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Theorem iscgrg 24636
Description: The congruence property for sequences of points. (Contributed by Thierry Arnoux, 3-Apr-2019.)
Hypotheses
Ref Expression
iscgrg.p  |-  P  =  ( Base `  G
)
iscgrg.m  |-  .-  =  ( dist `  G )
iscgrg.e  |-  .~  =  (cgrG `  G )
Assertion
Ref Expression
iscgrg  |-  ( G  e.  V  ->  ( A  .~  B  <->  ( ( A  e.  ( P  ^pm  RR )  /\  B  e.  ( P  ^pm  RR ) )  /\  ( dom  A  =  dom  B  /\  A. i  e.  dom  A A. j  e.  dom  A ( ( A `  i )  .-  ( A `  j )
)  =  ( ( B `  i ) 
.-  ( B `  j ) ) ) ) ) )
Distinct variable groups:    i, j, G    A, i, j    B, i, j
Allowed substitution hints:    P( i, j)    .~ ( i, j)    .- ( i, j)    V( i, j)

Proof of Theorem iscgrg
Dummy variables  a 
b  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iscgrg.e . . . 4  |-  .~  =  (cgrG `  G )
2 elex 3040 . . . . 5  |-  ( G  e.  V  ->  G  e.  _V )
3 fveq2 5879 . . . . . . . . . . . 12  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
4 iscgrg.p . . . . . . . . . . . 12  |-  P  =  ( Base `  G
)
53, 4syl6eqr 2523 . . . . . . . . . . 11  |-  ( g  =  G  ->  ( Base `  g )  =  P )
65oveq1d 6323 . . . . . . . . . 10  |-  ( g  =  G  ->  (
( Base `  g )  ^pm  RR )  =  ( P  ^pm  RR )
)
76eleq2d 2534 . . . . . . . . 9  |-  ( g  =  G  ->  (
a  e.  ( (
Base `  g )  ^pm  RR )  <->  a  e.  ( P  ^pm  RR ) ) )
86eleq2d 2534 . . . . . . . . 9  |-  ( g  =  G  ->  (
b  e.  ( (
Base `  g )  ^pm  RR )  <->  b  e.  ( P  ^pm  RR ) ) )
97, 8anbi12d 725 . . . . . . . 8  |-  ( g  =  G  ->  (
( a  e.  ( ( Base `  g
)  ^pm  RR )  /\  b  e.  (
( Base `  g )  ^pm  RR ) )  <->  ( a  e.  ( P  ^pm  RR )  /\  b  e.  ( P  ^pm  RR )
) ) )
10 fveq2 5879 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  ( dist `  g )  =  ( dist `  G
) )
11 iscgrg.m . . . . . . . . . . . . 13  |-  .-  =  ( dist `  G )
1210, 11syl6eqr 2523 . . . . . . . . . . . 12  |-  ( g  =  G  ->  ( dist `  g )  = 
.-  )
1312oveqd 6325 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
( a `  i
) ( dist `  g
) ( a `  j ) )  =  ( ( a `  i )  .-  (
a `  j )
) )
1412oveqd 6325 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
( b `  i
) ( dist `  g
) ( b `  j ) )  =  ( ( b `  i )  .-  (
b `  j )
) )
1513, 14eqeq12d 2486 . . . . . . . . . 10  |-  ( g  =  G  ->  (
( ( a `  i ) ( dist `  g ) ( a `
 j ) )  =  ( ( b `
 i ) (
dist `  g )
( b `  j
) )  <->  ( (
a `  i )  .-  ( a `  j
) )  =  ( ( b `  i
)  .-  ( b `  j ) ) ) )
16152ralbidv 2832 . . . . . . . . 9  |-  ( g  =  G  ->  ( A. i  e.  dom  a A. j  e.  dom  a ( ( a `
 i ) (
dist `  g )
( a `  j
) )  =  ( ( b `  i
) ( dist `  g
) ( b `  j ) )  <->  A. i  e.  dom  a A. j  e.  dom  a ( ( a `  i ) 
.-  ( a `  j ) )  =  ( ( b `  i )  .-  (
b `  j )
) ) )
1716anbi2d 718 . . . . . . . 8  |-  ( g  =  G  ->  (
( 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 ) ) )  <->  ( dom  a  =  dom  b  /\  A. i  e.  dom  a A. j  e.  dom  a ( ( a `  i
)  .-  ( a `  j ) )  =  ( ( b `  i )  .-  (
b `  j )
) ) ) )
189, 17anbi12d 725 . . . . . . 7  |-  ( g  =  G  ->  (
( ( 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 ) ) ) )  <->  ( (
a  e.  ( P 
^pm  RR )  /\  b  e.  ( P  ^pm  RR ) )  /\  ( dom  a  =  dom  b  /\  A. i  e. 
