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Theorem iscgra 24930
Description: Property for two angles ABC and DEF to be congruent. This is a modified version of the definition 11.3 of [Schwabhauser] p. 95. where the number of constructed points has been reduced to two. We chose this version rather than the textbook version because it is shorter and therefore simpler to work with. Theorem dfcgra2 24950 shows that those definitions are indeed equivalent. (Contributed by Thierry Arnoux, 31-Jul-2020.)
Hypotheses
Ref Expression
iscgra.p  |-  P  =  ( Base `  G
)
iscgra.i  |-  I  =  (Itv `  G )
iscgra.k  |-  K  =  (hlG `  G )
iscgra.g  |-  ( ph  ->  G  e. TarskiG )
iscgra.a  |-  ( ph  ->  A  e.  P )
iscgra.b  |-  ( ph  ->  B  e.  P )
iscgra.c  |-  ( ph  ->  C  e.  P )
iscgra.d  |-  ( ph  ->  D  e.  P )
iscgra.e  |-  ( ph  ->  E  e.  P )
iscgra.f  |-  ( ph  ->  F  e.  P )
Assertion
Ref Expression
iscgra  |-  ( ph  ->  ( <" A B C "> (cgrA `  G ) <" D E F ">  <->  E. x  e.  P  E. y  e.  P  ( <" A B C "> (cgrG `  G ) <" x E y ">  /\  x
( K `  E
) D  /\  y
( K `  E
) F ) ) )
Distinct variable groups:    x, A, y    x, B, y    x, C, y    x, D, y   
x, E, y    x, F, y    x, K, y    ph, x, y    x, G, y    x, I, y   
x, P, y

Proof of Theorem iscgra
Dummy variables  a 
b  g  k  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 464 . . . . . . . 8  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  a  =  <" A B C "> )
2 eqidd 2472 . . . . . . . . 9  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  x  =  x )
3 simpr 468 . . . . . . . . . 10  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  b  =  <" D E F "> )
43fveq1d 5881 . . . . . . . . 9  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( b `  1
)  =  ( <" D E F "> `  1
) )
5 eqidd 2472 . . . . . . . . 9  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  y  =  y )
62, 4, 5s3eqd 13019 . . . . . . . 8  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  <" x ( b `  1 ) y ">  =  <" x ( <" D E F "> `  1
) y "> )
71, 6breq12d 4408 . . . . . . 7  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( a (cgrG `  G ) <" x
( b `  1
) y ">  <->  <" A B C "> (cgrG `  G ) <" x ( <" D E F "> `  1
) y "> ) )
84fveq2d 5883 . . . . . . . 8  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( K `  (
b `  1 )
)  =  ( K `
 ( <" D E F "> `  1
) ) )
93fveq1d 5881 . . . . . . . 8  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( b `  0
)  =  ( <" D E F "> `  0
) )
102, 8, 9breq123d 4409 . . . . . . 7  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( x ( K `
 ( b ` 
1 ) ) ( b `  0 )  <-> 
x ( K `  ( <" D E F "> `  1
) ) ( <" D E F "> `  0
) ) )
