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Theorem isinag 24958
Description: Property for point  X to lie in the angle  <" A B C "> Defnition 11.23 of [Schwabhauser] p. 101. (Contributed by Thierry Arnoux, 15-Aug-2020.)
Hypotheses
Ref Expression
isinag.p  |-  P  =  ( Base `  G
)
isinag.i  |-  I  =  (Itv `  G )
isinag.k  |-  K  =  (hlG `  G )
isinag.x  |-  ( ph  ->  X  e.  P )
isinag.a  |-  ( ph  ->  A  e.  P )
isinag.b  |-  ( ph  ->  B  e.  P )
isinag.c  |-  ( ph  ->  C  e.  P )
isinag.g  |-  ( ph  ->  G  e.  V )
Assertion
Ref Expression
isinag  |-  ( ph  ->  ( X (inA `  G ) <" A B C ">  <->  ( ( A  =/=  B  /\  C  =/=  B  /\  X  =/= 
B )  /\  E. x  e.  P  (
x  e.  ( A I C )  /\  ( x  =  B  \/  x ( K `  B ) X ) ) ) ) )
Distinct variable groups:    x, A    x, B    x, C    x, G    x, P    x, X    ph, x
Allowed substitution hints:    I( x)    K( x)    V( x)

Proof of Theorem isinag
Dummy variables  p  t  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 468 . . . . . . . . 9  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  t  =  <" A B C "> )
21fveq1d 5881 . . . . . . . 8  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( t `  0
)  =  ( <" A B C "> `  0
) )
31fveq1d 5881 . . . . . . . 8  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( t `  1
)  =  ( <" A B C "> `  1
) )
42, 3neeq12d 2704 . . . . . . 7  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( ( t ` 
0 )  =/=  (
t `  1 )  <->  (
<" A B C "> `  0
)  =/=  ( <" A B C "> `  1
) ) )
51fveq1d 5881 . . . . . . . 8  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( t `  2
)  =  ( <" A B C "> `  2
) )
65, 3neeq12d 2704 . . . . . . 7  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( ( t ` 
2 )  =/=  (
t `  1 )  <->  (
<" A B C "> `  2
)  =/=  ( <" A B C "> `  1
) ) )
7 simpl 464 . . . . . . . 8  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  p  =  X )
87, 3neeq12d 2704 . . . . . . 7  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( p  =/=  (
t `  1 )  <->  X  =/=  ( <" A B C "> `  1
) ) )
94, 6, 83anbi123d 1365 . . . . . 6  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( ( ( t `
 0 )  =/=  ( t `  1
)  /\  ( t `  2 )  =/=  ( t `  1
)  /\  p  =/=  ( t `  1
) )  <->  ( ( <" A B C "> `  0
)  =/=  ( <" A B C "> `  1
)  /\  ( <" A B C "> `  2 )  =/=  ( <" A B C "> `  1
)  /\  X  =/=  ( <" A B C "> `  1
) ) ) )
10 eqidd 2472 . . . . . . . . 9  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  x  =  x )
