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Theorem usgra2wlkspthlem1 30308
Description: Lemma 1 for usgra2wlkspth 30310. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
Assertion
Ref Expression
usgra2wlkspthlem1  |-  ( ( F  e. Word  dom  E  /\  E : dom  E -1-1-> ran 
E  /\  ( # `  F
)  =  2 )  ->  ( ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/= 
B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )  ->  Fun  `' F ) )
Distinct variable groups:    i, E    i, F    P, i
Allowed substitution hints:    A( i)    B( i)

Proof of Theorem usgra2wlkspthlem1
StepHypRef Expression
1 wrdf 12252 . . 3  |-  ( F  e. Word  dom  E  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
2 simpl 457 . . . . . . . . . . . . 13  |-  ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E ) )  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
32adantr 465 . . . . . . . . . . . 12  |-  ( ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E ) )  /\  ( ( ( P `
 0 )  =  A  /\  ( P `
 ( # `  F
) )  =  B  /\  A  =/=  B
)  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } ) )  ->  F : ( 0..^ ( # `  F
) ) --> dom  E
)
4 oveq2 6111 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  ( 0..^ 2 ) )
5 fzo0to2pr 11626 . . . . . . . . . . . . . . . . 17  |-  ( 0..^ 2 )  =  {
0 ,  1 }
64, 5syl6req 2492 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  =  2  ->  { 0 ,  1 }  =  ( 0..^ ( # `  F
) ) )
76adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E )  ->  { 0 ,  1 }  =  ( 0..^ ( # `  F
) ) )
87adantl 466 . . . . . . . . . . . . . 14  |-  ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E ) )  ->  { 0 ,  1 }  =  ( 0..^ ( # `  F
) ) )
98adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E ) )  /\  ( ( ( P `
 0 )  =  A  /\  ( P `
 ( # `  F
) )  =  B  /\  A  =/=  B
)  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } ) )  ->  { 0 ,  1 }  =  ( 0..^ ( # `  F
) ) )
109feq2d 5559 . . . . . . . . . . . 12  |-  ( ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E ) )  /\  ( ( ( P `
 0 )  =  A  /\  ( P `
 ( # `  F
) )  =  B  /\  A  =/=  B
)  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } ) )  ->  ( F : { 0 ,  1 } --> dom  E  <->  F :
( 0..^ ( # `  F ) ) --> dom 
E ) )
113, 10mpbird 232 . . . . . . . . . . 11  |-  ( ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E ) )  /\  ( ( ( P `
 0 )  =  A  /\  ( P `
 ( # `  F
) )  =  B  /\  A  =/=  B
)  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } ) )  ->  F : {
0 ,  1 } --> dom  E )
12 preq1 3966 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( P `  0 )  =  A  ->  { ( P `  0 ) ,  ( P ` 
1 ) }  =  { A ,  ( P `
 1 ) } )
1312eqeq2d 2454 . . . . . . . . . . . . . . . . . . 19  |-  ( ( P `  0 )  =  A  ->  (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  <->  ( E `  ( F `  0
) )  =  { A ,  ( P `  1 ) } ) )
14 preq2 3967 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( P `  2 )  =  B  ->  { ( P `  1 ) ,  ( P ` 
2 ) }  =  { ( P ` 
1 ) ,  B } )
1514eqeq2d 2454 . . . . . . . . . . . . . . . . . . 19  |-  ( ( P `  2 )  =  B  ->  (
( E `  ( F `  1 )
)  =  { ( P `  1 ) ,  ( P ` 
2 ) }  <->  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  B }
) )
1613, 15bi2anan9 868 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( P `  0
)  =  A  /\  ( P `  2 )  =  B )  -> 
( ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) } )  <->  ( ( E `  ( F `  0 ) )  =  { A , 
( P `  1
) }  /\  ( E `  ( F `  1 ) )  =  { ( P `
 1 ) ,  B } ) ) )
