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Theorem List for Intuitionistic Logic Explorer - 12601-12700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsrngmulrd 12601 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 12602 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 12603 Index value of the df-sca 12546 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  (Scalar `  ndx )  =  5
 
Theoremscaid 12604 Utility theorem: index-independent form of scalar df-sca 12546. (Contributed by Mario Carneiro, 19-Jun-2014.)
 |- Scalar  = Slot  (Scalar `  ndx )
 
Theoremscaslid 12605 Slot property of Scalar. (Contributed by Jim Kingdon, 5-Feb-2023.)
 |-  (Scalar  = Slot  (Scalar `  ndx )  /\  (Scalar `  ndx )  e.  NN )
 
Theoremscandxnbasendx 12606 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 12607 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 12608 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 12609 Index value of the df-vsca 12547 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( .s `  ndx )  =  6
 
Theoremvscaid 12610 Utility theorem: index-independent form of scalar product df-vsca 12547. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |- 
 .s  = Slot  ( .s ` 
 ndx )
 
Theoremvscandxnbasendx 12611 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 12612 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 12613 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 12614 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 12615 Slot property of  .s. (Contributed by Jim Kingdon, 5-Feb-2023.)
 |-  ( .s  = Slot  ( .s `  ndx )  /\  ( .s `  ndx )  e.  NN )
 
Theoremlmodstrd 12616 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 12617 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 12618 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 12619 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 12620 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 12621 Index value of the df-ip 12548 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( .i `  ndx )  =  8
 
Theoremipid 12622 Utility theorem: index-independent form of df-ip 12548. (Contributed by Mario Carneiro, 6-Oct-2013.)
 |- 
 .i  = Slot  ( .i ` 
 ndx )
 
Theoremipslid 12623 Slot property of  .i. (Contributed by Jim Kingdon, 7-Feb-2023.)
 |-  ( .i  = Slot  ( .i `  ndx )  /\  ( .i `  ndx )  e.  NN )
 
Theoremipndxnbasendx 12624 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 12625 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 12626 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 12627 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 12628 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 12629 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 12630 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 12631 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 12632 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 12633 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 12634 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 12635 Index value of the df-tset 12549 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  (TopSet `  ndx )  =  9
 
Theoremtsetid 12636 Utility theorem: index-independent form of df-tset 12549. (Contributed by NM, 20-Oct-2012.)
 |- TopSet  = Slot  (TopSet `  ndx )
 
Theoremtsetslid 12637 Slot property of TopSet. (Contributed by Jim Kingdon, 9-Feb-2023.)
 |-  (TopSet  = Slot  (TopSet `  ndx )  /\  (TopSet `  ndx )  e.  NN )
 
Theoremtsetndxnn 12638 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 12639 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 12640 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 12641 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 12642 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 12643 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 12644 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 12645 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 12646 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 12647 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 12648 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 12649 Index value of the df-ple 12550 slot. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 9-Sep-2021.)
 |-  ( le `  ndx )  = ; 1 0
 
Theorempleid 12650 Utility theorem: self-referencing, index-independent form of df-ple 12550. (Contributed by NM, 9-Nov-2012.) (Revised by AV, 9-Sep-2021.)
 |- 
 le  = Slot  ( le ` 
 ndx )
 
Theorempleslid 12651 Slot property of  le. (Contributed by Jim Kingdon, 9-Feb-2023.)
 |-  ( le  = Slot  ( le `  ndx )  /\  ( le `  ndx )  e.  NN )
 
Theoremplendxnn 12652 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 12653 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 12654 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 12655 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 12656 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 12657 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 12658 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 12659 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 ) )
 
Theoremdsndx 12660 Index value of the df-ds 12552 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( dist `  ndx )  = ; 1
 2
 
Theoremdsid 12661 Utility theorem: index-independent form of df-ds 12552. (Contributed by Mario Carneiro, 23-Dec-2013.)
 |- 
 dist  = Slot  ( dist `  ndx )
 
Theoremdsslid 12662 Slot property of  dist. (Contributed by Jim Kingdon, 6-May-2023.)
 |-  ( dist  = Slot  ( dist ` 
 ndx )  /\  ( dist `  ndx )  e. 
 NN )
 
Theoremdsndxnn 12663 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 12664 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 12665 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 12666 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 12667 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 12668 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 12669 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 12670 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 12671 Index value of the df-unif 12553 slot. (Contributed by Thierry Arnoux, 17-Dec-2017.) (New usage is discouraged.)
 |-  ( UnifSet `  ndx )  = ; 1
 3
 
Theoremunifid 12672 Utility theorem: index-independent form of df-unif 12553. (Contributed by Thierry Arnoux, 17-Dec-2017.)
 |- 
 UnifSet  = Slot  ( UnifSet `  ndx )
 
Theoremunifndxnn 12673 The index of the slot for the uniform set in an extensible structure is a positive integer. (Contributed by AV, 28-Oct-2024.)
 |-  ( UnifSet `  ndx )  e. 
 NN
 
Theorembasendxltunifndx 12674 The index of the slot for the base set is less then the index of the slot for the uniform set in an extensible structure. (Contributed by AV, 28-Oct-2024.)
 |-  ( Base `  ndx )  < 
 ( UnifSet `  ndx )
 
Theoremunifndxnbasendx 12675 The slot for the uniform set is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
 |-  ( UnifSet `  ndx )  =/=  ( Base `  ndx )
 
Theoremunifndxntsetndx 12676 The slot for the uniform set is not the slot for the topology in an extensible structure. (Contributed by AV, 28-Oct-2024.)
 |-  ( UnifSet `  ndx )  =/=  (TopSet `  ndx )
 
Theoremslotsdifunifndx 12677 The index of the slot for the uniform set is not the index of other slots. (Contributed by AV, 10-Nov-2024.)
 |-  ( ( ( +g  ` 
 ndx )  =/=  ( UnifSet
 `  ndx )  /\  ( .r `  ndx )  =/=  ( UnifSet `  ndx )  /\  ( *r `  ndx )  =/=  ( UnifSet `  ndx ) )  /\  ( ( le `  ndx )  =/=  ( UnifSet `  ndx )  /\  ( dist `  ndx )  =/=  ( UnifSet `  ndx ) ) )
 
6.1.3  Definition of the structure product
 
Syntaxcrest 12678 Extend class notation with the function returning a subspace topology.
 classt
 
Syntaxctopn 12679 Extend class notation with the topology extractor function.
 class  TopOpen
 
Definitiondf-rest 12680* 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 12681 Define the topology extractor function. This differs from df-tset 12549 when a structure has been restricted using df-iress 12464; 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 12682 The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.)
 |-t  Fn  ( _V  X.  _V )
 
Theoremtopnfn 12683 The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  TopOpen 
 Fn  _V
 
Theoremrestval 12684* 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 12685* 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 12686 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 12687 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 12688 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 12689 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 12690 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 12691 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 12692 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 12693 Extend class notation with a function that converts a basis to its corresponding topology.
 class  topGen
 
Syntaxcpt 12694 Extend class notation with a function whose value is a product topology.
 class  Xt_
 
Syntaxc0g 12695 Extend class notation with group identity element.
 class  0g
 
Syntaxcgsu 12696 Extend class notation to include finitely supported group sums.
 class  gsumg
 
Definitiondf-0g 12697* Define group identity element. Remark: this definition is required here because the symbol  0g is already used in df-gsum 12698. 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 12698* 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 12699* 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 12700* 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
 ) ) } )
 )
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