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Theorem List for Intuitionistic Logic Explorer - 12901-13000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgrpbaseg 12901 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 ) )
 
Theoremgrpplusgg 12902 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 ) )
 
Theoremressplusgd 12903  +g is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  ( ph  ->  H  =  ( Gs  A ) )   &    |-  ( ph  ->  .+  =  ( +g  `  G ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  G  e.  W )   =>    |-  ( ph  ->  .+  =  ( +g  `  H ) )
 
Theoremmulrndx 12904 Index value of the df-mulr 12865 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( .r `  ndx )  =  3
 
Theoremmulridx 12905 Utility theorem: index-independent form of df-mulr 12865. (Contributed by Mario Carneiro, 8-Jun-2013.)
 |- 
 .r  = Slot  ( .r ` 
 ndx )
 
Theoremmulrslid 12906 Slot property of  .r. (Contributed by Jim Kingdon, 3-Feb-2023.)
 |-  ( .r  = Slot  ( .r `  ndx )  /\  ( .r `  ndx )  e.  NN )
 
Theoremplusgndxnmulrndx 12907 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 )
 
Theorembasendxnmulrndx 12908 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 )
 
Theoremrngstrg 12909 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 >. )
 
Theoremrngbaseg 12910 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 )
 )
 
Theoremrngplusgg 12911 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 )
 )
 
Theoremrngmulrg 12912 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 ) )
 
Theoremstarvndx 12913 Index value of the df-starv 12866 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( *r `  ndx )  =  4
 
Theoremstarvid 12914 Utility theorem: index-independent form of df-starv 12866. (Contributed by Mario Carneiro, 6-Oct-2013.)
 |-  *r  = Slot  ( *r `  ndx )
 
Theoremstarvslid 12915 Slot property of  *r. (Contributed by Jim Kingdon, 4-Feb-2023.)
 |-  ( *r  = Slot 
 ( *r `  ndx )  /\  ( *r `  ndx )  e.  NN )
 
Theoremstarvndxnbasendx 12916 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 )
 
Theoremstarvndxnplusgndx 12917 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 )
 
Theoremstarvndxnmulrndx 12918 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 )
 
Theoremressmulrg 12919  .r is unaffected by restriction. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  S  =  ( Rs  A )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( A  e.  V  /\  R  e.  W )  ->  .x.  =  ( .r `  S ) )
 
Theoremsrngstrd 12920 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 >.
 )
 
Theoremsrngbased 12921 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 ) )
 
Theoremsrngplusgd 12922 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 ) )
 
Theoremsrngmulrd 12923 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 ) )
 
Theoremsrnginvld 12924 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 ) )
 
Theoremscandx 12925 Index value of the df-sca 12867 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  (Scalar `  ndx )  =  5
 
Theoremscaid 12926 Utility theorem: index-independent form of scalar df-sca 12867. (Contributed by Mario Carneiro, 19-Jun-2014.)
 |- Scalar  = Slot  (Scalar `  ndx )
 
Theoremscaslid 12927 Slot property of Scalar. (Contributed by Jim Kingdon, 5-Feb-2023.)
 |-  (Scalar  = Slot  (Scalar `  ndx )  /\  (Scalar `  ndx )  e.  NN )
 
Theoremscandxnbasendx 12928 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 )
 
Theoremscandxnplusgndx 12929 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 )
 
Theoremscandxnmulrndx 12930 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 )
 
Theoremvscandx 12931 Index value of the df-vsca 12868 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( .s `  ndx )  =  6
 
Theoremvscaid 12932 Utility theorem: index-independent form of scalar product df-vsca 12868. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |- 
 .s  = Slot  ( .s ` 
 ndx )
 
Theoremvscandxnbasendx 12933 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 )
 
Theoremvscandxnplusgndx 12934 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 )
 
Theoremvscandxnmulrndx 12935 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 )
 
Theoremvscandxnscandx 12936 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 )
 
Theoremvscaslid 12937 Slot property of  .s. (Contributed by Jim Kingdon, 5-Feb-2023.)
 |-  ( .s  = Slot  ( .s `  ndx )  /\  ( .s `  ndx )  e.  NN )
 
Theoremlmodstrd 12938 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 >.
 )
 
Theoremlmodbased 12939 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 )
 )
 
Theoremlmodplusgd 12940 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 )
 )
 
Theoremlmodscad 12941 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 )
 )
 
Theoremlmodvscad 12942 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 ) )
 
Theoremipndx 12943 Index value of the df-ip 12869 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( .i `  ndx )  =  8
 
Theoremipid 12944 Utility theorem: index-independent form of df-ip 12869. (Contributed by Mario Carneiro, 6-Oct-2013.)
 |- 
 .i  = Slot  ( .i ` 
 ndx )
 
Theoremipslid 12945 Slot property of  .i. (Contributed by Jim Kingdon, 7-Feb-2023.)
 |-  ( .i  = Slot  ( .i `  ndx )  /\  ( .i `  ndx )  e.  NN )
 
Theoremipndxnbasendx 12946 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 )
 
Theoremipndxnplusgndx 12947 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 )
 
Theoremipndxnmulrndx 12948 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 )
 
Theoremslotsdifipndx 12949 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 ) )
 
Theoremipsstrd 12950 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 >.
 )
 