17163adant3 1008 . . . . . . . . . . . . . . . . 17  |-  ( ( ( P `  0
)  =  A  /\  ( P `  2 )  =  B  /\  A  =/=  B )  ->  (
( ( E `  ( F `  0 ) )  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } )  <->  ( ( E `  ( F `  0 ) )  =  { A , 
( P `  1
) }  /\  ( E `  ( F `  1 ) )  =  { ( P `
 1 ) ,  B } ) ) )
18 fveq2 5703 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( F `  0 )  =  ( F ` 
1 )  ->  ( E `  ( F `  0 ) )  =  ( E `  ( F `  1 ) ) )
1918eqeq1d 2451 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( F `  0 )  =  ( F ` 
1 )  ->  (
( E `  ( F `  0 )
)  =  { A ,  ( P ` 
1 ) }  <->  ( E `  ( F `  1
) )  =  { A ,  ( P `  1 ) } ) )
2019anbi1d 704 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( F `  0 )  =  ( F ` 
1 )  ->  (
( ( E `  ( F `  0 ) )  =  { A ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  B } )  <->  ( ( E `  ( F `  1 ) )  =  { A , 
( P `  1
) }  /\  ( E `  ( F `  1 ) )  =  { ( P `
 1 ) ,  B } ) ) )
2120anbi2d 703 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( F `  0 )  =  ( F ` 
1 )  ->  (
( ( ( ( P `  0 )  =  A  /\  ( P `  2 )  =  B )  /\  (
( # `  F )  =  2  /\  E : dom  E -1-1-> ran  E
) )  /\  (
( E `  ( F `  0 )
)  =  { A ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  B } ) )  <-> 
( ( ( ( P `  0 )  =  A  /\  ( P `  2 )  =  B )  /\  (
( # `  F )  =  2  /\  E : dom  E -1-1-> ran  E
) )  /\  (
( E `  ( F `  1 )
)  =  { A ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  B } ) ) ) )
22 eqtr2 2461 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( E `  ( F `  1 )
)  =  { A ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  B } )  ->  { A ,  ( P `
 1 ) }  =  { ( P `
 1 ) ,  B } )
23 fvex 5713 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( P `
 0 )  e. 
_V
2423a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( P `  0 )  =  A  ->  ( P `  0 )  e.  _V )
25 eleq1 2503 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( A  =  ( P ` 
0 )  ->  ( A  e.  _V  <->  ( P `  0 )  e. 
_V ) )
2625eqcoms 2446 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( P `  0 )  =  A  ->  ( A  e.  _V  <->  ( P `  0 )  e. 
_V ) )
2724, 26mpbird 232 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( P `  0 )  =  A  ->  A  e.  _V )
28 fvex 5713 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( P `
 1 )  e. 
_V
2928a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( P `  0 )  =  A  ->  ( P `  1 )  e.  _V )
3027, 29jca 532 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( P `  0 )  =  A  ->  ( A  e.  _V  /\  ( P `  1 )  e.  _V ) )
3128a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( P `  2 )  =  B  ->  ( P `  1 )  e.  _V )
32 fvex 5713 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( P `
 2 )  e. 
_V
3332a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( P `  2 )  =  B  ->  ( P `  2 )  e.  _V )
34 eleq1 2503 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( B  =  ( P ` 
2 )  ->  ( B  e.  _V  <->  ( P `  2 )  e. 
_V ) )
3534eqcoms 2446 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( P `  2 )  =  B  ->  ( B  e.  _V  <->  ( P `  2 )  e. 
_V ) )
3633, 35mpbird 232 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( P `  2 )  =  B  ->  B  e.  _V )
3731, 36jca 532 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( P `  2 )  =  B  ->  (
( P `  1
)  e.  _V  /\  B  e.  _V )
)
3830, 37anim12i 566 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( P `  0
)  =  A  /\  ( P `  2 )  =  B )  -> 
( ( A  e. 
_V  /\  ( P `  1 )  e. 