113fveq1d 5881 . . . . . . . 8  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( b `  2
)  =  ( <" D E F "> `  2
) )
125, 8, 11breq123d 4409 . . . . . . 7  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( y ( K `
 ( b ` 
1 ) ) ( b `  2 )  <-> 
y ( K `  ( <" D E F "> `  1
) ) ( <" D E F "> `  2
) ) )
137, 10, 123anbi123d 1365 . . . . . 6  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( ( a (cgrG `  G ) <" x
( b `  1
) y ">  /\  x ( K `  ( b `  1
) ) ( b `
 0 )  /\  y ( K `  ( b `  1
) ) ( b `
 2 ) )  <-> 
( <" A B C "> (cgrG `  G ) <" x
( <" D E F "> `  1
) y ">  /\  x ( K `  ( <" D E F "> `  1
) ) ( <" D E F "> `  0
)  /\  y ( K `  ( <" D E F "> `  1 ) ) ( <" D E F "> `  2
) ) ) )
14132rexbidv 2897 . . . . 5  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( E. x  e.  P  E. y  e.  P  ( a (cgrG `  G ) <" x
( b `  1
) y ">  /\  x ( K `  ( b `  1
) ) ( b `
 0 )  /\  y ( K `  ( b `  1
) ) ( b `
 2 ) )  <->  E. x  e.  P  E. y  e.  P  ( <" A B C "> (cgrG `  G ) <" x
( <" D E F "> `  1
) y ">  /\  x ( K `  ( <" D E F "> `  1
) ) ( <" D E F "> `  0
)  /\  y ( K `  ( <" D E F "> `  1 ) ) ( <" D E F "> `  2
) ) ) )
15 eqid 2471 . . . . 5  |-  { <. a ,  b >.  |  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  (
a (cgrG `  G
) <" x ( b `  1 ) y ">  /\  x
( K `  (
b `  1 )
) ( b ` 
0 )  /\  y
( K `  (
b `  1 )
) ( b ` 
2 ) ) ) }  =  { <. a ,  b >.  |  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  (
a (cgrG `  G
) <" x ( b `  1 ) y ">  /\  x
( K `  (
b `  1 )
) ( b ` 
0 )  /\  y
( K `  (
b `  1 )
) ( b ` 
2 ) ) ) }
1614, 15brab2a 4889 . . . 4  |-  ( <" A B C "> { <. a ,  b >.  |  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  (
a (cgrG `  G
) <" x ( b `  1 ) y ">  /\  x
( K `  (
b `  1 )
) ( b ` 
0 )  /\  y
( K `  (
b `  1 )
) ( b ` 
2 ) ) ) } <" D E F ">  <->  ( ( <" A B C ">  e.  ( P  ^m  ( 0..^ 3 ) )  /\  <" D E F ">  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  ( <" A B C "> (cgrG `  G ) <" x
( <" D E F "> `  1
) y ">  /\  x ( K `  ( <" D E F "> `  1
) ) ( <" D E F "> `  0
)  /\  y ( K `  ( <" D E F "> `  1 ) ) ( <" D E F "> `  2
) ) ) )
1716a1i 11 . . 3  |-  ( ph  ->  ( <" A B C "> { <. a ,  b >.  |  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  (
a (cgrG `  G
) <" x ( b `  1 ) y ">  /\  x
( K `  (
b `  1 )
) ( b ` 
0 )  /\  y
( K `  (
b `  1 )
) ( b ` 
2 ) ) ) } <" D E F ">  <->  ( ( <" A B C ">  e.  ( P  ^m  ( 0..^ 3 ) )  /\  <" D E F ">  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  ( <" A B C "> (cgrG `  G ) <" x
( <" D E F "> `  1
) y ">  /\  x ( K `  ( <" D E F "> `  1
) ) ( <" D E F "> `  0
)  /\  y ( K `  ( <" D E F "> `  1 ) ) ( <" D E F "> `  2
) ) ) ) )
18 eqidd 2472 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  P ) )  ->  x  =  x )
