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Theorem List for Intuitionistic Logic Explorer - 12501-12600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremopelstrsl 12501 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 ) )
 
Theoremopelstrbas 12502 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 )
 )
 
Theorem1strstrg 12503 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
 >. )
 
Theorem1strbas 12504 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 ) )
 
Theorem2strstrg 12505 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 ) 
 ->  G Struct  <. 1 ,  N >. )
 
Theorem2strbasg 12506 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 ) )
 
Theorem2stropg 12507 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 ) )
 
Theorem2strstr1g 12508 A constructed two-slot structure. Version of 2strstrg 12505 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 >. )
 
Theorem2strbas1g 12509 The base set of a constructed two-slot structure. Version of 2strbasg 12506 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 ) )
 
Theorem2strop1g 12510 The other slot of a constructed two-slot structure. Version of 2stropg 12507 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 ) )
 
Theorembasendxnplusgndx 12511 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 )
 
Theoremgrpstrg 12512 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 >. )
 
Theoremgrpbaseg 12513 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 12514 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 ) )
 
Theoremmulrndx 12515 Index value of the df-mulr 12481 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( .r `  ndx )  =  3
 
Theoremmulrid 12516 Utility theorem: index-independent form of df-mulr 12481. (Contributed by Mario Carneiro, 8-Jun-2013.)
 |- 
 .r  = Slot  ( .r ` 
 ndx )
 
Theoremmulrslid 12517 Slot property of  .r. (Contributed by Jim Kingdon, 3-Feb-2023.)
 |-  ( .r  = Slot  ( .r `  ndx )  /\  ( .r `  ndx )  e.  NN )
 
Theoremplusgndxnmulrndx 12518 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 12519 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 12520 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 12521 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 12522 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 12523 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 12524 Index value of the df-starv 12482 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( *r `  ndx )  =  4
 
Theoremstarvid 12525 Utility theorem: index-independent form of df-starv 12482. (Contributed by Mario Carneiro, 6-Oct-2013.)
 |-  *r  = Slot  ( *r `  ndx )
 
Theoremstarvslid 12526 Slot property of  *r. (Contributed by Jim Kingdon, 4-Feb-2023.)
 |-  ( *r  = Slot 
 ( *r `  ndx )  /\  ( *r `  ndx )  e.  NN )
 
Theoremsrngstrd 12527 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 12528 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 12529 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 12530 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 12531 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 12532 Index value of the df-sca 12483 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  (Scalar `  ndx )  =  5
 
Theoremscaid 12533 Utility theorem: index-independent form of scalar df-sca 12483. (Contributed by Mario Carneiro, 19-Jun-2014.)
 |- Scalar  = Slot  (Scalar `  ndx )
 
Theoremscaslid 12534 Slot property of Scalar. (Contributed by Jim Kingdon, 5-Feb-2023.)
 |-  (Scalar  = Slot  (Scalar `  ndx )  /\  (Scalar `  ndx )  e.  NN )
 
Theoremvscandx 12535 Index value of the df-vsca 12484 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( .s `  ndx )  =  6
 
Theoremvscaid 12536 Utility theorem: index-independent form of scalar product df-vsca 12484. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |- 
 .s  = Slot  ( .s ` 
 ndx )
 
Theoremvscaslid 12537 Slot property of  .s. (Contributed by Jim Kingdon, 5-Feb-2023.)
 |-  ( .s  = Slot  ( .s `  ndx )  /\  ( .s `  ndx )  e.  NN )
 
Theoremlmodstrd 12538 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 12539 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 12540 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 12541 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 12542 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 12543 Index value of the df-ip 12485 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( .i `  ndx )  =  8
 
Theoremipid 12544 Utility theorem: index-independent form of df-ip 12485. (Contributed by Mario Carneiro, 6-Oct-2013.)
 |- 
 .i  = Slot  ( .i ` 
 ndx )
 
Theoremipslid 12545 Slot property of  .i. (Contributed by Jim Kingdon, 7-Feb-2023.)
 |-  ( .i  = Slot  ( .i `  ndx )  /\  ( .i `  ndx )  e.  NN )
 
Theoremipsstrd 12546 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 12547 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 12548 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 12549 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 12550 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 12551 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 12552 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 ) )
 