_V )  /\  (
( P `  1
)  e.  _V  /\  B  e.  _V )
) )
3938adantr 465 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
2 )  =  B )  /\  ( (
# `  F )  =  2  /\  E : dom  E -1-1-> ran  E
) )  ->  (
( A  e.  _V  /\  ( P `  1
)  e.  _V )  /\  ( ( P ` 
1 )  e.  _V  /\  B  e.  _V )
) )
40 preq12bg 4063 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( A  e.  _V  /\  ( P `  1
)  e.  _V )  /\  ( ( P ` 
1 )  e.  _V  /\  B  e.  _V )
)  ->  ( { A ,  ( P `  1 ) }  =  { ( P `
 1 ) ,  B }  <->  ( ( A  =  ( P `  1 )  /\  ( P `  1 )  =  B )  \/  ( A  =  B  /\  ( P ` 
1 )  =  ( P `  1 ) ) ) ) )
4139, 40syl 16 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
2 )  =  B )  /\  ( (
# `  F )  =  2  /\  E : dom  E -1-1-> ran  E
) )  ->  ( { A ,  ( P `
 1 ) }  =  { ( P `
 1 ) ,  B }  <->  ( ( A  =  ( P `  1 )  /\  ( P `  1 )  =  B )  \/  ( A  =  B  /\  ( P ` 
1 )  =  ( P `  1 ) ) ) ) )
42 eqtr 2460 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( A  =  ( P `
 1 )  /\  ( P `  1 )  =  B )  ->  A  =  B )
43 simpl 457 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( A  =  B  /\  ( P `  1 )  =  ( P ` 
1 ) )  ->  A  =  B )
4442, 43jaoi 379 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( A  =  ( P `  1 )  /\  ( P ` 
1 )  =  B )  \/  ( A  =  B  /\  ( P `  1 )  =  ( P ` 
1 ) ) )  ->  A  =  B )
4541, 44syl6bi 228 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
2 )  =  B )  /\  ( (
# `  F )  =  2  /\  E : dom  E -1-1-> ran  E
) )  ->  ( { A ,  ( P `
 1 ) }  =  { ( P `
 1 ) ,  B }  ->  A  =  B ) )
4622, 45syl5com 30 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( E `  ( F `  1 )
)  =  { A ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  B } )  -> 
( ( ( ( P `  0 )  =  A  /\  ( P `  2 )  =  B )  /\  (
( # `  F )  =  2  /\  E : dom  E -1-1-> ran  E
) )  ->  A  =  B ) )
4746impcom 430 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( ( P `
 0 )  =  A  /\  ( P `
 2 )  =  B )  /\  (
( # `  F )  =  2  /\  E : dom  E -1-1-> ran  E
) )  /\  (
( E `  ( F `  1 )
)  =  { A ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  B } ) )  ->  A  =  B )
4821, 47syl6bi 228 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( F `  0 )  =  ( F ` 
1 )  ->  (
( ( ( ( P `  0 )  =  A  /\  ( P `  2 )  =  B )  /\  (
( # `  F )  =  2  /\  E : dom  E -1-1-> ran  E
) )  /\  (
( E `  ( F `  0 )
)  =  { A ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  B } ) )  ->  A  =  B ) )
4948com12 31 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( ( P `
 0 )  =  A  /\  ( P `
 2 )  =  B )  /\  (
( # `  F )  =  2  /\  E : dom  E -1-1-> ran  E
) )  /\  (
( E `  ( F `  0 )
)  =  { A ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  B } ) )  ->  ( ( F `
 0 )  =  ( F `  1
)  ->  A  =  B ) )
5049necon3d 2658 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( P `
 0 )  =  A  /\  ( P `
 2 )  =  B )  /\  (
( # `  F )  =  2  /\  E : dom  E -1-1-> ran  E
) )  /\  (
( E `  ( F `  0 )
)  =  { A ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  B } ) )  ->  ( A  =/= 
B  ->  ( F `  0 )  =/=  ( F `  1
) ) )
5150a1d 25 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( P `
 0 )  =  A  /\  ( P `
 2 )  =  B )  /\  (
( # `  F )  =  2  /\  E : dom  E -1-1-> ran  E
) )  /\  (
( E `  ( F `  0 )
)  =  { A ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  B } ) )  ->  ( F :
( 0..^ 2 ) --> dom  E  ->  ( A  =/=  B  ->  ( F `  0 )  =/=  ( F `  1
) ) ) )
5251exp31 604 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( P `  0
)  =  A  /\  ( P `  2 )  =  B )  -> 
( ( ( # `  F )  =  2  /\  E : dom  E
-1-1-> ran  E )  -> 
( ( ( E `
 ( F ` 
0 ) )  =  { A ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  B } )  ->  ( F : ( 0..^ 2 ) --> dom  E  ->  ( A  =/=  B  -> 
( F `  0
)  =/=  ( F `
 1 ) ) ) ) ) )
5352com25 91 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( P `  0
)  =  A  /\  ( P `  2 )  =  B )  -> 
( A  =/=  B  ->  ( ( ( E `
 ( F ` 
0 ) )  =  { A ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  B } )  ->  ( F : ( 0..^ 2 ) --> dom  E  ->  ( ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E )  ->  ( F `  0 )  =/=  ( F `  1