19 iscgra.e . . . . . . . . . 10  |-  ( ph  ->  E  e.  P )
20 s3fv1 13046 . . . . . . . . . 10  |-  ( E  e.  P  ->  ( <" D E F "> `  1
)  =  E )
2119, 20syl 17 . . . . . . . . 9  |-  ( ph  ->  ( <" D E F "> `  1
)  =  E )
2221adantr 472 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  P ) )  -> 
( <" D E F "> `  1
)  =  E )
23 eqidd 2472 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  P ) )  -> 
y  =  y )
2418, 22, 23s3eqd 13019 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  P ) )  ->  <" x ( <" D E F "> `  1
) y ">  =  <" x E y "> )
2524breq2d 4407 . . . . . 6  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  P ) )  -> 
( <" A B C "> (cgrG `  G ) <" x
( <" D E F "> `  1
) y ">  <->  <" A B C "> (cgrG `  G ) <" x E y "> ) )
2622fveq2d 5883 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  P ) )  -> 
( K `  ( <" D E F "> `  1
) )  =  ( K `  E ) )
27 iscgra.d . . . . . . . . 9  |-  ( ph  ->  D  e.  P )
28 s3fv0 13045 . . . . . . . . 9  |-  ( D  e.  P  ->  ( <" D E F "> `  0
)  =  D )
2927, 28syl 17 . . . . . . . 8  |-  ( ph  ->  ( <" D E F "> `  0
)  =  D )
3029adantr 472 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  P ) )  -> 
( <" D E F "> `  0
)  =  D )
3118, 26, 30breq123d 4409 . . . . . 6  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  P ) )  -> 
( x ( K `
 ( <" D E F "> `  1
) ) ( <" D E F "> `  0
)  <->  x ( K `
 E ) D ) )
32 iscgra.f . . . . . . . . 9  |-  ( ph  ->  F  e.  P )
33 s3fv2 13047 . . . . . . . . 9  |-  ( F  e.  P  ->  ( <" D E F "> `  2
)  =  F )
3432, 33syl 17 . . . . . . . 8  |-  ( ph  ->  ( <" D E F "> `  2
)  =  F )
3534adantr 472 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  P ) )  -> 
( <" D E F "> `  2
)  =  F )
3623, 26, 35breq123d 4409 . . . . . 6  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  P ) )  -> 
( y ( K `
 ( <" D E F "> `  1
) ) ( <" D E F "> `  2
)  <->  y ( K `
 E ) F ) )
3725, 31, 363anbi123d 1365 . . . . 5  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  P ) )  -> 
( ( <" A B C "> (cgrG `  G ) <" x
( <" D E F "> `  1
) y ">  /\  x ( K `  ( <" D E F "> `  1
) ) ( <" D E F "> `  0
)  /\  y ( K `  ( <" D E F "> `  1 ) ) ( <" D E F "> `  2
) )  <->  ( <" A B C "> (cgrG `  G ) <" x E y ">  /\  x
( K `  E
) D  /\  y
( K `  E
) F ) ) )
38372rexbidva 2896 . . . 4  |-  ( ph  ->  ( E. x  e.  P  E. y  e.  P  ( <" A B C "> (cgrG `  G ) <" x
( <" D E F "> `  1
) y ">  /\  x ( K `  ( <" D E F "> `  1
) ) ( <" D E F "> `  0
)  /\  y ( K `  ( <" D E F "> `  1 ) ) ( <" D E F "> `  2
) )  <->  E. x  e.  P  E. y  e.  P  ( <" A B C "> (cgrG `  G ) <" x E y ">  /\  x
( K `  E
) D  /\  y
( K `  E
) F ) ) )
