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Theorem isinag 24879
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 463 . . . . . . . . 9  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  t  =  <" A B C "> )
21fveq1d 5867 . . . . . . . 8  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( t `  0
)  =  ( <" A B C "> `  0
) )
31fveq1d 5867 . . . . . . . 8  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( t `  1
)  =  ( <" A B C "> `  1
) )
42, 3neeq12d 2685 . . . . . . 7  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( ( t ` 
0 )  =/=  (
t `  1 )  <->  (
<" A B C "> `  0
)  =/=  ( <" A B C "> `  1
) ) )
51fveq1d 5867 . . . . . . . 8  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( t `  2
)  =  ( <" A B C "> `  2
) )
65, 3neeq12d 2685 . . . . . . 7  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( ( t ` 
2 )  =/=  (
t `  1 )  <->  (
<" A B C "> `  2
)  =/=  ( <" A B C "> `  1
) ) )
7 simpl 459 . . . . . . . 8  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  p  =  X )
87, 3neeq12d 2685 . . . . . . 7  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( p  =/=  (
t `  1 )  <->  X  =/=  ( <" A B C "> `  1
) ) )
94, 6, 83anbi123d 1339 . . . . . 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 2452 . . . . . . . . 9  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  x  =  x )
112, 5oveq12d 6308 . . . . . . . . 9  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( ( t ` 
0 ) I ( t `  2 ) )  =  ( (
<" A B C "> `  0
) I ( <" A B C "> `  2
) ) )
1210, 11eleq12d 2523 . . . . . . . 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 2466 . . . . . . . . 9  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( x  =  ( t `  1 )  <-> 
x  =  ( <" A B C "> `  1
) ) )
143fveq2d 5869 . . . . . . . . . 10  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( K `  (
t `  1 )
)  =  ( K `
 ( <" A B C "> `  1
) ) )
1510, 14, 7breq123d 4416 . . . . . . . . 9  |-  ( ( p  =  X  /\  t  =  <" A B C "> )  ->  ( x ( K `
 ( t ` 
1 ) ) p  <-> 
x ( K `  ( <" A B C "> `  1
) ) X ) )
1613, 15orbi12d 716 . . . . . . . 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 717 . . . . . . 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 2901 . . . . . 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 717 . . . . 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 2451 . . . . 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 4884 . . . 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 241 . . . 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 12985 . . . . . . . 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 12986 . . . . . . . 8  |-  ( B  e.  P  ->  ( <" A B C "> `  1
)  =  B )
2927, 28syl 17 . . . . . . 7  |-  ( ph  ->  ( <" A B C "> `  1
)  =  B )
3026, 29neeq12d 2685 . . . . . 6  |-  ( ph  ->  ( ( <" A B C "> `  0
)  =/=  ( <" A B C "> `  1
)  <->  A  =/=  B
) )
31 isinag.c . . . . . . . 8  |-  ( ph  ->  C  e.  P )
32 s3fv2 12987 . . . . . . . 8  |-  ( C  e.  P  ->  ( <" A B C "> `  2
)  =  C )
3331, 32syl 17 . . . . . . 7  |-  ( ph  ->  ( <" A B C "> `  2
)  =  C )
3433, 29neeq12d 2685 . . . . . 6  |-  ( ph  ->  ( ( <" A B C "> `  2
)  =/=  ( <" A B C "> `  1
)  <->  C  =/=  B
) )
3529neeq2d 2684 . . . . . 6  |-  ( ph  ->  ( X  =/=  ( <" A B C "> `  1
)  <->  X  =/=  B
) )
3630, 34, 353anbi123d 1339 . . . . 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 6308 . . . . . . . 8  |-  ( ph  ->  ( ( <" A B C "> `  0
) I ( <" A B C "> `  2
) )  =  ( A I C ) )
3837eleq2d 2514 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( ( <" A B C "> `  0
) I ( <" A B C "> `  2
) )  <->  x  e.  ( A I C ) ) )
3929eqeq2d 2461 . . . . . . . 8  |-  ( ph  ->  ( x  =  (
<" A B C "> `  1
)  <->  x  =  B
) )
4029fveq2d 5869 . . . . . . . . 9  |-  ( ph  ->  ( K `  ( <" A B C "> `  1
) )  =  ( K `  B ) )
4140breqd 4413 . . . . . . . 8  |-  ( ph  ->  ( x ( K `
 ( <" A B C "> `  1
) ) X  <->  x ( K `  B ) X ) )
4239, 41orbi12d 716 . . . . . . 7  |-  ( ph  ->  ( ( x  =  ( <" A B C "> `  1
)  \/  x ( K `  ( <" A B C "> `  1
) ) X )  <-> 
( x  =  B  \/  x ( K `
 B ) X ) ) )