dom  a A. j  e.  dom  a ( ( a `  i ) 
.-  ( a `  j ) )  =  ( ( b `  i )  .-  (
b `  j )
) ) ) ) )
1918opabbidv 4459 . . . . . 6  |-  ( g  =  G  ->  { <. 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 ) ) ) ) }  =  { <. a ,  b
>.  |  ( (
a  e.  ( P 
^pm  RR )  /\  b  e.  ( P  ^pm  RR ) )  /\  ( dom  a  =  dom  b  /\  A. i  e. 
dom  a A. j  e.  dom  a ( ( a `  i ) 
.-  ( a `  j ) )  =  ( ( b `  i )  .-  (
b `  j )
) ) ) } )
20 df-cgrg 24635 . . . . . 6  |- 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 ) ) ) ) } )
21 df-xp 4845 . . . . . . . 8  |-  ( ( P  ^pm  RR )  X.  ( P  ^pm  RR ) )  =  { <. a ,  b >.  |  ( a  e.  ( P  ^pm  RR )  /\  b  e.  ( P  ^pm  RR )
) }
22 ovex 6336 . . . . . . . . 9  |-  ( P 
^pm  RR )  e.  _V
2322, 22xpex 6614 . . . . . . . 8  |-  ( ( P  ^pm  RR )  X.  ( P  ^pm  RR ) )  e.  _V
2421, 23eqeltrri 2546 . . . . . . 7  |-  { <. a ,  b >.  |  ( a  e.  ( P 
^pm  RR )  /\  b  e.  ( P  ^pm  RR ) ) }  e.  _V
25 simpl 464 . . . . . . . 8  |-  ( ( ( a  e.  ( P  ^pm  RR )  /\  b  e.  ( P  ^pm  RR ) )  /\  ( dom  a  =  dom  b  /\  A. i  e.  dom  a A. j  e.  dom  a ( ( a `  i
)  .-  ( a `  j ) )  =  ( ( b `  i )  .-  (
b `  j )
) ) )  -> 
( a  e.  ( P  ^pm  RR )  /\  b  e.  ( P  ^pm  RR ) ) )
2625ssopab2i 4729 . . . . . . 7  |-  { <. a ,  b >.  |  ( ( a  e.  ( P  ^pm  RR )  /\  b  e.  ( P  ^pm  RR ) )  /\  ( dom  a  =  dom  b  /\  A. i  e.  dom  a A. j  e.  dom  a ( ( a `  i
)  .-  ( a `  j ) )  =  ( ( b `  i )  .-  (
b `  j )
) ) ) } 
C_  { <. a ,  b >.  |  ( a  e.  ( P 
^pm  RR )  /\  b  e.  ( P  ^pm  RR ) ) }
2724, 26ssexi 4541 . . . . . 6  |-  { <. a ,  b >.  |  ( ( a  e.  ( P  ^pm  RR )  /\  b  e.  ( P  ^pm  RR ) )  /\  ( dom  a  =  dom  b  /\  A. i  e.  dom  a A. j  e.  dom  a ( ( a `  i
)  .-  ( a `  j ) )  =  ( ( b `  i )  .-  (
b `  j )
) ) ) }  e.  _V
2819, 20, 27fvmpt 5963 . . . . 5  |-  ( G  e.  _V  ->  (cgrG `  G )  =  { <. a ,  b >.  |  ( ( a  e.  ( P  ^pm  RR )  /\  b  e.  ( P  ^pm  RR ) )  /\  ( dom  a  =  dom  b  /\  A. i  e. 
dom  a A. j  e.  dom  a ( ( a `  i ) 
.-  ( a `  j ) )  =  ( ( b `  i )  .-  (
b `  j )
) ) ) } )
292, 28syl 17 . . . 4  |-  ( G  e.  V  ->  (cgrG `  G )  =  { <. a ,  b >.  |  ( ( a  e.  ( P  ^pm  RR )  /\  b  e.  ( P  ^pm  RR ) )  /\  ( dom  a  =  dom  b  /\  A. i  e. 
dom  a A. j  e.  dom  a ( ( a `  i ) 
.-  ( a `  j ) )  =  ( ( b `  i )  .-  (
b `  j )
) ) ) } )
301, 29syl5eq 2517 . . 3  |-  ( G  e.  V  ->  .~  =  { <. a ,  b
>.  |  ( (
a  e.  ( P 
^pm  RR )  /\  b  e.  ( P  ^pm  RR ) )  /\  ( dom  a  =  dom  b  /\  A. i  e. 