3938anbi2d 718 . . 3  |-  ( ph  ->  ( ( ( <" A B C ">  e.  ( P  ^m  ( 0..^ 3 ) )  /\  <" D E F ">  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  ( <" A B C "> (cgrG `  G ) <" x
( <" D E F "> `  1
) y ">  /\  x ( K `  ( <" D E F "> `  1
) ) ( <" D E F "> `  0
)  /\  y ( K `  ( <" D E F "> `  1 ) ) ( <" D E F "> `  2
) ) )  <->  ( ( <" A B C ">  e.  ( P  ^m  ( 0..^ 3 ) )  /\  <" D E F ">  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  ( <" A B C "> (cgrG `  G ) <" x E y ">  /\  x ( K `  E ) D  /\  y ( K `  E ) F ) ) ) )
4017, 39bitrd 261 . 2  |-  ( ph  ->  ( <" A B C "> { <. a ,  b >.  |  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  (
a (cgrG `  G
) <" x ( b `  1 ) y ">  /\  x
( K `  (
b `  1 )
) ( b ` 
0 )  /\  y
( K `  (
b `  1 )
) ( b ` 
2 ) ) ) } <" D E F ">  <->  ( ( <" A B C ">  e.  ( P  ^m  ( 0..^ 3 ) )  /\  <" D E F ">  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  ( <" A B C "> (cgrG `  G ) <" x E y ">  /\  x ( K `  E ) D  /\  y ( K `  E ) F ) ) ) )
41 iscgra.g . . . 4  |-  ( ph  ->  G  e. TarskiG )
42 elex 3040 . . . 4  |-  ( G  e. TarskiG  ->  G  e.  _V )
43 iscgra.p . . . . . . . 8  |-  P  =  ( Base `  G
)
44 iscgra.k . . . . . . . 8  |-  K  =  (hlG `  G )
45 simpl 464 . . . . . . . . . . . . 13  |-  ( ( p  =  P  /\  k  =  K )  ->  p  =  P )
4645eqcomd 2477 . . . . . . . . . . . 12  |-  ( ( p  =  P  /\  k  =  K )  ->  P  =  p )
4746oveq1d 6323 . . . . . . . . . . 11  |-  ( ( p  =  P  /\  k  =  K )  ->  ( P  ^m  (
0..^ 3 ) )  =  ( p  ^m  ( 0..^ 3 ) ) )
4847eleq2d 2534 . . . . . . . . . 10  |-  ( ( p  =  P  /\  k  =  K )  ->  ( a  e.  ( P  ^m  ( 0..^ 3 ) )  <->  a  e.  ( p  ^m  (
0..^ 3 ) ) ) )
4947eleq2d 2534 . . . . . . . . . 10  |-  ( ( p  =  P  /\  k  =  K )  ->  ( b  e.  ( P  ^m  ( 0..^ 3 ) )  <->  b  e.  ( p  ^m  (
0..^ 3 ) ) ) )
5048, 49anbi12d 725 . . . . . . . . 9  |-  ( ( p  =  P  /\  k  =  K )  ->  ( ( a  e.  ( P  ^m  (
0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  <-> 
( a  e.  ( p  ^m  ( 0..^ 3 ) )  /\  b  e.  ( p  ^m  ( 0..^ 3 ) ) ) ) )
51 simpr 468 . . . . . . . . . . . . . . 15  |-  ( ( p  =  P  /\  k  =  K )  ->  k  =  K )
52 eqidd 2472 . . . . . . . . . . . . . . 15  |-  ( ( p  =  P  /\  k  =  K )  ->  ( b `  1
)  =  ( b `
 1 ) )
5351, 52fveq12d 5885 . . . . . . . . . . . . . 14  |-  ( ( p  =  P  /\  k  =  K )  ->  ( k `  (
b `  1 )
)  =  ( K `
 ( b ` 
1 ) ) )
5453breqd 4406 . . . . . . . . . . . . 13  |-  ( ( p  =  P  /\  k  =  K )  ->  ( x ( k `
 ( b ` 
1 ) ) ( b `  0 )  <-> 
x ( K `  ( b `  1
) ) ( b `
 0 ) ) )
5553breqd 4406 . . . . . . . . . . . . 13  |-  ( ( p  =  P  /\  k  =  K )  ->  ( y ( k `
 ( b ` 
1 ) ) ( b `  2 )  <-> 
y ( K `  ( b `  1
) ) ( b `
 2 ) ) )
5654, 553anbi23d 1368 . . . . . . . . . . . 12  |-  ( ( p  =  P  /\  k  =  K )  ->  ( ( a (cgrG `  g ) <" x
