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Definition df-cgrg 22964
Description: Define the relation congruence bewteen shapes. Definition 4.4 of [Schwabhauser] p. 35. Ideally, we would define this for functions of any set, but we will used words (functions over  NN) in most cases. (Contributed by Thierry Arnoux, 3-Apr-2019.)
Assertion
Ref Expression
df-cgrg  |- cgrG  =  ( g  e.  _V  |->  {
<. a ,  b >.  |  ( ( a  e.  ( ( Base `  g )  ^pm  RR )  /\  b  e.  ( ( Base `  g
)  ^pm  RR )
)  /\  ( dom  a  =  dom  b  /\  A. i  e.  dom  a A. j  e.  dom  a ( ( a `
 i ) (
dist `  g )
( a `  j
) )  =  ( ( b `  i
) ( dist `  g
) ( b `  j ) ) ) ) } )
Distinct variable group:    a, b, g, i, j

Detailed syntax breakdown of Definition df-cgrg
StepHypRef Expression
1 ccgrg 22963 . 2  class cgrG
2 vg . . 3  setvar  g
3 cvv 2972 . . 3  class  _V
4 va . . . . . . . 8  setvar  a
54cv 1368 . . . . . . 7  class  a
62cv 1368 . . . . . . . . 9  class  g
7 cbs 14174 . . . . . . . . 9  class  Base
86, 7cfv 5418 . . . . . . . 8  class  ( Base `  g )
9 cr 9281 . . . . . . . 8  class  RR
10 cpm 7215 . . . . . . . 8  class  ^pm
118, 9, 10co 6091 . . . . . . 7  class  ( (
Base `  g )  ^pm  RR )
125, 11wcel 1756 . . . . . 6  wff  a  e.  ( ( Base `  g
)  ^pm  RR )
13 vb . . . . . . . 8  setvar  b
1413cv 1368 . . . . . . 7  class  b
1514, 11wcel 1756 . . . . . 6  wff  b  e.  ( ( Base `  g
)  ^pm  RR )
1612, 15wa 369 . . . . 5  wff  ( a  e.  ( ( Base `  g )  ^pm  RR )  /\  b  e.  ( ( Base `  g
)  ^pm  RR )
)
175cdm 4840 . . . . . . 7  class  dom  a
1814cdm 4840 . . . . . . 7  class  dom  b
1917, 18wceq 1369 . . . . . 6  wff  dom  a  =  dom  b
20 vi . . . . . . . . . . . 12  setvar  i
2120cv 1368 . . . . . . . . . . 11  class  i
2221, 5cfv 5418 . . . . . . . . . 10  class  ( a `
 i )
23 vj . . . . . . . . . . . 12  setvar  j
2423cv 1368 . . . . . . . . . . 11  class  j
2524, 5cfv 5418 . . . . . . . . . 10  class  ( a `
 j )
26 cds 14247 . . . . . . . . . . 11  class  dist
276, 26cfv 5418 . . . . . . . . . 10  class  ( dist `  g )
2822, 25, 27co 6091 . . . . . . . . 9  class  ( ( a `  i ) ( dist `  g
) ( a `  j ) )
2921, 14cfv 5418 . . . . . . . . . 10  class  ( b `
 i )
3024, 14cfv 5418 . . . . . . . . . 10  class  ( b `
 j )
3129, 30, 27co 6091 . . . . . . . . 9  class  ( ( b `  i ) ( dist `  g
) ( b `  j ) )
3228, 31wceq 1369 . . . . . . . 8  wff  ( ( a `  i ) ( dist `  g
) ( a `  j ) )  =  ( ( b `  i ) ( dist `  g ) ( b `
 j ) )
3332, 23, 17wral 2715 . . . . . . 7  wff  A. j  e.  dom  a ( ( a `  i ) ( dist `  g
) ( a `  j ) )  =  ( ( b `  i ) ( dist `  g ) ( b `
 j ) )
3433, 20, 17wral 2715 . . . . . 6  wff  A. i  e.  dom  a A. j  e.  dom  a ( ( a `  i ) ( dist `  g
) ( a `  j ) )  =  ( ( b `  i ) ( dist `  g ) ( b `
 j ) )
3519, 34wa 369 . . . . 5  wff  ( dom  a  =  dom  b  /\  A. i  e.  dom  a A. j  e.  dom  a ( ( a `
 i ) (
dist `  g )
( a `  j
) )  =  ( ( b `  i
) ( dist `  g
) ( b `  j ) ) )
3616, 35wa 369 . . . 4  wff  ( ( a  e.  ( (
Base `  g )  ^pm  RR )  /\  b  e.  ( ( Base `  g
)  ^pm  RR )
)  /\  ( dom  a  =  dom  b  /\  A. i  e.  dom  a A. j  e.  dom  a ( ( a `
 i ) (
dist `  g )
( a `  j
) )  =  ( ( b `  i
) ( dist `  g
) ( b `  j ) ) ) )
3736, 4, 13copab 4349 . . 3  class  { <. a ,  b >.  |  ( ( a  e.  ( ( Base `  g
)  ^pm  RR )  /\  b  e.  (
( Base `  g )  ^pm  RR ) )  /\  ( dom  a  =  dom  b  /\  A. i  e. 
dom  a A. j  e.  dom  a ( ( a `  i ) ( dist `  g
) ( a `  j ) )  =  ( ( b `  i ) ( dist `  g ) ( b `
 j ) ) ) ) }
382, 3, 37cmpt 4350 . 2  class  ( g  e.  _V  |->  { <. a ,  b >.  |  ( ( a  e.  ( ( Base `  g
)  ^pm  RR )  /\  b  e.  (
( Base `  g )  ^pm  RR ) )  /\  ( dom  a  =  dom  b  /\  A. i  e. 
dom  a A. j  e.  dom  a ( ( a `  i ) ( dist `  g
) ( a `  j ) )  =  ( ( b `  i ) ( dist `  g ) ( b `
 j ) ) ) ) } )
391, 38wceq 1369 1  wff cgrG  =  ( g  e.  _V  |->  {
<. a ,  b >.  |  ( ( a  e.  ( ( Base `  g )  ^pm  RR )  /\  b  e.  ( ( Base `  g
)  ^pm  RR )
)  /\  ( dom  a  =  dom  b  /\  A. i  e.  dom  a A. j  e.  dom  a ( ( a `
 i ) (
dist `  g )
( a `  j
) )  =  ( ( b `  i
) ( dist `  g
) ( b `  j ) ) ) ) } )
Colors of variables: wff setvar class
This definition is referenced by:  iscgrg  22965  ercgrg  22969
  Copyright terms: Public domain W3C validator