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Theorem iscgra 24851
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 24871 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 459 . . . . . . . 8  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  a  =  <" A B C "> )
2 eqidd 2452 . . . . . . . . 9  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  x  =  x )
3 simpr 463 . . . . . . . . . 10  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  b  =  <" D E F "> )
43fveq1d 5867 . . . . . . . . 9  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( b `  1
)  =  ( <" D E F "> `  1
) )
5 eqidd 2452 . . . . . . . . 9  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  y  =  y )
62, 4, 5s3eqd 12959 . . . . . . . 8  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  <" x ( b `  1 ) y ">  =  <" x ( <" D E F "> `  1
) y "> )
71, 6breq12d 4415 . . . . . . 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 5869 . . . . . . . 8  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( K `  (
b `  1 )
)  =  ( K `
 ( <" D E F "> `  1
) ) )
93fveq1d 5867 . . . . . . . 8  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( b `  0
)  =  ( <" D E F "> `  0
) )
102, 8, 9breq123d 4416 . . . . . . 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 5867 . . . . . . . 8  |-  ( ( a  =  <" A B C ">  /\  b  =  <" D E F "> )  ->  ( b `  2
)  =  ( <" D E F "> `  2
) )
125, 8, 11breq123d 4416 . . . . . . 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 1339 . . . . . 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 2908 . . . . 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 2451 . . . . 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 4884 . . . 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 2452 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  P ) )  ->  x  =  x )
19 iscgra.e . . . . . . . . . 10  |-  ( ph  ->  E  e.  P )
20 s3fv1 12986 . . . . . . . . . 10  |-  ( E  e.  P  ->  ( <" D E F "> `  1
)  =  E )
2119, 20syl 17 . . . . . . . . 9  |-  ( ph  ->  ( <" D E F "> `  1
)  =  E )
2221adantr 467 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  P ) )  -> 
( <" D E F "> `  1
)  =  E )
23 eqidd 2452 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  P ) )  -> 
y  =  y )
2418, 22, 23s3eqd 12959 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  P ) )  ->  <" x ( <" D E F "> `  1
) y ">  =  <" x E y "> )
2524breq2d 4414 . . . . . 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 5869 . . . . . . 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 12985 . . . . . . . . 9  |-  ( D  e.  P  ->  ( <" D E F "> `  0
)  =  D )
2927, 28syl 17 . . . . . . . 8  |-  ( ph  ->  ( <" D E F "> `  0
)  =  D )
3029adantr 467 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  P ) )  -> 
( <" D E F "> `  0
)  =  D )
3118, 26, 30breq123d 4416 . . . . . 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 12987 . . . . . . . . 9  |-  ( F  e.  P  ->  ( <" D E F "> `  2
)  =  F )
3432, 33syl 17 . . . . . . . 8  |-  ( ph  ->  ( <" D E F "> `  2
)  =  F )
3534adantr 467 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  P  /\  y  e.  P ) )  -> 
( <" D E F "> `  2
)  =  F )
3623, 26, 35breq123d 4416 . . . . . 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 1339 . . . . 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 2907 . . . 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 710 . . 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 257 . 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 3054 . . . 4  |-  ( G  e. TarskiG  ->  G  e.  _V )
43 iscgra.p . . . . . . . 8  |-  P  =  ( Base `  G
)
44 iscgra.k . . . . . . . 8  |-  K  =  (hlG `  G )
45 simpl 459 . . . . . . . . . . . . 13  |-  ( ( p  =  P  /\  k  =  K )  ->  p  =  P )
4645eqcomd 2457 . . . . . . . . . . . 12  |-  ( ( p  =  P  /\  k  =  K )  ->  P  =  p )
4746oveq1d 6305 . . . . . . . . . . 11  |-  ( ( p  =  P  /\  k  =  K )  ->  ( P  ^m  (
0..^ 3 ) )  =  ( p  ^m  ( 0..^ 3 ) ) )
4847eleq2d 2514 . . . . . . . . . 10  |-  ( ( p  =  P  /\  k  =  K )  ->  ( a  e.  ( P  ^m  ( 0..^ 3 ) )  <->  a  e.  ( p  ^m  (
0..^ 3 ) ) ) )
4947eleq2d 2514 . . . . . . . . . 10  |-  ( ( p  =  P  /\  k  =  K )  ->  ( b  e.  ( P  ^m  ( 0..^ 3 ) )  <->  b  e.  ( p  ^m  (
0..^ 3 ) ) ) )
5048, 49anbi12d 717 . . . . . . . . 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 463 . . . . . . . . . . . . . . 15  |-  ( ( p  =  P  /\  k  =  K )  ->  k  =  K )
52 eqidd 2452 . . . . . . . . . . . . . . 15  |-  ( ( p  =  P  /\  k  =  K )  ->  ( b `  1
)  =  ( b `
 1 ) )
5351, 52fveq12d 5871 . . . . . . . . . . . . . 14  |-  ( ( p  =  P  /\  k  =  K )  ->  ( k `  (
b `  1 )
)  =  ( K `
 ( b ` 
1 ) ) )