112, 5oveq12d 6326 . . . . . . . . 9  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( ( t ` 
0 ) I ( t `  2 ) )  =  ( (
<" A B C "> `  0
) I ( <" A B C "> `  2
) ) )
1210, 11eleq12d 2543 . . . . . . . 8  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( x  e.  ( ( t `  0
) I ( t `
 2 ) )  <-> 
x  e.  ( (
<" A B C "> `  0
) I ( <" A B C "> `  2
) ) ) )
1310, 3eqeq12d 2486 . . . . . . . . 9  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( x  =  ( t `  1 )  <-> 
x  =  ( <" A B C "> `  1
) ) )
143fveq2d 5883 . . . . . . . . . 10  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( K `  (
t `  1 )
)  =  ( K `
 ( <" A B C "> `  1
) ) )
1510, 14, 7breq123d 4409 . . . . . . . . 9  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( x ( K `
 ( t ` 
1 ) ) p  <-> 
x ( K `  ( <" A B C "> `  1
) ) X ) )
1613, 15orbi12d 724 . . . . . . . 8  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( ( x  =  ( t `  1
)  \/  x ( K `  ( t `
 1 ) ) p )  <->  ( x  =  ( <" A B C "> `  1
)  \/  x ( K `  ( <" A B C "> `  1
) ) X ) ) )
1712, 16anbi12d 725 . . . . . . 7  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( ( x  e.  ( ( t ` 
0 ) I ( t `  2 ) )  /\  ( x  =  ( t ` 
1 )  \/  x
( K `  (
t `  1 )
) p ) )  <-> 
( x  e.  ( ( <" A B C "> `  0
) I ( <" A B C "> `  2
) )  /\  (
x  =  ( <" A B C "> `  1
)  \/  x ( K `  ( <" A B C "> `  1
) ) X ) ) ) )
1817rexbidv 2892 . . . . . 6  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( E. x  e.  P  ( x  e.  ( ( t ` 
0 ) I ( t `  2 ) )  /\  ( x  =  ( t ` 
1 )  \/  x
( K `  (
t `  1 )
) p ) )  <->  E. x  e.  P  ( x  e.  (
( <" A B C "> `  0
) I ( <" A B C "> `  2
) )  /\  (
x  =  ( <" A B C "> `  1
)  \/  x ( K `  ( <" A B C "> `  1
) ) X ) ) ) )
199, 18anbi12d 725 . . . . 5  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( ( ( ( t `  0 )  =/=  ( t ` 
1 )  /\  (
t `  2 )  =/=  ( t `  1
)  /\  p  =/=  ( t `  1
) )  /\  E. x  e.  P  (
x  e.  ( ( t `  0 ) I ( t ` 
2 ) )  /\  ( x  =  (
t `  1 )  \/  x ( K `  ( t `  1
) ) p ) ) )  <->  ( (
( <" A B C "> `  0
)  =/=  ( <" A B C "> `  1
)  /\  ( <" A B C "> `  2 )  =/=  ( <" A B C "> `  1
)  /\  X  =/=  ( <" A B C "> `  1
) )  /\  E. x  e.  P  (
x  e.  ( (
<" A B C "> `  0
) I ( <" A B C "> `  2
) )  /\  (
x  =  ( <" A B C "> `  1
)  \/  x ( K `  ( <" A B C "> `  1
) ) X ) ) ) ) )
20 eqid 2471 . . . . 5  |-  { <. p ,  t >.  |  ( ( p  e.  P  /\  t  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  ( ( ( t `
 0 )  =/=  ( t `  1
)  /\  ( t `  2 )  =/=  ( t `  1