Theoremtsetndx 12553 Index value of the df-tset 12486 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  (TopSet `  ndx )  =  9
 
Theoremtsetid 12554 Utility theorem: index-independent form of df-tset 12486. (Contributed by NM, 20-Oct-2012.)
 |- TopSet  = Slot  (TopSet `  ndx )
 
Theoremtsetslid 12555 Slot property of TopSet. (Contributed by Jim Kingdon, 9-Feb-2023.)
 |-  (TopSet  = Slot  (TopSet `  ndx )  /\  (TopSet `  ndx )  e.  NN )
 
Theoremtopgrpstrd 12556 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 12557 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 12558 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 12559 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 12560 Index value of the df-ple 12487 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
 |-  ( le `  ndx )  = ; 1 0
 
Theorempleid 12561 Utility theorem: self-referencing, index-independent form of df-ple 12487. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.)
 |- 
 le  = Slot  ( le ` 
 ndx )
 
Theorempleslid 12562 Slot property of  le. (Contributed by Jim Kingdon, 9-Feb-2023.)
 |-  ( le  = Slot  ( le `  ndx )  /\  ( le `  ndx )  e.  NN )
 
Theoremdsndx 12563 Index value of the df-ds 12489 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( dist `  ndx )  = ; 1
 2
 
Theoremdsid 12564 Utility theorem: index-independent form of df-ds 12489. (Contributed by Mario Carneiro, 23-Dec-2013.)
 |- 
 dist  = Slot  ( dist `  ndx )
 
Theoremdsslid 12565 Slot property of  dist. (Contributed by Jim Kingdon, 6-May-2023.)
 |-  ( dist  = Slot  ( dist ` 
 ndx )  /\  ( dist `  ndx )  e. 
 NN )
 
6.1.3  Definition of the structure product
 
Syntaxcrest 12566 Extend class notation with the function returning a subspace topology.
 classt
 
Syntaxctopn 12567 Extend class notation with the topology extractor function.
 class  TopOpen
 
Definitiondf-rest 12568* Function returning the subspace topology induced by the topology  y and the set  x. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.)
 |-t  =  ( j  e.  _V ,  x  e.  _V  |->  ran  ( y  e.  j  |->  ( y  i^i  x ) ) )
 
Definitiondf-topn 12569 Define the topology extractor function. This differs from df-tset 12486 when a structure has been restricted using df-ress 12411; in this case the TopSet component will still have a topology over the larger set, and this function fixes this by restricting the topology as well. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  TopOpen  =  ( w  e. 
 _V  |->  ( (TopSet `  w )t  ( Base `  w )
 ) )
 
Theoremrestfn 12570 The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.)
 |-t  Fn  ( _V  X.  _V )
 
Theoremtopnfn 12571 The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  TopOpen 
 Fn  _V
 
Theoremrestval 12572* The subspace topology induced by the topology  J on the set  A. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.)
 |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( Jt  A )  =  ran  ( x  e.  J  |->  ( x  i^i  A ) ) )
 
Theoremelrest 12573* The predicate "is an open set of a subspace topology". (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  ( ( J  e.  V  /\  B  e.  W )  ->  ( A  e.  ( Jt  B )  <->  E. x  e.  J  A  =  ( x  i^i  B ) ) )
 
Theoremelrestr 12574 Sufficient condition for being an open set in a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  ( ( J  e.  V  /\  S  e.  W  /\  A  e.  J ) 
 ->  ( A  i^i  S )  e.  ( Jt  S ) )
 
Theoremrestid2 12575 The subspace topology over a subset of the base set is the original topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ( Jt  A )  =  J )
 
Theoremrestsspw 12576 The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( Jt  A )  C_  ~P A
 
Theoremrestid 12577 The subspace topology of the base set is the original topology. (Contributed by Jeff Hankins, 9-Jul-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( J  e.  V  ->  ( Jt  X )  =  J )
 
Theoremtopnvalg 12578 Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by Jim Kingdon, 11-Feb-2023.)
 |-  B  =  ( Base `  W )   &    |-  J  =  (TopSet `  W )   =>    |-  ( W  e.  V  ->  ( Jt  B )  =  (
 TopOpen `  W ) )
 