dom  a A. j  e.  dom  a ( ( a `  i ) 
.-  ( a `  j ) )  =  ( ( b `  i )  .-  (
b `  j )
) ) ) } )
3130breqd 4406 . 2  |-  ( G  e.  V  ->  ( A  .~  B  <->  A { <. a ,  b >.  |  ( ( a  e.  ( P  ^pm  RR )  /\  b  e.  ( P  ^pm  RR ) )  /\  ( dom  a  =  dom  b  /\  A. i  e. 
dom  a A. j  e.  dom  a ( ( a `  i ) 
.-  ( a `  j ) )  =  ( ( b `  i )  .-  (
b `  j )
) ) ) } B ) )
32 dmeq 5040 . . . . . 6  |-  ( a  =  A  ->  dom  a  =  dom  A )
3332eqeq1d 2473 . . . . 5  |-  ( a  =  A  ->  ( dom  a  =  dom  b 
<->  dom  A  =  dom  b ) )
3432adantr 472 . . . . . . 7  |-  ( ( a  =  A  /\  i  e.  dom  a )  ->  dom  a  =  dom  A )
35 simpll 768 . . . . . . . . . 10  |-  ( ( ( a  =  A  /\  i  e.  dom  a )  /\  j  e.  dom  a )  -> 
a  =  A )
3635fveq1d 5881 . . . . . . . . 9  |-  ( ( ( a  =  A  /\  i  e.  dom  a )  /\  j  e.  dom  a )  -> 
( a `  i
)  =  ( A `
 i ) )
3735fveq1d 5881 . . . . . . . . 9  |-  ( ( ( a  =  A  /\  i  e.  dom  a )  /\  j  e.  dom  a )  -> 
( a `  j
)  =  ( A `
 j ) )
3836, 37oveq12d 6326 . . . . . . . 8  |-  ( ( ( a  =  A  /\  i  e.  dom  a )  /\  j  e.  dom  a )  -> 
( ( a `  i )  .-  (
a `  j )
)  =  ( ( A `  i ) 
.-  ( A `  j ) ) )
3938eqeq1d 2473 . . . . . . 7  |-  ( ( ( a  =  A  /\  i  e.  dom  a )  /\  j  e.  dom  a )  -> 
( ( ( a `
 i )  .-  ( a `  j
) )  =  ( ( b `  i
)  .-  ( b `  j ) )  <->  ( ( A `  i )  .-  ( A `  j
) )  =  ( ( b `  i
)  .-  ( b `  j ) ) ) )
4034, 39raleqbidva 2989 . . . . . 6  |-  ( ( a  =  A  /\  i  e.  dom  a )  ->  ( A. j  e.  dom  a ( ( a `  i ) 
.-  ( a `  j ) )  =  ( ( b `  i )  .-  (
b `  j )
)  <->  A. j  e.  dom  A ( ( A `  i )  .-  ( A `  j )
)  =  ( ( b `  i ) 
.-  ( b `  j ) ) ) )
4132, 40raleqbidva 2989 . . . . 5  |-  ( a  =  A  ->  ( A. i  e.  dom  a A. j  e.  dom  a ( ( a `
 i )  .-  ( a `  j
) )  =  ( ( b `  i
)  .-  ( b `  j ) )  <->  A. i  e.  dom  A A. j  e.  dom  A ( ( A `  i ) 
.-  ( A `  j ) )  =  ( ( b `  i )  .-  (
b `  j )
) ) )
4233, 41anbi12d 725 . . . 4  |-  ( a  =  A  ->  (
( dom  a  =  dom  b  /\  A. i  e.  dom  a A. j  e.  dom  a ( ( a `  i ) 
.-  ( a `  j ) )  =  ( ( b `  i )  .-  (
b `  j )
) )  <->  ( dom  A  =  dom  b  /\  A. i  e.  dom  A A. j  e.  dom  A ( ( A `  i )  .-  ( A `  j )
)  =  ( ( b `  i ) 
.-  ( b `  j ) ) ) ) )
43 dmeq 5040 . . . . . 6  |-  ( b  =  B  ->  dom  b  =  dom  B )
4443eqeq2d 2481 . . . . 5  |-  ( b  =  B  ->  ( dom  A  =  dom  b  <->  dom 
A  =  dom  B
) )
45 fveq1 5878 . . . . . . . 8  |-  ( b  =  B  ->  (
b `  i )  =  ( B `  i ) )
46 fveq1 5878 . . . . . . . 8  |-  ( b  =  B  ->  (
b `  j )  =  ( B `  j ) )
4745, 46oveq12d 6326 . . . . . . 7  |-  ( b  =  B  ->  (
( b `  i
)  .-  ( b `  j ) )  =  ( ( B `  i )  .-  ( B `  j )
) )