)  /\  p  =/=  ( t `  1
) )  /\  E. x  e.  P  (
x  e.  ( ( t `  0 ) I ( t ` 
2 ) )  /\  ( x  =  (
t `  1 )  \/  x ( K `  ( t `  1
) ) p ) ) ) ) }  =  { <. p ,  t >.  |  ( ( p  e.  P  /\  t  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  ( ( ( t `
 0 )  =/=  ( t `  1
)  /\  ( t `  2 )  =/=  ( t `  1
)  /\  p  =/=  ( t `  1
) )  /\  E. x  e.  P  (
x  e.  ( ( t `  0 ) I ( t ` 
2 ) )  /\  ( x  =  (
t `  1 )  \/  x ( K `  ( t `  1
) ) p ) ) ) ) }
2119, 20brab2a 4889 . . . 4  |-  ( X { <. p ,  t
>.  |  ( (
p  e.  P  /\  t  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  (
( ( t ` 
0 )  =/=  (
t `  1 )  /\  ( t `  2
)  =/=  ( t `
 1 )  /\  p  =/=  ( t ` 
1 ) )  /\  E. x  e.  P  ( x  e.  ( ( t `  0 ) I ( t ` 
2 ) )  /\  ( x  =  (
t `  1 )  \/  x ( K `  ( t `  1
) ) p ) ) ) ) }
<" A B C ">  <->  ( ( X  e.  P  /\  <" A B C ">  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  ( ( (
<" A B C "> `  0
)  =/=  ( <" A B C "> `  1
)  /\  ( <" A B C "> `  2 )  =/=  ( <" A B C "> `  1
)  /\  X  =/=  ( <" A B C "> `  1
) )  /\  E. x  e.  P  (
x  e.  ( (
<" A B C "> `  0
) I ( <" A B C "> `  2
) )  /\  (
x  =  ( <" A B C "> `  1
)  \/  x ( K `  ( <" A B C "> `  1
) ) X ) ) ) ) )
2221a1i 11 . . 3  |-  ( ph  ->  ( X { <. p ,  t >.  |  ( ( p  e.  P  /\  t  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  ( ( ( t `
 0 )  =/=  ( t `  1
)  /\  ( t `  2 )  =/=  ( t `  1
)  /\  p  =/=  ( t `  1
) )  /\  E. x  e.  P  (
x  e.  ( ( t `  0 ) I ( t ` 
2 ) )  /\  ( x  =  (
t `  1 )  \/  x ( K `  ( t `  1
) ) p ) ) ) ) }
<" A B C ">  <->  ( ( X  e.  P  /\  <" A B C ">  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  ( ( (
<" A B C "> `  0
)  =/=  ( <" A B C "> `  1
)  /\  ( <" A B C "> `  2 )  =/=  ( <" A B C "> `  1
)  /\  X  =/=  ( <" A B C "> `  1
) )  /\  E. x  e.  P  (
x  e.  ( (
<" A B C "> `  0
) I ( <" A B C "> `  2
) )  /\  (
x  =  ( <" A B C "> `  1
)  \/  x ( K `  ( <" A B C "> `  1
) ) X ) ) ) ) ) )
23 biidd 245 . . . 4  |-  ( ph  ->  ( ( X  e.  P  /\  <" A B C ">  e.  ( P  ^m  (
0..^ 3 ) ) )  <->  ( X  e.  P  /\  <" A B C ">  e.  ( P  ^m  (
0..^ 3 ) ) ) ) )
24 isinag.a . . . . . . . 8  |-  ( ph  ->  A  e.  P )
25 s3fv0 13045 . . . . . . . 8  |-  ( A  e.  P  ->  ( <" A B C "> `  0
)  =  A )
2624, 25syl 17 . . . . . . 7  |-  ( ph  ->  ( <" A B C "> `  0
)  =  A )
27 isinag.b . . . . . . . 8  |-  ( ph  ->  B  e.  P )
28 s3fv1 13046 . . . . . . . 8  |-  ( B  e.  P  ->  ( <" A B C "> `  1
)  =  B )
