P. 133 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13201-13300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	grpplusgg 13201	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 13202	+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 13203	Index value of the df-mulr 13164 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( .r ` ndx ) = 3
Theorem	mulridx 13204	Utility theorem: index-independent form of df-mulr 13164. (Contributed by Mario Carneiro, 8-Jun-2013.)
|- .r = Slot ( .r ` ndx )
Theorem	mulrslid 13205	Slot property of .r. (Contributed by Jim Kingdon, 3-Feb-2023.)
|- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
Theorem	plusgndxnmulrndx 13206	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 13207	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 13208	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 13209	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 13210	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 13211	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 13212	Index value of the df-starv 13165 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( *r ` ndx ) = 4
Theorem	starvid 13213	Utility theorem: index-independent form of df-starv 13165. (Contributed by Mario Carneiro, 6-Oct-2013.)
|- *r = Slot ( *r ` ndx )
Theorem	starvslid 13214	Slot property of *r. (Contributed by Jim Kingdon, 4-Feb-2023.)
|- ( *r = Slot ( *r ` ndx ) /\ ( *r ` ndx ) e. NN )
Theorem	starvndxnbasendx 13215	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 13216	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 13217	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 13218	.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 13219	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 13220	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 13221	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 13222	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 13223	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 13224	Index value of the df-sca 13166 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- (Scalar ` ndx ) = 5
Theorem	scaid 13225	Utility theorem: index-independent form of scalar df-sca 13166. (Contributed by Mario Carneiro, 19-Jun-2014.)
|- Scalar = Slot (Scalar ` ndx )
Theorem	scaslid 13226	Slot property of Scalar. (Contributed by Jim Kingdon, 5-Feb-2023.)
|- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
Theorem	scandxnbasendx 13227	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 13228	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 13229	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 13230	Index value of the df-vsca 13167 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( .s ` ndx ) = 6
Theorem	vscaid 13231	Utility theorem: index-independent form of scalar product df-vsca 13167. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
|- .s = Slot ( .s ` ndx )
Theorem	vscandxnbasendx 13232	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 13233	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 13234	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 13235	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 13236	Slot property of .s. (Contributed by Jim Kingdon, 5-Feb-2023.)
|- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
Theorem	lmodstrd 13237	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 13238	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 13239	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 13240	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 13241	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 13242	Index value of the df-ip 13168 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( .i ` ndx ) = 8
Theorem	ipid 13243	Utility theorem: index-independent form of df-ip 13168. (Contributed by Mario Carneiro, 6-Oct-2013.)
|- .i = Slot ( .i ` ndx )
Theorem	ipslid 13244	Slot property of .i. (Contributed by Jim Kingdon, 7-Feb-2023.)
|- ( .i = Slot ( .i ` ndx ) /\ ( .i ` ndx ) e. NN )
Theorem	ipndxnbasendx 13245	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 13246	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 13247	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 13248	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 13249	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 13250	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 13251	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 13252	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 13253	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 13254	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 13255	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 13256	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 13257	.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 13258	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 13259	Index value of the df-tset 13169 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- (TopSet ` ndx ) = 9
Theorem	tsetid 13260	Utility theorem: index-independent form of df-tset 13169. (Contributed by NM, 20-Oct-2012.)
|- TopSet = Slot (TopSet ` ndx )
Theorem	tsetslid 13261	Slot property of TopSet. (Contributed by Jim Kingdon, 9-Feb-2023.)
|- (TopSet = Slot (TopSet ` ndx ) /\ (TopSet ` ndx ) e. NN )
Theorem	tsetndxnn 13262	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 13263	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 13264	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 13265	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 13266	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 13267	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 13268	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 13269	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 13270	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 13271	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 13272	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 13273	Index value of the df-ple 13170 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
|- ( le ` ndx ) = ; 1 0
Theorem	pleid 13274	Utility theorem: self-referencing, index-independent form of df-ple 13170. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.)
|- le = Slot ( le ` ndx )
Theorem	pleslid 13275	Slot property of le. (Contributed by Jim Kingdon, 9-Feb-2023.)
|- ( le = Slot ( le ` ndx ) /\ ( le ` ndx ) e. NN )
Theorem	plendxnn 13276	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
Theorem	basendxltplendx 13277	The index value of the Base slot is less than the index value of the le slot. (Contributed by AV, 30-Oct-2024.)
|- ( Base ` ndx ) < ( le ` ndx )
Theorem	plendxnbasendx 13278	The slot for the order is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 30-Oct-2024.)
|- ( le ` ndx ) =/= ( Base ` ndx )
Theorem	plendxnplusgndx 13279	The slot for the "less than or equal to" ordering is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
|- ( le ` ndx ) =/= ( +g ` ndx )
Theorem	plendxnmulrndx 13280	The slot for the "less than or equal to" ordering is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 1-Nov-2024.)