Theoremtopnidg 12579 Value of the topology extractor function when the topology is defined over the same set as the base. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  B  =  ( Base `  W )   &    |-  J  =  (TopSet `  W )   =>    |-  ( ( W  e.  V  /\  J  C_  ~P B )  ->  J  =  (
 TopOpen `  W ) )
 
Theoremtopnpropgd 12580 The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Jim Kingdon, 13-Feb-2023.)
 |-  ( ph  ->  ( Base `  K )  =  ( Base `  L )
 )   &    |-  ( ph  ->  (TopSet `  K )  =  (TopSet `  L ) )   &    |-  ( ph  ->  K  e.  V )   &    |-  ( ph  ->  L  e.  W )   =>    |-  ( ph  ->  ( TopOpen `  K )  =  (
 TopOpen `  L ) )
 
Syntaxctg 12581 Extend class notation with a function that converts a basis to its corresponding topology.
 class  topGen
 
Syntaxcpt 12582 Extend class notation with a function whose value is a product topology.
 class  Xt_
 
Syntaxc0g 12583 Extend class notation with group identity element.
 class  0g
 
Syntaxcgsu 12584 Extend class notation to include finitely supported group sums.
 class  gsumg
 
Definitiondf-0g 12585* Define group identity element. Remark: this definition is required here because the symbol  0g is already used in df-gsum 12586. The related theorems will be provided later. (Contributed by NM, 20-Aug-2011.)
 |- 
 0g  =  ( g  e.  _V  |->  ( iota
 e ( e  e.  ( Base `  g )  /\  A. x  e.  ( Base `  g ) ( ( e ( +g  `  g ) x )  =  x  /\  ( x ( +g  `  g
 ) e )  =  x ) ) ) )
 
Definitiondf-gsum 12586* Define the group sum for the structure  G of a finite sequence of elements whose values are defined by the expression  B and whose set of indices is  A. It may be viewed as a product (if 
G is a multiplication), a sum (if 
G is an addition) or any other operation. The variable  k is normally a free variable in  B (i.e.,  B can be thought of as  B ( k )). The definition is meaningful in different contexts, depending on the size of the index set  A and each demanding different properties of  G.

1. If  A  =  (/) and  G has an identity element, then the sum equals this identity.

2. If  A  =  ( M ... N ) and 
G is any magma, then the sum is the sum of the elements, evaluated left-to-right, i.e.,  ( B ( 1 )  +  B
( 2 ) )  +  B ( 3 ), etc.

3. If  A is a finite set (or is nonzero for finitely many indices) and  G is a commutative monoid, then the sum adds up these elements in some order, which is then uniquely defined.

4. If  A is an infinite set and  G is a Hausdorff topological group, then there is a meaningful sum, but  gsumg cannot handle this case. (Contributed by FL, 5-Sep-2010.) (Revised by FL, 17-Oct-2011.) (Revised by Mario Carneiro, 7-Dec-2014.)

 |- 
 gsumg  =  ( w  e.  _V ,  f  e.  _V  |->  [_
 { x  e.  ( Base `  w )  | 
 A. y  e.  ( Base `  w ) ( ( x ( +g  `  w ) y )  =  y  /\  (
 y ( +g  `  w ) x )  =  y ) }  /  o ]_ if ( ran  f  C_  o ,  ( 0g
 `  w ) ,  if ( dom  f  e.  ran  ... ,  ( iota
 x E. m E. n  e.  ( ZZ>= `  m ) ( dom  f  =  ( m
 ... n )  /\  x  =  (  seq m ( ( +g  `  w ) ,  f
 ) `  n )
 ) ) ,  ( iota x E. g [. ( `' f " ( _V  \  o ) )  /  y ]. ( g : ( 1 ... ( `  y ) ) -1-1-onto-> y  /\  x  =  (  seq 1 ( ( +g  `  w ) ,  (
 f  o.  g ) ) `  ( `  y
 ) ) ) ) ) ) )
 