2927, 28syl 17 . . . . . . 7  |-  ( ph  ->  ( <" A B C "> `  1
)  =  B )
3026, 29neeq12d 2704 . . . . . 6  |-  ( ph  ->  ( ( <" A B C "> `  0
)  =/=  ( <" A B C "> `  1
)  <->  A  =/=  B
) )
31 isinag.c . . . . . . . 8  |-  ( ph  ->  C  e.  P )
32 s3fv2 13047 . . . . . . . 8  |-  ( C  e.  P  ->  ( <" A B C "> `  2
)  =  C )
3331, 32syl 17 . . . . . . 7  |-  ( ph  ->  ( <" A B C "> `  2
)  =  C )
3433, 29neeq12d 2704 . . . . . 6  |-  ( ph  ->  ( ( <" A B C "> `  2
)  =/=  ( <" A B C "> `  1
)  <->  C  =/=  B
) )
3529neeq2d 2703 . . . . . 6  |-  ( ph  ->  ( X  =/=  ( <" A B C "> `  1
)  <->  X  =/=  B
) )
3630, 34, 353anbi123d 1365 . . . . 5  |-  ( ph  ->  ( ( ( <" A B C "> `  0
)  =/=  ( <" A B C "> `  1
)  /\  ( <" A B C "> `  2 )  =/=  ( <" A B C "> `  1
)  /\  X  =/=  ( <" A B C "> `  1
) )  <->  ( A  =/=  B  /\  C  =/= 
B  /\  X  =/=  B ) ) )
3726, 33oveq12d 6326 . . . . . . . 8  |-  ( ph  ->  ( ( <" A B C "> `  0
) I ( <" A B C "> `  2
) )  =  ( A I C ) )
3837eleq2d 2534 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( ( <" A B C "> `  0
) I ( <" A B C "> `  2
) )  <->  x  e.  ( A I C ) ) )
3929eqeq2d 2481 . . . . . . . 8  |-  ( ph  ->  ( x  =  (
<" A B C "> `  1
)  <->  x  =  B
) )
4029fveq2d 5883 . . . . . . . . 9  |-  ( ph  ->  ( K `  ( <" A B C "> `  1
) )  =  ( K `  B ) )
4140breqd 4406 . . . . . . . 8  |-  ( ph  ->  ( x ( K `
 ( <" A B C "> `  1
) ) X  <->  x ( K `  B ) X ) )
4239, 41orbi12d 724 . . . . . . 7  |-  ( ph  ->  ( ( x  =  ( <" A B C "> `  1
)  \/  x ( K `  ( <" A B C "> `  1
) ) X )  <-> 
( x  =  B  \/  x ( K `
 B ) X ) ) )
4338, 42anbi12d 725 . . . . . 6  |-  ( ph  ->  ( ( x  e.  ( ( <" A B C "> `  0
) I ( <" A B C "> `  2
) )  /\  (
x  =  ( <" A B C "> `  1
)  \/  x ( K `  ( <" A B C "> `  1
) ) X ) )  <->  ( x  e.  ( A I C )  /\  ( x  =  B  \/  x
( K `  B
) X ) ) ) )
4443rexbidv 2892 . . . . 5  |-  ( ph  ->  ( E. x  e.  P  ( x  e.  ( ( <" A B C "> `  0
) I ( <" A B C "> `  2
) )  /\  (
x  =  ( <" A B C "> `  1
)  \/  x ( K `  ( <" A B C "> `  1
) ) X ) )  <->  E. x  e.  P  ( x  e.  ( A I C )  /\  ( x  =  B  \/  x ( K `  B ) X ) ) ) )
4536, 44anbi12d 725 . . . 4  |-  ( ph  ->  ( ( ( (
<" A B C "> `  0
)  =/=  ( <" A B C "> `  1
)  /\  ( <" A B C "> `  2 )  =/=  ( <" A B C "> `  1
)  /\  X  =/=  ( <" A B C "> `  1
) )  /\  E. x  e.  P  (
x  e.  ( (
<" A B C "> `  0
) I ( <" A B C "> `  2
) )  /\  (
x  =  ( <" A B C "> `  1
)  \/  x ( K `  ( <" A B C "> `  1
) ) X ) ) )  <->  ( ( A  =/=  B  /\  C  =/=  B  /\  X  =/= 