4847eqeq2d 2481 . . . . . 6  |-  ( b  =  B  ->  (
( ( A `  i )  .-  ( A `  j )
)  =  ( ( b `  i ) 
.-  ( b `  j ) )  <->  ( ( A `  i )  .-  ( A `  j
) )  =  ( ( B `  i
)  .-  ( B `  j ) ) ) )
49482ralbidv 2832 . . . . 5  |-  ( b  =  B  ->  ( A. i  e.  dom  A A. j  e.  dom  A ( ( A `  i )  .-  ( A `  j )
)  =  ( ( b `  i ) 
.-  ( b `  j ) )  <->  A. i  e.  dom  A A. j  e.  dom  A ( ( A `  i ) 
.-  ( A `  j ) )  =  ( ( B `  i )  .-  ( B `  j )
) ) )
5044, 49anbi12d 725 . . . 4  |-  ( b  =  B  ->  (
( dom  A  =  dom  b  /\  A. i  e.  dom  A A. j  e.  dom  A ( ( A `  i ) 
.-  ( A `  j ) )  =  ( ( b `  i )  .-  (
b `  j )
) )  <->  ( dom  A  =  dom  B  /\  A. i  e.  dom  A A. j  e.  dom  A ( ( A `  i )  .-  ( A `  j )
)  =  ( ( B `  i ) 
.-  ( B `  j ) ) ) ) )
5142, 50sylan9bb 714 . . 3  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( dom  a  =  dom  b  /\  A. i  e.  dom  a A. j  e.  dom  a ( ( a `  i
)  .-  ( a `  j ) )  =  ( ( b `  i )  .-  (
b `  j )
) )  <->  ( dom  A  =  dom  B  /\  A. i  e.  dom  A A. j  e.  dom  A ( ( A `  i )  .-  ( A `  j )
)  =  ( ( B `  i ) 
.-  ( B `  j ) ) ) ) )
52 eqid 2471 . . 3  |-  { <. a ,  b >.  |  ( ( a  e.  ( P  ^pm  RR )  /\  b  e.  ( P  ^pm  RR ) )  /\  ( dom  a  =  dom  b  /\  A. i  e.  dom  a A. j  e.  dom  a ( ( a `  i
)  .-  ( a `  j ) )  =  ( ( b `  i )  .-  (
b `  j )
) ) ) }  =  { <. a ,  b >.  |  ( ( a  e.  ( P  ^pm  RR )  /\  b  e.  ( P  ^pm  RR ) )  /\  ( dom  a  =  dom  b  /\  A. i  e.  dom  a A. j  e.  dom  a ( ( a `  i
)  .-  ( a `  j ) )  =  ( ( b `  i )  .-  (
b `  j )
) ) ) }
5351, 52brab2a 4889 . 2  |-  ( A { <. a ,  b
>.  |  ( (
a  e.  ( P 
^pm  RR )  /\  b  e.  ( P  ^pm  RR ) )  /\  ( dom  a  =  dom  b  /\  A. i  e. 
dom  a A. j  e.  dom  a ( ( a `  i ) 
.-  ( a `  j ) )  =  ( ( b `  i )  .-  (
b `  j )
) ) ) } B  <->  ( ( A  e.  ( P  ^pm  RR )  /\  B  e.  ( P  ^pm  RR ) )  /\  ( dom  A  =  dom  B  /\  A. i  e.  dom  A A. j  e.  dom  A ( ( A `  i )  .-  ( A `  j )
)  =  ( ( B `  i ) 
.-  ( B `  j ) ) ) ) )
5431, 53syl6bb 269 1  |-  ( G  e.  V  ->  ( A  .~  B  <->  ( ( A  e.  ( P  ^pm  RR )  /\  B  e.  ( P  ^pm  RR ) )  /\  ( dom  A  =  dom  B  /\  A. i  e.  dom  A A. j  e.  dom  A ( ( A `  i )  .-  ( A `  j )
)  =  ( ( B `  i ) 
.-  ( B `  j ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   _Vcvv 3031   class class class wbr 4395   {copab 4453    X. cxp 4837   dom cdm 4839   ` cfv 5589  (class class class)co 6308    ^pm cpm 7491   RRcr 9556   Basecbs 15199   distcds 15277  cgrGccgrg 24634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-cgrg 24635
This theorem is referenced by:  iscgrgd  24637  ercgrg  24641
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