Definitiondf-topgen 12587* Define a function that converts a basis to its corresponding topology. Equivalent to the definition of a topology generated by a basis in [Munkres] p. 78. (Contributed by NM, 16-Jul-2006.)
 |-  topGen  =  ( x  e. 
 _V  |->  { y  |  y 
 C_  U. ( x  i^i  ~P y ) } )
 
Definitiondf-pt 12588* Define the product topology on a collection of topologies. For convenience, it is defined on arbitrary collections of sets, expressed as a function from some index set to the subbases of each factor space. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |- 
 Xt_  =  ( f  e.  _V  |->  ( topGen `  { x  |  E. g ( ( g  Fn  dom  f  /\  A. y  e.  dom  f ( g `  y )  e.  (
 f `  y )  /\  E. z  e.  Fin  A. y  e.  ( dom  f  \  z ) ( g `  y
 )  =  U. (
 f `  y )
 )  /\  x  =  X_ y  e.  dom  f
 ( g `  y
 ) ) } )
 )
 
Syntaxcprds 12589 The function constructing structure products.
 class  X_s
 
Syntaxcpws 12590 The function constructing structure powers.
 class  ^s
 
Definitiondf-prds 12591* Define a structure product. This can be a product of groups, rings, modules, or ordered topological fields; any unused components will have garbage in them but this is usually not relevant for the purpose of inheriting the structures present in the factors. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.)
 |-  X_s  =  ( s  e.  _V ,  r  e.  _V  |->  [_ X_ x  e.  dom  r ( Base `  (
 r `  x )
 )  /  v ]_ [_ ( f  e.  v ,  g  e.  v  |->  X_ x  e.  dom  r ( ( f `
  x ) ( Hom  `  ( r `  x ) ) ( g `  x ) ) )  /  h ]_ ( ( { <. (
 Base `  ndx ) ,  v >. ,  <. ( +g  ` 
 ndx ) ,  (
 f  e.  v ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( ( f `  x ) ( +g  `  ( r `  x ) ) ( g `
  x ) ) ) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( ( f `
  x ) ( .r `  ( r `
  x ) ) ( g `  x ) ) ) )
 >. }  u.  { <. (Scalar `  ndx ) ,  s >. ,  <. ( .s `  ndx ) ,  ( f  e.  ( Base `  s
 ) ,  g  e.  v  |->  ( x  e. 
 dom  r  |->  ( f ( .s `  (
 r `  x )
 ) ( g `  x ) ) ) ) >. ,  <. ( .i
 `  ndx ) ,  (
 f  e.  v ,  g  e.  v  |->  ( s  gsumg  ( x  e.  dom  r  |->  ( ( f `
  x ) ( .i `  ( r `
  x ) ) ( g `  x ) ) ) ) ) >. } )  u.  ( { <. (TopSet `  ndx ) ,  ( Xt_ `  ( TopOpen  o.  r )
 ) >. ,  <. ( le ` 
 ndx ) ,  { <. f ,  g >.  |  ( { f ,  g }  C_  v  /\  A. x  e.  dom  r ( f `  x ) ( le `  ( r `  x ) ) ( g `
  x ) ) } >. ,  <. ( dist ` 
 ndx ) ,  (
 f  e.  v ,  g  e.  v  |->  sup ( ( ran  ( x  e.  dom  r  |->  ( ( f `  x ) ( dist `  (
 r `  x )
 ) ( g `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) >. }  u.  { <. ( Hom  `  ndx ) ,  h >. , 
 <. (comp `  ndx ) ,  ( a  e.  (
 v  X.  v ) ,  c  e.  v  |->  ( d  e.  (
 c h ( 2nd `  a ) ) ,  e  e.  ( h `
  a )  |->  ( x  e.  dom  r  |->  ( ( d `  x ) ( <. ( ( 1st `  a
 ) `  x ) ,  ( ( 2nd `  a
 ) `  x ) >. (comp `  ( r `  x ) ) ( c `  x ) ) ( e `  x ) ) ) ) ) >. } )
 ) )
 
Theoremreldmprds 12592 The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.)
 |- 
 Rel  dom  X_s
 
Definitiondf-pws 12593* Define a structure power, which is just a structure product where all the factors are the same. (Contributed by Mario Carneiro, 11-Jan-2015.)
 |- 
 ^s  =  ( r  e. 
 _V ,  i  e. 
 _V  |->  ( (Scalar `  r
 ) X_s ( i  X.  {
 r } ) ) )
 
PART 7  BASIC ALGEBRAIC STRUCTURES
 
7.1  Monoids
 
7.1.1  Magmas

According to Wikipedia ("Magma (algebra)", 08-Jan-2020, https://en.wikipedia.org/wiki/magma_(algebra)) "In abstract algebra, a magma [...] is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with a single binary operation. The binary operation must be closed by definition but no other properties are imposed.".