B )  /\  E. x  e.  P  (
x  e.  ( A I C )  /\  ( x  =  B  \/  x ( K `  B ) X ) ) ) ) )
4623, 45anbi12d 725 . . 3  |-  ( ph  ->  ( ( ( X  e.  P  /\  <" A B C ">  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  ( ( ( <" A B C "> `  0
)  =/=  ( <" A B C "> `  1
)  /\  ( <" A B C "> `  2 )  =/=  ( <" A B C "> `  1
)  /\  X  =/=  ( <" A B C "> `  1
) )  /\  E. x  e.  P  (
x  e.  ( (
<" A B C "> `  0
) I ( <" A B C "> `  2
) )  /\  (
x  =  ( <" A B C "> `  1
)  \/  x ( K `  ( <" A B C "> `  1
) ) X ) ) ) )  <->  ( ( X  e.  P  /\  <" A B C ">  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  ( ( A  =/=  B  /\  C  =/=  B  /\  X  =/= 
B )  /\  E. x  e.  P  (
x  e.  ( A I C )  /\  ( x  =  B  \/  x ( K `  B ) X ) ) ) ) ) )
4722, 46bitrd 261 . 2  |-  ( ph  ->  ( X { <. p ,  t >.  |  ( ( p  e.  P  /\  t  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  ( ( ( t `
 0 )  =/=  ( t `  1
)  /\  ( t `  2 )  =/=  ( t `  1
)  /\  p  =/=  ( t `  1
) )  /\  E. x  e.  P  (
x  e.  ( ( t `  0 ) I ( t ` 
2 ) )  /\  ( x  =  (
t `  1 )  \/  x ( K `  ( t `  1
) ) p ) ) ) ) }
<" A B C ">  <->  ( ( X  e.  P  /\  <" A B C ">  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  ( ( A  =/=  B  /\  C  =/=  B  /\  X  =/= 
B )  /\  E. x  e.  P  (
x  e.  ( A I C )  /\  ( x  =  B  \/  x ( K `  B ) X ) ) ) ) ) )
48 isinag.g . . . 4  |-  ( ph  ->  G  e.  V )
49 elex 3040 . . . 4  |-  ( G  e.  V  ->  G  e.  _V )
50 fveq2 5879 . . . . . . . . . 10  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
51 isinag.p . . . . . . . . . 10  |-  P  =  ( Base `  G
)
5250, 51syl6eqr 2523 . . . . . . . . 9  |-  ( g  =  G  ->  ( Base `  g )  =  P )
5352eleq2d 2534 . . . . . . . 8  |-  ( g  =  G  ->  (
p  e.  ( Base `  g )  <->  p  e.  P ) )
5452oveq1d 6323 . . . . . . . . 9  |-  ( g  =  G  ->  (
( Base `  g )  ^m  ( 0..^ 3 ) )  =  ( P  ^m  ( 0..^ 3 ) ) )
5554eleq2d 2534 . . . . . . . 8  |-  ( g  =  G  ->  (
t  e.  ( (
Base `  g )  ^m  ( 0..^ 3 ) )  <->  t  e.  ( P  ^m  ( 0..^ 3 ) ) ) )
5653, 55anbi12d 725 . . . . . . 7  |-  ( g  =  G  ->  (
( p  e.  (
Base `  g )  /\  t  e.  (
( Base `  g )  ^m  ( 0..^ 3 ) ) )  <->  ( p  e.  P  /\  t  e.  ( P  ^m  (
0..^ 3 ) ) ) ) )
57 fveq2 5879 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  (Itv `  g )  =  (Itv
`  G ) )
58 isinag.i . . . . . . . . . . . . 13  |-  I  =  (Itv `  G )
5957, 58syl6eqr 2523 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (Itv `  g )  =  I )
6059oveqd 6325 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