( b `  1
) y ">  /\  x ( k `  ( b `  1
) ) ( b `
 0 )  /\  y ( k `  ( b `  1
) ) ( b `
 2 ) )  <-> 
( a (cgrG `  g ) <" x
( b `  1
) y ">  /\  x ( K `  ( b `  1
) ) ( b `
 0 )  /\  y ( K `  ( b `  1
) ) ( b `
 2 ) ) ) )
5756bicomd 206 . . . . . . . . . . 11  |-  ( ( p  =  P  /\  k  =  K )  ->  ( ( a (cgrG `  g ) <" x
( b `  1
) y ">  /\  x ( K `  ( b `  1
) ) ( b `
 0 )  /\  y ( K `  ( b `  1
) ) ( b `
 2 ) )  <-> 
( a (cgrG `  g ) <" x
( b `  1
) y ">  /\  x ( k `  ( b `  1
) ) ( b `
 0 )  /\  y ( k `  ( b `  1
) ) ( b `
 2 ) ) ) )
5846, 57rexeqbidv 2988 . . . . . . . . . 10  |-  ( ( p  =  P  /\  k  =  K )  ->  ( E. y  e.  P  ( a (cgrG `  g ) <" x
( b `  1
) y ">  /\  x ( K `  ( b `  1
) ) ( b `
 0 )  /\  y ( K `  ( b `  1
) ) ( b `
 2 ) )  <->  E. y  e.  p  ( a (cgrG `  g ) <" x
( b `  1
) y ">  /\  x ( k `  ( b `  1
) ) ( b `
 0 )  /\  y ( k `  ( b `  1
) ) ( b `
 2 ) ) ) )
5946, 58rexeqbidv 2988 . . . . . . . . 9  |-  ( ( p  =  P  /\  k  =  K )  ->  ( E. x  e.  P  E. y  e.  P  ( a (cgrG `  g ) <" x
( b `  1
) y ">  /\  x ( K `  ( b `  1
) ) ( b `
 0 )  /\  y ( K `  ( b `  1
) ) ( b `
 2 ) )  <->  E. x  e.  p  E. y  e.  p  ( a (cgrG `  g ) <" x
( b `  1
) y ">  /\  x ( k `  ( b `  1
) ) ( b `
 0 )  /\  y ( k `  ( b `  1
) ) ( b `
 2 ) ) ) )
6050, 59anbi12d 725 . . . . . . . 8  |-  ( ( p  =  P  /\  k  =  K )  ->  ( ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  ( a (cgrG `  g ) <" x
( b `  1
) y ">  /\  x ( K `  ( b `  1
) ) ( b `
 0 )  /\  y ( K `  ( b `  1
) ) ( b `
 2 ) ) )  <->  ( ( a  e.  ( p  ^m  ( 0..^ 3 ) )  /\  b  e.  ( p  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  p  E. y  e.  p  ( a (cgrG `  g ) <" x
( b `  1
) y ">  /\  x ( k `  ( b `  1
) ) ( b `
 0 )  /\  y ( k `  ( b `  1
) ) ( b `
 2 ) ) ) ) )
6143, 44, 60sbcie2s 15244 . . . . . . 7  |-  ( g  =  G  ->  ( [. ( Base `  g
)  /  p ]. [. (hlG `  g )  /  k ]. (
( a  e.  ( p  ^m  ( 0..^ 3 ) )  /\  b  e.  ( p  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  p  E. y  e.  p  (
a (cgrG `  g
) <" x ( b `  1 ) y ">  /\  x
( k `  (
b `  1 )
) ( b ` 
0 )  /\  y
( k `  (
b `  1 )
) ( b ` 
2 ) ) )  <-> 
( ( a  e.  ( P  ^m  (
0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  ( a (cgrG `  g ) <" x
( b `  1
) y ">  /\  x ( K `  ( b `  1
) ) ( b `
 0 )  /\  y ( K `  ( b `  1
) ) ( b `
 2 ) ) ) ) )
6261opabbidv 4459 . . . . . 6  |-  ( g  =  G  ->  { <. a ,  b >.  |  [. ( Base `  g )  /  p ]. [. (hlG `  g )  /  k ]. ( ( a  e.  ( p  ^m  (
0..^ 3 ) )  /\  b  e.  ( p  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  p  E. y  e.  p  ( a (cgrG `  g ) <" x
( b `  1
) y ">  /\  x ( k `  ( b `  1
) ) ( b `
 0 )  /\  y ( k `  ( b `  1
) ) ( b `