4338, 42anbi12d 717 . . . . . 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 2901 . . . . 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 717 . . . 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 717 . . 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 257 . 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 3054 . . . 4  |-  ( G  e.  V  ->  G  e.  _V )
50 fveq2 5865 . . . . . . . . . 10  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
51 isinag.p . . . . . . . . . 10  |-  P  =  ( Base `  G
)
5250, 51syl6eqr 2503 . . . . . . . . 9  |-  ( g  =  G  ->  ( Base `  g )  =  P )
5352eleq2d 2514 . . . . . . . 8  |-  ( g  =  G  ->  (
p  e.  ( Base `  g )  <->  p  e.  P ) )
5452oveq1d 6305 . . . . . . . . 9  |-  ( g  =  G  ->  (
( Base `  g )  ^m  ( 0..^ 3 ) )  =  ( P  ^m  ( 0..^ 3 ) ) )
5554eleq2d 2514 . . . . . . . 8  |-  ( g  =  G  ->  (
t  e.  ( (
Base `  g )  ^m  ( 0..^ 3 ) )  <->  t  e.  ( P  ^m  ( 0..^ 3 ) ) ) )
5653, 55anbi12d 717 . . . . . . 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 5865 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  (Itv `  g )  =  (Itv
`  G ) )
58 isinag.i . . . . . . . . . . . . 13  |-  I  =  (Itv `  G )
5957, 58syl6eqr 2503 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (Itv `  g )  =  I )
6059oveqd 6307 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
( t `  0
) (Itv `  g
) ( t ` 
2 ) )  =  ( ( t ` 
0 ) I ( t `  2 ) ) )
6160eleq2d 2514 . . . . . . . . . 10  |-  ( g  =  G  ->  (
x  e.  ( ( t `  0 ) (Itv `  g )
( t `  2
) )  <->  x  e.  ( ( t ` 
0 ) I ( t `  2 ) ) ) )
62 fveq2 5865 . . . . . . . . . . . . . 14  |-  ( g  =  G  ->  (hlG `  g )  =  (hlG
`  G ) )
63 isinag.k . . . . . . . . . . . . . 14  |-  K  =  (hlG `  G )
6462, 63syl6eqr 2503 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  (hlG `  g )  =  K )
6564fveq1d 5867 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (
(hlG `  g ) `  ( t `  1
) )  =  ( K `  ( t `
 1 ) ) )
6665breqd 4413 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
x ( (hlG `  g ) `  (
t `  1 )
) p  <->  x ( K `  ( t `  1 ) ) p ) )
6766orbi2d 708 . . . . . . . . . 10  |-  ( g  =  G  ->  (
( x  =  ( t `  1 )  \/  x ( (hlG
`  g ) `  ( t `  1
) ) p )  <-> 
( x  =  ( t `  1 )  \/  x ( K `
 ( t ` 
1 ) ) p ) ) )
6861, 67anbi12d 717 . . . . . . . . 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 3002 . . . . . . . 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 710 . . . . . . 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 717 . . . . . 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 4466 . . . . 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 24878 . . . . 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 5875 . . . . . . . 8  |-  ( Base `  G )  e.  _V
7551, 74eqeltri 2525 . . . . . . 7  |-  P  e. 
_V
76 ovex 6318 . . . . . . 7  |-  ( P  ^m  ( 0..^ 3 ) )  e.  _V
7775, 76xpex 6595 . . . . . 6  |-  ( P  X.  ( P  ^m  ( 0..^ 3 ) ) )  e.  _V
78 opabssxp 4909 . . . . . 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 4548 . . . . 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 5948 . . . 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 4413 . 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 12966 . . . . . 6  |-  ( ph  ->  <" A B C ">  e. Word  P )
85 s3len 12988 . . . . . . 7  |-  ( # `  <" A B C "> )  =  3
8685a1i 11 . . . . . 6  |-  ( ph  ->  ( # `  <" A B C "> )  =  3
)
8784, 86jca 535 . . . . 5  |-  ( ph  ->  ( <" A B C ">  e. Word  P  /\  ( # `  <" A B C "> )  =  3
) )
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 ) ) ) )
9075, 88, 89mp2an 678 . . . . 5  |-  ( (
<" A B C ">  e. Word  P  /\  ( # `  <" A B C "> )  =  3
)  <->  <" A B C ">  e.  ( P  ^m  (
0..^ 3 ) ) )
9187, 90sylib 200 . . . 4  |-  ( ph  ->  <" A B C ">  e.  ( P  ^m  (
0..^ 3 ) ) )
9283, 91jca 535 . . 3  |-  ( ph  ->  ( X  e.  P  /\  <" A B C ">  e.  ( P  ^m  (
0..^ 3 ) ) ) )
9392biantrurd 511 . 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 289 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 188    \/ wo 370    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   E.wrex 2738   _Vcvv 3045   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  Itvcitv 24484  hlGchlg 24645  inAcinag 24876
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-inag 24878
This theorem is referenced by:  inagswap  24880  inaghl  24881
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