( t `  0
) (Itv `  g
) ( t ` 
2 ) )  =  ( ( t ` 
0 ) I ( t `  2 ) ) )
6160eleq2d 2534 . . . . . . . . . 10  |-  ( g  =  G  ->  (
x  e.  ( ( t `  0 ) (Itv `  g )
( t `  2
) )  <->  x  e.  ( ( t ` 
0 ) I ( t `  2 ) ) ) )
62 fveq2 5879 . . . . . . . . . . . . . 14  |-  ( g  =  G  ->  (hlG `  g )  =  (hlG
`  G ) )
63 isinag.k . . . . . . . . . . . . . 14  |-  K  =  (hlG `  G )
6462, 63syl6eqr 2523 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  (hlG `  g )  =  K )
6564fveq1d 5881 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (
(hlG `  g ) `  ( t `  1
) )  =  ( K `  ( t `
 1 ) ) )
6665breqd 4406 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
x ( (hlG `  g ) `  (
t `  1 )
) p  <->  x ( K `  ( t `  1 ) ) p ) )
6766orbi2d 716 . . . . . . . . . 10  |-  ( g  =  G  ->  (
( x  =  ( t `  1 )  \/  x ( (hlG
`  g ) `  ( t `  1
) ) p )  <-> 
( x  =  ( t `  1 )  \/  x ( K `
 ( t ` 
1 ) ) p ) ) )
6861, 67anbi12d 725 . . . . . . . . 9  |-  ( g  =  G  ->  (
( x  e.  ( ( t `  0
) (Itv `  g
) ( t ` 
2 ) )  /\  ( x  =  (
t `  1 )  \/  x ( (hlG `  g ) `  (
t `  1 )
) p ) )  <-> 
( x  e.  ( ( t `  0
) I ( t `
 2 ) )  /\  ( x  =  ( t `  1
)  \/  x ( K `  ( t `
 1 ) ) p ) ) ) )
6952, 68rexeqbidv 2988 . . . . . . . 8  |-  ( g  =  G  ->  ( E. x  e.  ( Base `  g ) ( x  e.  ( ( t `  0 ) (Itv `  g )
( t `  2
) )  /\  (
x  =  ( t `
 1 )  \/  x ( (hlG `  g ) `  (
t `  1 )
) p ) )  <->  E. x  e.  P  ( x  e.  (
( t `  0
) I ( t `
 2 ) )  /\  ( x  =  ( t `  1
)  \/  x ( K `  ( t `
 1 ) ) p ) ) ) )
7069anbi2d 718 . . . . . . 7  |-  ( g  =  G  ->  (
( ( ( t `
 0 )  =/=  ( t `  1
)  /\  ( t `  2 )  =/=  ( t `  1
)  /\  p  =/=  ( t `  1
) )  /\  E. x  e.  ( Base `  g ) ( x  e.  ( ( t `
 0 ) (Itv
`  g ) ( t `  2 ) )  /\  ( x  =  ( t ` 
1 )  \/  x
( (hlG `  g
) `  ( t `  1 ) ) p ) ) )  <-> 
( ( ( t `
 0 )  =/=  ( t `  1
)  /\  ( t `  2 )  =/=  ( t `  1
)  /\  p  =/=  ( t `  1
) )  /\  E. x  e.  P  (
x  e.  ( ( t `  0 ) I ( t ` 
2 ) )  /\  ( x  =  (
t `  1 )  \/  x ( K `  ( t `  1
) ) p ) ) ) ) )
7156, 70anbi12d 725 . . . . . 6  |-  ( g  =  G  ->  (
( ( p  e.  ( Base `  g
)  /\  t  e.  ( ( Base `  g
)  ^m  ( 0..^ 3 ) ) )  /\  ( ( ( t `  0 )  =/=  ( t ` 
1 )  /\  (
t `  2 )  =/=  ( t `  1
)  /\  p  =/=  ( t `  1
) )  /\  E. x  e.  ( Base `  g ) ( x  e.  ( ( t `
 0 ) (Itv
`  g ) ( t `  2 ) )  /\  ( x  =  ( t ` 
1 )  \/  x
( (hlG `  g
) `  ( t `  1 ) ) p ) ) ) )  <->  ( ( p  e.  P  /\  t  e.  ( P  ^m  (
0..^ 3 ) ) )  /\  ( ( ( t `  0
)  =/=  ( t `