 2 ) ) ) }  =  { <. a ,  b >.  |  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  ( a (cgrG `  g ) <" x
( b `  1
) y ">  /\  x ( K `  ( b `  1
) ) ( b `
 0 )  /\  y ( K `  ( b `  1
) ) ( b `
 2 ) ) ) } )
63 fveq2 5879 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (cgrG `  g )  =  (cgrG `  G ) )
6463breqd 4406 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
a (cgrG `  g
) <" x ( b `  1 ) y ">  <->  a (cgrG `  G ) <" x
( b `  1
) y "> ) )
65643anbi1d 1369 . . . . . . . . . 10  |-  ( g  =  G  ->  (
( a (cgrG `  g ) <" x
( b `  1
) y ">  /\  x ( K `  ( b `  1
) ) ( b `
 0 )  /\  y ( K `  ( b `  1
) ) ( b `
 2 ) )  <-> 
( a (cgrG `  G ) <" x
( b `  1
) y ">  /\  x ( K `  ( b `  1
) ) ( b `
 0 )  /\  y ( K `  ( b `  1
) ) ( b `
 2 ) ) ) )
6665rexbidv 2892 . . . . . . . . 9  |-  ( g  =  G  ->  ( E. y  e.  P  ( a (cgrG `  g ) <" x
( b `  1
) y ">  /\  x ( K `  ( b `  1
) ) ( b `
 0 )  /\  y ( K `  ( b `  1
) ) ( b `
 2 ) )  <->  E. y  e.  P  ( a (cgrG `  G ) <" x
( b `  1
) y ">  /\  x ( K `  ( b `  1
) ) ( b `
 0 )  /\  y ( K `  ( b `  1
) ) ( b `
 2 ) ) ) )
6766rexbidv 2892 . . . . . . . 8  |-  ( g  =  G  ->  ( E. x  e.  P  E. y  e.  P  ( a (cgrG `  g ) <" x
( b `  1
) y ">  /\  x ( K `  ( b `  1
) ) ( b `
 0 )  /\  y ( K `  ( b `  1
) ) ( b `
 2 ) )  <->  E. x  e.  P  E. y  e.  P  ( a (cgrG `  G ) <" x
( b `  1
) y ">  /\  x ( K `  ( b `  1
) ) ( b `
 0 )  /\  y ( K `  ( b `  1
) ) ( b `
 2 ) ) ) )
6867anbi2d 718 . . . . . . 7  |-  ( g  =  G  ->  (
( ( a  e.  ( P  ^m  (
0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  ( a (cgrG `  g ) <" x
( b `  1
) y ">  /\  x ( K `  ( b `  1
) ) ( b `
 0 )  /\  y ( K `  ( b `  1
) ) ( b `
 2 ) ) )  <->  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  ( a (cgrG `  G ) <" x
( b `  1
) y ">  /\  x ( K `  ( b `  1
) ) ( b `
 0 )  /\  y ( K `  ( b `  1
) ) ( b `
 2 ) ) ) ) )
6968opabbidv 4459 . . . . . 6  |-  ( g  =  G  ->  { <. a ,  b >.  |  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  (
a (cgrG `  g
) <" x ( b `  1 ) y ">  /\  x
( K `  (
b `  1 )
) ( b ` 
0 )  /\  y
( K `  (
b `  1 )
) ( b ` 
2 ) ) ) }  =  { <. a ,  b >.  |  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  (
a (cgrG `  G
) <" x ( b `  1 ) y ">  /\  x
( K `  (
b `  1 )
) ( b ` 
0 )  /\  y
( K `  (
b `  1 )
) ( b ` 
2 ) ) ) } )
7062, 69eqtrd 2505 . . . . 5  |-  ( g  =  G  ->  { <. a ,  b >.  |  [. ( Base `  g )  /  p ]. [. (hlG `  g )  /  k ]. ( ( a  e.  ( p  ^m  (
0..^ 3 ) )  /\  b  e.  ( p  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  p  E. y  e.  p  ( a (cgrG `  g ) <" x
( b `  1
) y ">  /\  x ( k `  ( b `  1
) ) ( b `
 0 )  /\  y ( k `  ( b `  1
) ) ( b `
