem	mulridx 13428	Utility theorem: index-independent form of df-mulr 13388. (Contributed by Mario Carneiro, 8-Jun-2013.)
|- .r = Slot ( .r ` ndx )
Theorem	mulrslid 13429	Slot property of .r. (Contributed by Jim Kingdon, 3-Feb-2023.)
|- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
Theorem	plusgndxnmulrndx 13430	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 13431	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 13432	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 13433	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 13434	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 13435	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 13436	Index value of the df-starv 13389 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( *r ` ndx ) = 4
Theorem	starvid 13437	Utility theorem: index-independent form of df-starv 13389. (Contributed by Mario Carneiro, 6-Oct-2013.)
|- *r = Slot ( *r ` ndx )
Theorem	starvslid 13438	Slot property of *r. (Contributed by Jim Kingdon, 4-Feb-2023.)
|- ( *r = Slot ( *r ` ndx ) /\ ( *r ` ndx ) e. NN )
Theorem	starvndxnbasendx 13439	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 13440	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 13441	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 13442	.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 13443	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 13444	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 13445	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 13446	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 13447	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 13448	Index value of the df-sca 13390 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- (Scalar ` ndx ) = 5
Theorem	scaid 13449	Utility theorem: index-independent form of scalar df-sca 13390. (Contributed by Mario Carneiro, 19-Jun-2014.)
|- Scalar = Slot (Scalar ` ndx )
Theorem	scaslid 13450	Slot property of Scalar. (Contributed by Jim Kingdon, 5-Feb-2023.)
|- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
Theorem	scandxnbasendx 13451	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 13452	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 13453	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 13454	Index value of the df-vsca 13391 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( .s ` ndx ) = 6
Theorem	vscaid 13455	Utility theorem: index-independent form of scalar product df-vsca 13391. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- .s = Slot ( .s ` ndx )
Theorem	vscandxnbasendx 13456	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 13457	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 13458	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 13459	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 13460	Slot property of .s. (Contributed by Jim Kingdon, 5-Feb-2023.)
|- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
Theorem	lmodstrd 13461	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 13462	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 13463	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 13464	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 13465	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 13466	Index value of the df-ip 13392 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( .i ` ndx ) = 8
Theorem	ipid 13467	Utility theorem: index-independent form of df-ip 13392. (Contributed by Mario Carneiro, 6-Oct-2013.)
|- .i = Slot ( .i ` ndx )
Theorem	ipslid 13468	Slot property of .i. (Contributed by Jim Kingdon, 7-Feb-2023.)
|- ( .i = Slot ( .i ` ndx ) /\ ( .i ` ndx ) e. NN )
Theorem	ipndxnbasendx 13469	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 13470	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 13471	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 13472	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 13473	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 13474	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 13475	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 13476	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 13477	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 13478	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 13479	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 13480	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 13481	.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 13482	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 13483	Index value of the df-tset 13393 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- (TopSet ` ndx ) = 9
Theorem	tsetid 13484	Utility theorem: index-independent form of df-tset 13393. (Contributed by NM, 20-Oct-2012.)
|- TopSet = Slot (TopSet ` ndx )
Theorem	tsetslid 13485	Slot property of TopSet. (Contributed by Jim Kingdon, 9-Feb-2023.)
|- (TopSet = Slot (TopSet ` ndx ) /\ (TopSet ` ndx ) e. NN )
Theorem	tsetndxnn 13486	The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 31-Oct-2024.)
|- (TopSet ` ndx ) e. NN
Theorem	basendxlttsetndx 13487	The index of the slot for the base set is less then the index of the slot for the topology in an extensible structure. (Contributed by AV, 31-Oct-2024.)
|- ( Base ` ndx ) < (TopSet ` ndx )
Theorem	tsetndxnbasendx 13488	The slot for the topology is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 31-Oct-2024.)
|- (TopSet ` ndx ) =/= ( Base ` ndx )
Theorem	tsetndxnplusgndx 13489	The slot for the topology is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
|- (TopSet ` ndx ) =/= ( +g ` ndx )
Theorem	tsetndxnmulrndx 13490	The slot for the topology is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.)
|- (TopSet ` ndx ) =/= ( .r ` ndx )
Theorem	tsetndxnstarvndx 13491	The slot for the topology is not the slot for the involution in an extensible structure. (Contributed by AV, 11-Nov-2024.)
|- (TopSet ` ndx ) =/= ( *r ` ndx )
Theorem	slotstnscsi 13492	The slots Scalar, .s and .i are different from the slot TopSet. (Contributed by AV, 29-Oct-2024.)
|- ( (TopSet ` ndx ) =/= (Scalar ` ndx ) /\ (TopSet ` ndx ) =/= ( .s ` ndx ) /\ (TopSet ` ndx ) =/= ( .i ` ndx ) )
Theorem	topgrpstrd 13493	A constructed topological group is a structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
|- W = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , J >. }   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> J e. X )   =>    |- ( ph -> W Struct <. 1 , 9 >. )
Theorem	topgrpbasd 13494	The base set of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
|- W = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , J >. }   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> J e. X )   =>    |- ( ph -> B = ( Base ` W ) )
Theorem	topgrpplusgd 13495	The additive operation of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
|- W = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , J >. }   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> J e. X )   =>    |- ( ph -> .+ = ( +g ` W ) )
Theorem	topgrptsetd 13496	The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
|- W = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , J >. }   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> J e. X )   =>    |- ( ph -> J = (TopSet ` W ) )
Theorem	plendx 13497	Index value of the df-ple 13394 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
|- ( le ` ndx ) = ; 1 0
Theorem	pleid 13498	Utility theorem: self-referencing, index-independent form of df-ple 13394. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.)
|- le = Slot ( le ` ndx )
Theorem	pleslid 13499	Slot property of le. (Contributed by Jim Kingdon, 9-Feb-2023.)
|- ( le = Slot ( le ` ndx ) /\ ( le ` ndx ) e. NN )
Theorem	plendxnn 13500	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 ) e. NN