|- ( le ` ndx ) =/= ( .r ` ndx )
Theorem	plendxnscandx 13281	The slot for the "less than or equal to" ordering is not the slot for the scalar in an extensible structure. (Contributed by AV, 1-Nov-2024.)
|- ( le ` ndx ) =/= (Scalar ` ndx )
Theorem	plendxnvscandx 13282	The slot for the "less than or equal to" ordering is not the slot for the scalar product in an extensible structure. (Contributed by AV, 1-Nov-2024.)
|- ( le ` ndx ) =/= ( .s ` ndx )
Theorem	slotsdifplendx 13283	The index of the slot for the distance is not the index of other slots. (Contributed by AV, 11-Nov-2024.)
|- ( ( *r ` ndx ) =/= ( le ` ndx ) /\ (TopSet ` ndx ) =/= ( le ` ndx ) )
Theorem	ocndx 13284	Index value of the df-ocomp 13171 slot. (Contributed by Mario Carneiro, 25-Oct-2015.) (New usage is discouraged.)
|- ( oc ` ndx ) = ; 1 1
Theorem	ocid 13285	Utility theorem: index-independent form of df-ocomp 13171. (Contributed by Mario Carneiro, 25-Oct-2015.)
|- oc = Slot ( oc ` ndx )
Theorem	basendxnocndx 13286	The slot for the orthocomplementation is not the slot for the base set in an extensible structure. (Contributed by AV, 11-Nov-2024.)
|- ( Base ` ndx ) =/= ( oc ` ndx )
Theorem	plendxnocndx 13287	The slot for the orthocomplementation is not the slot for the order in an extensible structure. (Contributed by AV, 11-Nov-2024.)
|- ( le ` ndx ) =/= ( oc ` ndx )
Theorem	dsndx 13288	Index value of the df-ds 13172 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( dist ` ndx ) = ; 1 2
Theorem	dsid 13289	Utility theorem: index-independent form of df-ds 13172. (Contributed by Mario Carneiro, 23-Dec-2013.)
|- dist = Slot ( dist ` ndx )
Theorem	dsslid 13290	Slot property of dist. (Contributed by Jim Kingdon, 6-May-2023.)
|- ( dist = Slot ( dist ` ndx ) /\ ( dist ` ndx ) e. NN )
Theorem	dsndxnn 13291	The index of the slot for the distance in an extensible structure is a positive integer. (Contributed by AV, 28-Oct-2024.)
|- ( dist ` ndx ) e. NN
Theorem	basendxltdsndx 13292	The index of the slot for the base set is less then the index of the slot for the distance in an extensible structure. (Contributed by AV, 28-Oct-2024.)
|- ( Base ` ndx ) < ( dist ` ndx )
Theorem	dsndxnbasendx 13293	The slot for the distance is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 28-Oct-2024.)
|- ( dist ` ndx ) =/= ( Base ` ndx )
Theorem	dsndxnplusgndx 13294	The slot for the distance function is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
|- ( dist ` ndx ) =/= ( +g ` ndx )
Theorem	dsndxnmulrndx 13295	The slot for the distance function is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.)
|- ( dist ` ndx ) =/= ( .r ` ndx )
Theorem	slotsdnscsi 13296	The slots Scalar, .s and .i are different from the slot dist. (Contributed by AV, 29-Oct-2024.)
|- ( ( dist ` ndx ) =/= (Scalar ` ndx ) /\ ( dist ` ndx ) =/= ( .s ` ndx ) /\ ( dist ` ndx ) =/= ( .i ` ndx ) )
Theorem	dsndxntsetndx 13297	The slot for the distance function is not the slot for the topology in an extensible structure. (Contributed by AV, 29-Oct-2024.)
|- ( dist ` ndx ) =/= (TopSet ` ndx )
Theorem	slotsdifdsndx 13298	The index of the slot for the distance is not the index of other slots. (Contributed by AV, 11-Nov-2024.)
|- ( ( *r ` ndx ) =/= ( dist ` ndx ) /\ ( le ` ndx ) =/= ( dist ` ndx ) )
Theorem	unifndx 13299	Index value of the df-unif 13173 slot. (Contributed by Thierry Arnoux, 17-Dec-2017.) (New usage is discouraged.)
|- ( UnifSet ` ndx ) = ; 1 3
Theorem	unifid 13300	Utility theorem: index-independent form of df-unif 13173. (Contributed by Thierry Arnoux, 17-Dec-2017.)
|- UnifSet = Slot ( UnifSet ` ndx )