 2 ) ) ) }  =  { <. a ,  b >.  |  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  ( a (cgrG `  G ) <" x
( b `  1
) y ">  /\  x ( K `  ( b `  1
) ) ( b `
 0 )  /\  y ( K `  ( b `  1
) ) ( b `
 2 ) ) ) } )
71 df-cgra 24929 . . . . 5  |- cgrA  =  ( g  e.  _V  |->  {
<. a ,  b >.  |  [. ( Base `  g
)  /  p ]. [. (hlG `  g )  /  k ]. (
( a  e.  ( p  ^m  ( 0..^ 3 ) )  /\  b  e.  ( p  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  p  E. y  e.  p  (
a (cgrG `  g
) <" x ( b `  1 ) y ">  /\  x
( k `  (
b `  1 )
) ( b ` 
0 )  /\  y
( k `  (
b `  1 )
) ( b ` 
2 ) ) ) } )
72 ovex 6336 . . . . . . 7  |-  ( P  ^m  ( 0..^ 3 ) )  e.  _V
7372, 72xpex 6614 . . . . . 6  |-  ( ( P  ^m  ( 0..^ 3 ) )  X.  ( P  ^m  (
0..^ 3 ) ) )  e.  _V
74 opabssxp 4914 . . . . . 6  |-  { <. a ,  b >.  |  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  (
a (cgrG `  G
) <" x ( b `  1 ) y ">  /\  x
( K `  (
b `  1 )
) ( b ` 
0 )  /\  y
( K `  (
b `  1 )
) ( b ` 
2 ) ) ) }  C_  ( ( P  ^m  ( 0..^ 3 ) )  X.  ( P  ^m  ( 0..^ 3 ) ) )
7573, 74ssexi 4541 . . . . 5  |-  { <. a ,  b >.  |  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  (
a (cgrG `  G
) <" x ( b `  1 ) y ">  /\  x
( K `  (
b `  1 )
) ( b ` 
0 )  /\  y
( K `  (
b `  1 )
) ( b ` 
2 ) ) ) }  e.  _V
7670, 71, 75fvmpt 5963 . . . 4  |-  ( G  e.  _V  ->  (cgrA `  G )  =  { <. a ,  b >.  |  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  ( a (cgrG `  G ) <" x
( b `  1
) y ">  /\  x ( K `  ( b `  1
) ) ( b `
 0 )  /\  y ( K `  ( b `  1
) ) ( b `
 2 ) ) ) } )
7741, 42, 763syl 18 . . 3  |-  ( ph  ->  (cgrA `  G )  =  { <. a ,  b
>.  |  ( (
a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  (
0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  ( a
(cgrG `  G ) <" x ( b `
 1 ) y ">  /\  x
( K `  (
b `  1 )
) ( b ` 
0 )  /\  y
( K `  (
b `  1 )
) ( b ` 
2 ) ) ) } )
7877breqd 4406 . 2  |-  ( ph  ->  ( <" A B C "> (cgrA `  G ) <" D E F ">  <->  <" A B C "> { <. a ,  b >.  |  ( ( a  e.  ( P  ^m  ( 0..^ 3 ) )  /\  b  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  (
a (cgrG `  G
) <" x ( b `  1 ) y ">  /\  x
( K `  (
b `  1 )
) ( b ` 
0 )  /\  y
( K `  (
b `  1 )
) ( b ` 
2 ) ) ) } <" D E F "> )
)
79 iscgra.a . . . . . . 7  |-  ( ph  ->  A  e.  P )
80 iscgra.b . . . . . . 7  |-  ( ph  ->  B  e.  P )
81 iscgra.c . . . . . . 7  |-  ( ph  ->  C  e.  P )
8279, 80, 81s3cld 13026 . . . . . 6  |-  ( ph  ->  <" A B C ">  e. Word  P )
83 s3len 13048 . . . . . . 7  |-  ( # `  <" A B C "> )  =  3
8483a1i 11 . . . . . 6  |-  ( ph  ->  ( # `  <" A B C "> )  =  3
)
8582, 84jca 541 . . . . 5  |-  ( ph  ->  ( <" A B C ">  e. Word  P  /\  ( # `  <" A B C "> )  =  3
) )
86 fvex 5889 . . . . . . 7  |-  ( Base `  G )  e.  _V
8743, 86eqeltri 2545 . . . . . 6  |-  P  e. 