Theoremipsbased 12951 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 )
 )
 
Theoremipsaddgd 12952 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 )
 )
 
Theoremipsmulrd 12953 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 ) )
 
Theoremipsscad 12954 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 )
 )
 
Theoremipsvscad 12955 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 ) )
 
Theoremipsipd 12956 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 ) )
 
Theoremressscag 12957 Scalar is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |-  H  =  ( Gs  A )   &    |-  F  =  (Scalar `  G )   =>    |-  ( ( G  e.  X  /\  A  e.  V )  ->  F  =  (Scalar `  H ) )
 
Theoremressvscag 12958  .s is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |-  H  =  ( Gs  A )   &    |-  .x.  =  ( .s `  G )   =>    |-  ( ( G  e.  X  /\  A  e.  V )  ->  .x.  =  ( .s `  H ) )
 
Theoremressipg 12959 The inner product is unaffected by restriction. (Contributed by Thierry Arnoux, 16-Jun-2019.)
 |-  H  =  ( Gs  A )   &    |-  .,  =  ( .i `  G )   =>    |-  ( ( G  e.  X  /\  A  e.  V )  ->  .,  =  ( .i `  H ) )
 
Theoremtsetndx 12960 Index value of the df-tset 12870 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  (TopSet `  ndx )  =  9
 
Theoremtsetid 12961 Utility theorem: index-independent form of df-tset 12870. (Contributed by NM, 20-Oct-2012.)
 |- TopSet  = Slot  (TopSet `  ndx )
 
Theoremtsetslid 12962 Slot property of TopSet. (Contributed by Jim Kingdon, 9-Feb-2023.)
 |-  (TopSet  = Slot  (TopSet `  ndx )  /\  (TopSet `  ndx )  e.  NN )
 
Theoremtsetndxnn 12963 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
 
Theorembasendxlttsetndx 12964 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 )
 
Theoremtsetndxnbasendx 12965 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 )
 
Theoremtsetndxnplusgndx 12966 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 )
 
Theoremtsetndxnmulrndx 12967 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 )
 
Theoremtsetndxnstarvndx 12968 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 )
 
Theoremslotstnscsi 12969 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 ) )
 
Theoremtopgrpstrd 12970 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 >.
 )
 
Theoremtopgrpbasd 12971 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 )
 )
 
Theoremtopgrpplusgd 12972 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 )
 )
 
Theoremtopgrptsetd 12973 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 )
 )
 
Theoremplendx 12974 Index value of the df-ple 12871 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
 |-  ( le `  ndx )  = ; 1 0
 
Theorempleid 12975 Utility theorem: self-referencing, index-independent form of df-ple 12871. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.)
 |- 
 le  = Slot  ( le ` 
 ndx )
 
Theorempleslid 12976 Slot property of  le. (Contributed by Jim Kingdon, 9-Feb-2023.)
 |-  ( le  = Slot  ( le `  ndx )  /\  ( le `  ndx )  e.  NN )
 
Theoremplendxnn 12977 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
 
Theorembasendxltplendx 12978 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 )
 
Theoremplendxnbasendx 12979 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 )
 
Theoremplendxnplusgndx 12980 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 )
 
Theoremplendxnmulrndx 12981 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 )
 
Theoremplendxnscandx 12982 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 )
 
Theoremplendxnvscandx 12983 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 )
 
Theoremslotsdifplendx 12984 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 ) )
 
Theoremocndx 12985 Index value of the df-ocomp 12872 slot. (Contributed by Mario Carneiro, 25-Oct-2015.) (New usage is discouraged.)
 |-  ( oc `  ndx )  = ; 1 1
 
Theoremocid 12986 Utility theorem: index-independent form of df-ocomp 12872. (Contributed by Mario Carneiro, 25-Oct-2015.)
 |- 
 oc  = Slot  ( oc ` 
 ndx )
 
Theorembasendxnocndx 12987 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 )
 
Theoremplendxnocndx 12988 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 )
 
Theoremdsndx 12989 Index value of the df-ds 12873 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( dist `  ndx )  = ; 1
 2
 
Theoremdsid 12990 Utility theorem: index-independent form of df-ds 12873. (Contributed by Mario Carneiro, 23-Dec-2013.)
 |- 
 dist  = Slot  ( dist `  ndx )
 
Theoremdsslid 12991 Slot property of  dist. (Contributed by Jim Kingdon, 6-May-2023.)
 |-  ( dist  = Slot  ( dist ` 
 ndx )  /\  ( dist `  ndx )  e. 
 NN )
 
Theoremdsndxnn 12992 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
 
Theorembasendxltdsndx 12993 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 )
 
Theoremdsndxnbasendx 12994 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 )
 
Theoremdsndxnplusgndx 12995 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 )
 
Theoremdsndxnmulrndx 12996 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 )
 
Theoremslotsdnscsi 12997 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 ) )
 
Theoremdsndxntsetndx 12998 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 )
 
Theoremslotsdifdsndx 12999 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 ) )
 
Theoremunifndx 13000 Index value of the df-unif 12874 slot. (Contributed by Thierry Arnoux, 17-Dec-2017.) (New usage is discouraged.)
 |-  ( UnifSet `  ndx )  = ; 1
 3
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