5453breqd 4413 . . . . . . . . . . . . 13  |-  ( ( p  =  P  /\  k  =  K )  ->  ( x ( k `
 ( b ` 
1 ) ) ( b `  0 )  <-> 
x ( K `  ( b `  1
) ) ( b `
 0 ) ) )
5553breqd 4413 . . . . . . . . . . . . 13  |-  ( ( p  =  P  /\  k  =  K )  ->  ( y ( k `
 ( b ` 
1 ) ) ( b `  2 )  <-> 
y ( K `  ( b `  1
) ) ( b `
 2 ) ) )
5654, 553anbi23d 1342 . . . . . . . . . . . 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 205 . . . . . . . . . . 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 3002 . . . . . . . . . 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 3002 . . . . . . . . 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 717 . . . . . . . 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 15166 . . . . . . 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 4466 . . . . . 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 5865 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (cgrG `  g )  =  (cgrG `  G ) )
6463breqd 4413 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
a (cgrG `  g
) <" x ( b `  1 ) y ">  <->  a (cgrG `  G ) <" x
( b `  1
) y "> ) )
65643anbi1d 1343 . . . . . . . . . 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 2901 . . . . . . . . 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 2901 . . . . . . . 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 710 . . . . . . 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 4466 . . . . . 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 2485 . . . . 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 24850 . . . . 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 6318 . . . . . . 7  |-  ( P  ^m  ( 0..^ 3 ) )  e.  _V
7372, 72xpex 6595 . . . . . 6  |-  ( ( P  ^m  ( 0..^ 3 ) )  X.  ( P  ^m  (
0..^ 3 ) ) )  e.  _V
74 opabssxp 4909 . . . . . 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 4548 . . . . 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 5948 . . . 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 4413 . 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 12966 . . . . . 6  |-  ( ph  ->  <" A B C ">  e. Word  P )
83 s3len 12988 . . . . . . 7  |-  ( # `  <" A B C "> )  =  3
8483a1i 11 . . . . . 6  |-  ( ph  ->  ( # `  <" A B C "> )  =  3
)
8582, 84jca 535 . . . . 5  |-  ( ph  ->  ( <" A B C ">  e. Word  P  /\  ( # `  <" A B C "> )  =  3
) )
86 fvex 5875 . . . . . . 7  |-  ( Base `  G )  e.  _V
8743, 86eqeltri 2525 . . . . . 6  |-  P  e. 
_V
88 3nn0 10887 . . . . . 6  |-  3  e.  NN0
89 wrdmap 12698 . . . . . 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 678 . . . . 5  |-  ( (
<" A B C ">  e. Word  P  /\  ( # `  <" A B C "> )  =  3
)  <->  <" A B C ">  e.  ( P  ^m  (
0..^ 3 ) ) )
9185, 90sylib 200 . . . 4  |-  ( ph  ->  <" A B C ">  e.  ( P  ^m  (
0..^ 3 ) ) )
9227, 19, 32s3cld 12966 . . . . . 6  |-  ( ph  ->  <" D E F ">  e. Word  P )
93 s3len 12988 . . . . . . 7  |-  ( # `  <" D E F "> )  =  3
9493a1i 11 . . . . . 6  |-  ( ph  ->  ( # `  <" D E F "> )  =  3
)
9592, 94jca 535 . . . . 5  |-  ( ph  ->  ( <" D E F ">  e. Word  P  /\  ( # `  <" D E F "> )  =  3
) )
96 wrdmap 12698 . . . . . 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 678 . . . . 5  |-  ( (
<" D E F ">  e. Word  P  /\  ( # `  <" D E F "> )  =  3
)  <->  <" D E F ">  e.  ( P  ^m  (
0..^ 3 ) ) )
9895, 97sylib 200 . . . 4  |-  ( ph  ->  <" D E F ">  e.  ( P  ^m  (
0..^ 3 ) ) )
9991, 98jca 535 . . 3  |-  ( ph  ->  ( <" A B C ">  e.  ( P  ^m  (
0..^ 3 ) )  /\  <" D E F ">  e.  ( P  ^m  (
0..^ 3 ) ) ) )
10099biantrurd 511 . 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 289 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 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   E.wrex 2738   _Vcvv 3045   [.wsbc 3267   class class class wbr 4402   {copab 4460    X. cxp 4832   ` cfv 5582  (class class class)co 6290    ^m cmap 7472   0cc0 9539   1c1 9540   2c2 10659   3c3 10660   NN0cn0 10869  ..^cfzo 11915   #chash 12515  Word cword 12656   <"cs3 12938   Basecbs 15121  TarskiGcstrkg 24478  Itvcitv 24484  cgrGccgrg 24555  hlGchlg 24645  cgrAccgra 24849
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-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  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-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-hash 12516  df-word 12664  df-concat 12666  df-s1 12667  df-s2 12944  df-s3 12945  df-cgra 24850
This theorem is referenced by:  iscgra1  24852  iscgrad  24853  cgrane1  24854  cgrane2  24855  cgrane3  24856  cgrane4  24857  cgrahl1  24858  cgrahl2  24859  cgracgr  24860  cgraswap  24862  cgracom  24864  cgratr  24865  cgrabtwn  24867  cgrahl  24868  sacgr  24872
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