_V
88 3nn0 10911 . . . . . 6  |-  3  e.  NN0
89 wrdmap 12749 . . . . . 6  |-  ( ( P  e.  _V  /\  3  e.  NN0 )  -> 
( ( <" A B C ">  e. Word  P  /\  ( # `  <" A B C "> )  =  3
)  <->  <" A B C ">  e.  ( P  ^m  (
0..^ 3 ) ) ) )
9087, 88, 89mp2an 686 . . . . 5  |-  ( (
<" A B C ">  e. Word  P  /\  ( # `  <" A B C "> )  =  3
)  <->  <" A B C ">  e.  ( P  ^m  (
0..^ 3 ) ) )
9185, 90sylib 201 . . . 4  |-  ( ph  ->  <" A B C ">  e.  ( P  ^m  (
0..^ 3 ) ) )
9227, 19, 32s3cld 13026 . . . . . 6  |-  ( ph  ->  <" D E F ">  e. Word  P )
93 s3len 13048 . . . . . . 7  |-  ( # `  <" D E F "> )  =  3
9493a1i 11 . . . . . 6  |-  ( ph  ->  ( # `  <" D E F "> )  =  3
)
9592, 94jca 541 . . . . 5  |-  ( ph  ->  ( <" D E F ">  e. Word  P  /\  ( # `  <" D E F "> )  =  3
) )
96 wrdmap 12749 . . . . . 6  |-  ( ( P  e.  _V  /\  3  e.  NN0 )  -> 
( ( <" D E F ">  e. Word  P  /\  ( # `  <" D E F "> )  =  3
)  <->  <" D E F ">  e.  ( P  ^m  (
0..^ 3 ) ) ) )
9787, 88, 96mp2an 686 . . . . 5  |-  ( (
<" D E F ">  e. Word  P  /\  ( # `  <" D E F "> )  =  3
)  <->  <" D E F ">  e.  ( P  ^m  (
0..^ 3 ) ) )
9895, 97sylib 201 . . . 4  |-  ( ph  ->  <" D E F ">  e.  ( P  ^m  (
0..^ 3 ) ) )
9991, 98jca 541 . . 3  |-  ( ph  ->  ( <" A B C ">  e.  ( P  ^m  (
0..^ 3 ) )  /\  <" D E F ">  e.  ( P  ^m  (
0..^ 3 ) ) ) )
10099biantrurd 516 . 2  |-  ( ph  ->  ( E. x  e.  P  E. y  e.  P  ( <" A B C "> (cgrG `  G ) <" x E y ">  /\  x ( K `  E ) D  /\  y ( K `  E ) F )  <-> 
( ( <" A B C ">  e.  ( P  ^m  (
0..^ 3 ) )  /\  <" D E F ">  e.  ( P  ^m  (
0..^ 3 ) ) )  /\  E. x  e.  P  E. y  e.  P  ( <" A B C "> (cgrG `  G ) <" x E y ">  /\  x
( K `  E
) D  /\  y
( K `  E
) F ) ) ) )
10140, 78, 1003bitr4d 293 1  |-  ( ph  ->  ( <" A B C "> (cgrA `  G ) <" D E F ">  <->  E. x  e.  P  E. y  e.  P  ( <" A B C "> (cgrG `  G ) <" x E y ">  /\  x
( K `  E
) D  /\  y
( K `  E
) F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   E.wrex 2757   _Vcvv 3031   [.wsbc 3255   class class class wbr 4395   {copab 4453    X. cxp 4837   ` cfv 5589  (class class class)co 6308    ^m cmap 7490   0cc0 9557   1c1 9558   2c2 10681   3c3 10682   NN0cn0 10893  ..^cfzo 11942   #chash 12553  Word cword 12703   <"cs3 12997   Basecbs 15199  TarskiGcstrkg 24557  Itvcitv 24563  cgrGccgrg 24634  hlGchlg 24724  cgrAccgra 24928
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-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  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-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-concat 12713  df-s1 12714  df-s2 13003  df-s3 13004  df-cgra 24929
This theorem is referenced by:  iscgra1  24931  iscgrad  24932  cgrane1  24933  cgrane2  24934  cgrane3  24935  cgrane4  24936  cgrahl1  24937  cgrahl2  24938  cgracgr  24939  cgraswap  24941  cgracom  24943  cgratr  24944  cgrabtwn  24946  cgrahl  24947  sacgr  24951
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