Since the concept of a "binary operation" is used in different variants, these differences are explained in more detail in the following:

With df-mpo 5855, binary operations are defined by a rule, and with df-ov 5853, the value of a binary operation applied to two operands can be expressed. In both cases, the two operands can belong to different sets, and the result can be an element of a third set. However, according to Wikipedia "Binary operation", see https://en.wikipedia.org/wiki/Binary_operation 5853 (19-Jan-2020), "... a binary operation on a set  S is a mapping of the elements of the Cartesian product 
S  X.  S to S:  f : S  X.  S --> S. Because the result of performing the operation on a pair of elements of S is again an element of S, the operation is called a closed binary operation on S (or sometimes expressed as having the property of closure).". To distinguish this more restrictive definition (in Wikipedia and most of the literature) from the general case, binary operations mapping the elements of the Cartesian product  S  X.  S are more precisely called internal binary operations. If, in addition, the result is also contained in the set  S, the operation should be called closed internal binary operation. Therefore, a "binary operation on a set  S" according to Wikipedia is a "closed internal binary operation" in a more precise terminology. If the sets are different, the operation is explicitly called external binary operation (see Wikipedia https://en.wikipedia.org/wiki/Binary_operation#External_binary_operations 5853).

The definition of magmas (Mgm, see df-mgm 12597) concentrates on the closure property of the associated operation, and poses no additional restrictions on it. In this way, it is most general and flexible.

 
Syntaxcplusf 12594 Extend class notation with group addition as a function.
 class  +f
 
Syntaxcmgm 12595 Extend class notation with class of all magmas.
 class Mgm
 
Definitiondf-plusf 12596* Define group addition function. Usually we will use  +g directly instead of  +f, and they have the same behavior in most cases. The main advantage of  +f for any magma is that it is a guaranteed function (mgmplusf 12607), while  +g only has closure (mgmcl 12600). (Contributed by Mario Carneiro, 14-Aug-2015.)
 |- 
 +f  =  ( g  e.  _V  |->  ( x  e.  ( Base `  g ) ,  y  e.  ( Base `  g )  |->  ( x ( +g  `  g ) y ) ) )
 
Definitiondf-mgm 12597* A magma is a set equipped with an everywhere defined internal operation. Definition 1 in [BourbakiAlg1] p. 1, or definition of a groupoid in section I.1 of [Bruck] p. 1. Note: The term "groupoid" is now widely used to refer to other objects: (small) categories all of whose morphisms are invertible, or groups with a partial function replacing the binary operation. Therefore, we will only use the term "magma" for the present notion in set.mm. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
 |- Mgm 
 =  { g  | 
 [. ( Base `  g
 )  /  b ]. [. ( +g  `  g
 )  /  o ]. A. x  e.  b  A. y  e.  b  ( x o y )  e.  b }
 
Theoremismgm 12598* The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  (
 +g  `  M )   =>    |-  ( M  e.  V  ->  ( M  e. Mgm  <->  A. x  e.  B  A. y  e.  B  ( x  .o.  y )  e.  B ) )
 
Theoremismgmn0 12599* The predicate "is a magma" for a structure with a nonempty base set. (Contributed by AV, 29-Jan-2020.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  (
 +g  `  M )   =>    |-  ( A  e.  B  ->  ( M  e. Mgm  <->  A. x  e.  B  A. y  e.  B  ( x  .o.  y )  e.  B ) )
 
Theoremmgmcl 12600 Closure of the operation of a magma. (Contributed by FL, 14-Sep-2010.) (Revised by AV, 13-Jan-2020.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  (
 +g  `  M )   =>    |-  (
 ( M  e. Mgm  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .o.  Y )  e.  B )
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