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Theorem iscgrg 22970
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 2986 . . . . 5  |-  ( G  e.  V  ->  G  e.  _V )
3 fveq2 5696 . . . . . . . . . . . 12  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
4 iscgrg.p . . . . . . . . . . . 12  |-  P  =  ( Base `  G
)
53, 4syl6eqr 2493 . . . . . . . . . . 11  |-  ( g  =  G  ->  ( Base `  g )  =  P )
65oveq1d 6111 . . . . . . . . . 10  |-  ( g  =  G  ->  (
( Base `  g )  ^pm  RR )  =  ( P  ^pm  RR )
)
76eleq2d 2510 . . . . . . . . 9  |-  ( g  =  G  ->  (
a  e.  ( (
Base `  g )  ^pm  RR )  <->  a  e.  ( P  ^pm  RR ) ) )
86eleq2d 2510 . . . . . . . . 9  |-  ( g  =  G  ->  (
b  e.  ( (
Base `  g )  ^pm  RR )  <->  b  e.  ( P  ^pm  RR ) ) )
97, 8anbi12d 710 . . . . . . . 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 5696 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  ( dist `  g )  =  ( dist `  G
) )
11 iscgrg.m . . . . . . . . . . . . 13  |-  .-  =  ( dist `  G )
1210, 11syl6eqr 2493 . . . . . . . . . . . 12  |-  ( g  =  G  ->  ( dist `  g )  = 
.-  )
1312oveqd 6113 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
( a `  i
) ( dist `  g
) ( a `  j ) )  =  ( ( a `  i )  .-  (
a `  j )
) )
1412oveqd 6113 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
( b `  i
) ( dist `  g
) ( b `  j ) )  =  ( ( b `  i )  .-  (
b `  j )
) )
1513, 14eqeq12d 2457 . . . . . . . . . 10  |-  ( g  =  G  ->  (
( ( a `  i ) ( dist `  g ) ( a `
 j ) )  =  ( ( b `
 i ) (
dist `  g )
( b `  j
) )  <->  ( (
a `  i )  .-  ( a `  j
) )  =  ( ( b `  i
)  .-  ( b `  j ) ) ) )
16152ralbidv 2762 . . . . . . . . 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 703 . . . . . . . 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 710 . . . . . . 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 4360 . . . . . 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 22969 . . . . . 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 4851 . . . . . . . 8  |-  ( ( P  ^pm  RR )  X.  ( P  ^pm  RR ) )  =  { <. a ,  b >.  |  ( a  e.  ( P  ^pm  RR )  /\  b  e.  ( P  ^pm  RR )
) }
22 ovex 6121 . . . . . . . . 9  |-  ( P 
^pm  RR )  e.  _V
2322, 22xpex 6513 . . . . . . . 8  |-  ( ( P  ^pm  RR )  X.  ( P  ^pm  RR ) )  e.  _V
2421, 23eqeltrri 2514 . . . . . . 7  |-  { <. a ,  b >.  |  ( a  e.  ( P 
^pm  RR )  /\  b  e.  ( P  ^pm  RR ) ) }  e.  _V
25 simpl 457 . . . . . . . 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 4621 . . . . . . 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 4442 . . . . . 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 5779 . . . . 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 )
) ) ) } )
29 id 22 . . . . 5  |-  ( (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 )
) ) ) }  ->  (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 )
) ) ) } )
302, 28, 293syl 20 . . . 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 )
) ) ) } )
311, 30syl5eq 2487 . . 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 )
) ) ) } )
3231breqd 4308 . 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 ) )
33 dmeq 5045 . . . . . 6  |-  ( a  =  A  ->  dom  a  =  dom  A )
3433eqeq1d 2451 . . . . 5  |-  ( a  =  A  ->  ( dom  a  =  dom  b 
<->  dom  A  =  dom  b ) )
3533adantr 465 . . . . . . 7  |-  ( ( a  =  A  /\  i  e.  dom  a )  ->  dom  a  =  dom  A )
36 simpll 753 . . . . . . . . . 10  |-  ( ( ( a  =  A  /\  i  e.  dom  a )  /\  j  e.  dom  a )  -> 
a  =  A )
3736fveq1d 5698 . . . . . . . . 9  |-  ( ( ( a  =  A  /\  i  e.  dom  a )  /\  j  e.  dom  a )  -> 
( a `  i
)  =  ( A `
 i ) )
3836fveq1d 5698 . . . . . . . . 9  |-  ( ( ( a  =  A  /\  i  e.  dom  a )  /\  j  e.  dom  a )  -> 
( a `  j
)  =  ( A `
 j ) )
3937, 38oveq12d 6114 . . . . . . . 8  |-  ( ( ( a  =  A  /\  i  e.  dom  a )  /\  j  e.  dom  a )  -> 
( ( a `  i )  .-  (
a `  j )
)  =  ( ( A `  i ) 
.-  ( A `  j ) ) )
4039eqeq1d 2451 . . . . . . 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 ) ) ) )
4135, 40raleqbidva 2938 . . . . . 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 ) ) ) )
4233, 41raleqbidva 2938 . . . . 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 )
) ) )
4334, 42anbi12d 710 . . . 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 ) ) ) ) )
44 dmeq 5045 . . . . . 6  |-  ( b  =  B  ->  dom  b  =  dom  B )
4544eqeq2d 2454 . . . . 5  |-  ( b  =  B  ->  ( dom  A  =  dom  b  <->  dom 
A  =  dom  B
) )
46 fveq1 5695 . . . . . . . 8  |-  ( b  =  B  ->  (
b `  i )  =  ( B `  i ) )
47 fveq1 5695 . . . . . . . 8  |-  ( b  =  B  ->  (
b `  j )  =  ( B `  j ) )
4846, 47oveq12d 6114 . . . . . . 7  |-  ( b  =  B  ->  (
( b `  i
)  .-  ( b `  j ) )  =  ( ( B `  i )  .-  ( B `  j )
) )
4948eqeq2d 2454 . . . . . 6  |-  ( b  =  B  ->  (
( ( A `  i )  .-  ( A `  j )
)  =  ( ( b `  i ) 
.-  ( b `  j ) )  <->  ( ( A `  i )  .-  ( A `  j
) )  =  ( ( B `  i
)  .-  ( B `  j ) ) ) )
50492ralbidv 2762 . . . . 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 )
) ) )
5145, 50anbi12d 710 . . . 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 ) ) ) ) )
5243, 51sylan9bb 699 . . 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 ) ) ) ) )
53 eqid 2443 . . 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 )
) ) ) }
5452, 53brab2a 4893 . 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 ) ) ) ) )
5532, 54syl6bb 261 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 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   _Vcvv 2977   class class class wbr 4297   {copab 4354    X. cxp 4843   dom cdm 4845   ` cfv 5423  (class class class)co 6096    ^pm cpm 7220   RRcr 9286   Basecbs 14179   distcds 14252  cgrGccgrg 22968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5386  df-fun 5425  df-fv 5431  df-ov 6099  df-cgrg 22969
This theorem is referenced by:  iscgrgd  22971  ercgrg  22974
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