 1 )  /\  ( t `  2
)  =/=  ( t `
 1 )  /\  p  =/=  ( t ` 
1 ) )  /\  E. x  e.  P  ( x  e.  ( ( t `  0 ) I ( t ` 
2 ) )  /\  ( x  =  (
t `  1 )  \/  x ( K `  ( t `  1
) ) p ) ) ) ) ) )
7271opabbidv 4459 . . . . 5  |-  ( g  =  G  ->  { <. p ,  t >.  |  ( ( p  e.  (
Base `  g )  /\  t  e.  (
( Base `  g )  ^m  ( 0..^ 3 ) ) )  /\  (
( ( t ` 
0 )  =/=  (
t `  1 )  /\  ( t `  2
)  =/=  ( t `
 1 )  /\  p  =/=  ( t ` 
1 ) )  /\  E. x  e.  ( Base `  g ) ( x  e.  ( ( t `
 0 ) (Itv
`  g ) ( t `  2 ) )  /\  ( x  =  ( t ` 
1 )  \/  x
( (hlG `  g
) `  ( t `  1 ) ) p ) ) ) ) }  =  { <. p ,  t >.  |  ( ( p  e.  P  /\  t  e.  ( P  ^m  (
0..^ 3 ) ) )  /\  ( ( ( t `  0
)  =/=  ( t `
 1 )  /\  ( t `  2
)  =/=  ( t `
 1 )  /\  p  =/=  ( t ` 
1 ) )  /\  E. x  e.  P  ( x  e.  ( ( t `  0 ) I ( t ` 
2 ) )  /\  ( x  =  (
t `  1 )  \/  x ( K `  ( t `  1
) ) p ) ) ) ) } )
73 df-inag 24957 . . . . 5  |- inA  =  ( g  e.  _V  |->  {
<. p ,  t >.  |  ( ( p  e.  ( Base `  g
)  /\  t  e.  ( ( Base `  g
)  ^m  ( 0..^ 3 ) ) )  /\  ( ( ( t `  0 )  =/=  ( t ` 
1 )  /\  (
t `  2 )  =/=  ( t `  1
)  /\  p  =/=  ( t `  1
) )  /\  E. x  e.  ( Base `  g ) ( x  e.  ( ( t `
 0 ) (Itv
`  g ) ( t `  2 ) )  /\  ( x  =  ( t ` 
1 )  \/  x
( (hlG `  g
) `  ( t `  1 ) ) p ) ) ) ) } )
74 fvex 5889 . . . . . . . 8  |-  ( Base `  G )  e.  _V
7551, 74eqeltri 2545 . . . . . . 7  |-  P  e. 
_V
76 ovex 6336 . . . . . . 7  |-  ( P  ^m  ( 0..^ 3 ) )  e.  _V
7775, 76xpex 6614 . . . . . 6  |-  ( P  X.  ( P  ^m  ( 0..^ 3 ) ) )  e.  _V
78 opabssxp 4914 . . . . . 6  |-  { <. p ,  t >.  |  ( ( p  e.  P  /\  t  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  ( ( ( t `
 0 )  =/=  ( t `  1
)  /\  ( t `  2 )  =/=  ( t `  1
)  /\  p  =/=  ( t `  1
) )  /\  E. x  e.  P  (
x  e.  ( ( t `  0 ) I ( t ` 
2 ) )  /\  ( x  =  (
t `  1 )  \/  x ( K `  ( t `  1
) ) p ) ) ) ) } 
C_  ( P  X.  ( P  ^m  (
0..^ 3 ) ) )
7977, 78ssexi 4541 . . . . 5  |-  { <. p ,  t >.  |  ( ( p  e.  P  /\  t  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  ( ( ( t `
 0 )  =/=  ( t `  1
)  /\  ( t `  2 )  =/=  ( t `  1
)  /\  p  =/=  ( t `  1
) )  /\  E. x  e.  P  (
x  e.  ( ( t `  0 ) I ( t ` 
2 ) )  /\  ( x  =  (
t `  1 )  \/  x ( K `  ( t `  1
) ) p ) ) ) ) }  e.  _V
8072, 73, 79fvmpt 5963 . . . 4  |-  ( G  e.  _V  ->  (inA `  G )  =  { <. p ,  t >.  |  ( ( p  e.  P  /\  t  e.  ( P  ^m  (
0..^ 3 ) ) )  /\  ( ( ( t `  0
)  =/=  ( t `
 1 )  /\  ( t `  2
)  =/=  ( t `
 1 )  /\  p  =/=  ( t ` 
1 ) )  /\  E. x  e.  P  ( x  e.  ( ( t `  0 ) I ( t ` 
