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Theorem iscgrg 24557
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 3054 . . . . 5  |-  ( G  e.  V  ->  G  e.  _V )
3 fveq2 5865 . . . . . . . . . . . 12  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
4 iscgrg.p . . . . . . . . . . . 12  |-  P  =  ( Base `  G
)
53, 4syl6eqr 2503 . . . . . . . . . . 11  |-  ( g  =  G  ->  ( Base `  g )  =  P )
65oveq1d 6305 . . . . . . . . . 10  |-  ( g  =  G  ->  (
( Base `  g )  ^pm  RR )  =  ( P  ^pm  RR )
)
76eleq2d 2514 . . . . . . . . 9  |-  ( g  =  G  ->  (
a  e.  ( (
Base `  g )  ^pm  RR )  <->  a  e.  ( P  ^pm  RR ) ) )
86eleq2d 2514 . . . . . . . . 9  |-  ( g  =  G  ->  (
b  e.  ( (
Base `  g )  ^pm  RR )  <->  b  e.  ( P  ^pm  RR ) ) )
97, 8anbi12d 717 . . . . . . . 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 5865 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  ( dist `  g )  =  ( dist `  G
) )
11 iscgrg.m . . . . . . . . . . . . 13  |-  .-  =  ( dist `  G )
1210, 11syl6eqr 2503 . . . . . . . . . . . 12  |-  ( g  =  G  ->  ( dist `  g )  = 
.-  )
1312oveqd 6307 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
( a `  i
) ( dist `  g
) ( a `  j ) )  =  ( ( a `  i )  .-  (
a `  j )
) )
1412oveqd 6307 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
( b `  i
) ( dist `  g
) ( b `  j ) )  =  ( ( b `  i )  .-  (
b `  j )
) )
1513, 14eqeq12d 2466 . . . . . . . . . 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 710 . . . . . . . 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 717 . . . . . . 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 4466 . . . . . 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 24556 . . . . . 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 4840 . . . . . . . 8  |-  ( ( P  ^pm  RR )  X.  ( P  ^pm  RR ) )  =  { <. a ,  b >.  |  ( a  e.  ( P  ^pm  RR )  /\  b  e.  ( P  ^pm  RR )
) }
22 ovex 6318 . . . . . . . . 9  |-  ( P 
^pm  RR )  e.  _V
2322, 22xpex 6595 . . . . . . . 8  |-  ( ( P  ^pm  RR )  X.  ( P  ^pm  RR ) )  e.  _V
2421, 23eqeltrri 2526 . . . . . . 7  |-  { <. a ,  b >.  |  ( a  e.  ( P 
^pm  RR )  /\  b  e.  ( P  ^pm  RR ) ) }  e.  _V
25 simpl 459 . . . . . . . 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 4548 . . . . . 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 5948 . . . . 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 2497 . . 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 4413 . 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 5035 . . . . . 6  |-  ( a  =  A  ->  dom  a  =  dom  A )
3332eqeq1d 2453 . . . . 5  |-  ( a  =  A  ->  ( dom  a  =  dom  b 
<->  dom  A  =  dom  b ) )
3432adantr 467 . . . . . . 7  |-  ( ( a  =  A  /\  i  e.  dom  a )  ->  dom  a  =  dom  A )
35 simpll 760 . . . . . . . . . 10  |-  ( ( ( a  =  A  /\  i  e.  dom  a )  /\  j  e.  dom  a )  -> 
a  =  A )
3635fveq1d 5867 . . . . . . . . 9  |-  ( ( ( a  =  A  /\  i  e.  dom  a )  /\  j  e.  dom  a )  -> 
( a `  i
)  =  ( A `
 i ) )
3735fveq1d 5867 . . . . . . . . 9  |-  ( ( ( a  =  A  /\  i  e.  dom  a )  /\  j  e.  dom  a )  -> 
( a `  j
)  =  ( A `
 j ) )
3836, 37oveq12d 6308 . . . . . . . 8  |-  ( ( ( a  =  A  /\  i  e.  dom  a )  /\  j  e.  dom  a )  -> 
( ( a `  i )  .-  (
a `  j )
)  =  ( ( A `  i ) 
.-  ( A `  j ) ) )
3938eqeq1d 2453 . . . . . . 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 3003 . . . . . 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 3003 . . . . 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 717 . . . 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 5035 . . . . . 6  |-  ( b  =  B  ->  dom  b  =  dom  B )
4443eqeq2d 2461 . . . . 5  |-  ( b  =  B  ->  ( dom  A  =  dom  b  <->  dom 
A  =  dom  B
) )
45 fveq1 5864 . . . . . . . 8  |-  ( b  =  B  ->  (
b `  i )  =  ( B `  i ) )
46 fveq1 5864 . . . . . . . 8  |-  ( b  =  B  ->  (
b `  j )  =  ( B `  j ) )
4745, 46oveq12d 6308 . . . . . . 7  |-  ( b  =  B  ->  (
( b `  i
)  .-  ( b `  j ) )  =  ( ( B `  i )  .-  ( B `  j )
) )
4847eqeq2d 2461 . . . . . 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 717 . . . 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 706 . . 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 2451 . . 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 4884 . 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 265 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 188    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   _Vcvv 3045   class class class wbr 4402   {copab 4460    X. cxp 4832   dom cdm 4834   ` cfv 5582  (class class class)co 6290    ^pm cpm 7473   RRcr 9538   Basecbs 15121   distcds 15199  cgrGccgrg 24555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-iota 5546  df-fun 5584  df-fv 5590  df-ov 6293  df-cgrg 24556
This theorem is referenced by:  iscgrgd  24558  ercgrg  24562
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