2 ) )  /\  ( x  =  (
t `  1 )  \/  x ( K `  ( t `  1
) ) p ) ) ) ) } )
8148, 49, 803syl 18 . . 3  |-  ( ph  ->  (inA `  G )  =  { <. p ,  t
>.  |  ( (
p  e.  P  /\  t  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  (
( ( t ` 
0 )  =/=  (
t `  1 )  /\  ( t `  2
)  =/=  ( t `
 1 )  /\  p  =/=  ( t ` 
1 ) )  /\  E. x  e.  P  ( x  e.  ( ( t `  0 ) I ( t ` 
2 ) )  /\  ( x  =  (
t `  1 )  \/  x ( K `  ( t `  1
) ) p ) ) ) ) } )
8281breqd 4406 . 2  |-  ( ph  ->  ( X (inA `  G ) <" A B C ">  <->  X { <. p ,  t >.  |  ( ( p  e.  P  /\  t  e.  ( P  ^m  (
0..^ 3 ) ) )  /\  ( ( ( t `  0
)  =/=  ( t `
 1 )  /\  ( t `  2
)  =/=  ( t `
 1 )  /\  p  =/=  ( t ` 
1 ) )  /\  E. x  e.  P  ( x  e.  ( ( t `  0 ) I ( t ` 
2 ) )  /\  ( x  =  (
t `  1 )  \/  x ( K `  ( t `  1
) ) p ) ) ) ) }
<" A B C "> ) )
83 isinag.x . . . 4  |-  ( ph  ->  X  e.  P )
8424, 27, 31s3cld 13026 . . . . . 6  |-  ( ph  ->  <" A B C ">  e. Word  P )
85 s3len 13048 . . . . . . 7  |-  ( # `  <" A B C "> )  =  3
8685a1i 11 . . . . . 6  |-  ( ph  ->  ( # `  <" A B C "> )  =  3
)
8784, 86jca 541 . . . . 5  |-  ( ph  ->  ( <" A B C ">  e. Word  P  /\  ( # `  <" A B C "> )  =  3
) )
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 ) ) ) )
9075, 88, 89mp2an 686 . . . . 5  |-  ( (
<" A B C ">  e. Word  P  /\  ( # `  <" A B C "> )  =  3
)  <->  <" A B C ">  e.  ( P  ^m  (
0..^ 3 ) ) )
9187, 90sylib 201 . . . 4  |-  ( ph  ->  <" A B C ">  e.  ( P  ^m  (
0..^ 3 ) ) )
9283, 91jca 541 . . 3  |-  ( ph  ->  ( X  e.  P  /\  <" A B C ">  e.  ( P  ^m  (
0..^ 3 ) ) ) )
9392biantrurd 516 . 2  |-  ( ph  ->  ( ( ( A  =/=  B  /\  C  =/=  B  /\  X  =/= 
B )  /\  E. x  e.  P  (
x  e.  ( A I C )  /\  ( x  =  B  \/  x ( K `  B ) X ) ) )  <->  ( ( X  e.  P  /\  <" A B C ">  e.  ( P  ^m  ( 0..^ 3 ) ) )  /\  ( ( A  =/=  B  /\  C  =/=  B  /\  X  =/= 
B )  /\  E. x  e.  P  (
x  e.  ( A I C )  /\  ( x  =  B  \/  x ( K `  B ) X ) ) ) ) ) )
9447, 82, 933bitr4d 293 1  |-  ( ph  ->  ( X (inA `  G ) <" A B C ">  <->  ( ( A  =/=  B  /\  C  =/=  B  /\  X  =/= 
B )  /\  E. x  e.  P  (
x  e.  ( A I C )  /\  ( x  =  B  \/  x ( K `  B ) X ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757   _Vcvv 3031   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  Itvcitv 24563  hlGchlg 24724  inAcinag 24955
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-inag 24957
This theorem is referenced by:  inagswap  24959  inaghl  24960
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