) ) ) ) ) )
54533impia 1184 . . . . . . . . . . . . . . . . 17  |-  ( ( ( P `  0
)  =  A  /\  ( P `  2 )  =  B  /\  A  =/=  B )  ->  (
( ( E `  ( F `  0 ) )  =  { A ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  B } )  -> 
( F : ( 0..^ 2 ) --> dom 
E  ->  ( (
( # `  F )  =  2  /\  E : dom  E -1-1-> ran  E
)  ->  ( F `  0 )  =/=  ( F `  1
) ) ) ) )
5517, 54sylbid 215 . . . . . . . . . . . . . . . 16  |-  ( ( ( P `  0
)  =  A  /\  ( P `  2 )  =  B  /\  A  =/=  B )  ->  (
( ( E `  ( F `  0 ) )  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } )  -> 
( F : ( 0..^ 2 ) --> dom 
E  ->  ( (
( # `  F )  =  2  /\  E : dom  E -1-1-> ran  E
)  ->  ( F `  0 )  =/=  ( F `  1
) ) ) ) )
5655imp 429 . . . . . . . . . . . . . . 15  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
2 )  =  B  /\  A  =/=  B
)  /\  ( ( E `  ( F `  0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  /\  ( E `  ( F `  1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) )  ->  ( F :
( 0..^ 2 ) --> dom  E  ->  (
( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E )  ->  ( F `  0 )  =/=  ( F `  1
) ) ) )
5756com13 80 . . . . . . . . . . . . . 14  |-  ( ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E )  ->  ( F : ( 0..^ 2 ) --> dom  E  ->  ( ( ( ( P `
 0 )  =  A  /\  ( P `
 2 )  =  B  /\  A  =/= 
B )  /\  (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) )  ->  ( F ` 
0 )  =/=  ( F `  1 )
) ) )
584feq2d 5559 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  =  2  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  <->  F :
( 0..^ 2 ) --> dom  E ) )
59 fveq2 5703 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  =  2  ->  ( P `  ( # `  F
) )  =  ( P `  2 ) )
6059eqeq1d 2451 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  F )  =  2  ->  (
( P `  ( # `
 F ) )  =  B  <->  ( P `  2 )  =  B ) )
61603anbi2d 1294 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  F )  =  2  ->  (
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/=  B )  <->  ( ( P `  0 )  =  A  /\  ( P `  2 )  =  B  /\  A  =/= 
B ) ) )
624, 5syl6eq 2491 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  {
0 ,  1 } )
6362raleqdv 2935 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  F )  =  2  ->  ( A. i  e.  (
0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  <->  A. i  e.  {
0 ,  1 }  ( E `  ( F `  i )
)  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) )
64 c0ex 9392 . . . . . . . . . . . . . . . . . . . 20  |-  0  e.  _V
65 1ex 9393 . . . . . . . . . . . . . . . . . . . 20  |-  1  e.  _V
66 fveq2 5703 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i  =  0  ->  ( F `  i )  =  ( F ` 
0 ) )
6766fveq2d 5707 . . . . . . . . . . . . . . . . . . . . 21  |-  ( i  =  0  ->  ( E `  ( F `  i ) )  =  ( E `  ( F `  0 )
) )
68 fveq2 5703 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i  =  0  ->  ( P `  i )  =  ( P ` 
0 ) )
69 oveq1 6110 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( i  =  0  ->  (
i  +  1 )  =  ( 0  +  1 ) )
7069fveq2d 5707 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( i  =  0  ->  ( P `  ( i  +  1 ) )  =  ( P `  ( 0  +  1 ) ) )
71 0p1e1 10445 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 0  +  1 )  =  1
7271fveq2i 5706 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( P `
 ( 0  +  1 ) )  =  ( P `  1
)
7370, 72syl6eq 2491 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i  =  0  ->  ( P `  ( i  +  1 ) )  =  ( P ` 
1 ) )
7468, 73preq12d 3974 . . . . . . . . . . . . . . . . . . . . 21  |-  ( i  =  0  ->  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  =  { ( P ` 
0 ) ,  ( P `  1 ) } )
7567, 74eqeq12d 2457 . . . . . . . . . . . . . . . . . . . 20  |-  ( i  =  0  ->  (
( E `  ( F `  i )
)  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  <->  ( E `  ( F `  0
) )  =  {
( P `  0
) ,  ( P `
 1 ) } ) )
76 fveq2 5703 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i  =  1  ->  ( F `  i )  =  ( F ` 
1 ) )
7776fveq2d 5707 . . . . . . . . . . . . . . . . . . . . 21  |-  ( i  =  1  ->  ( E `  ( F `  i ) )  =  ( E `  ( F `  1 )
) )
78 fveq2 5703 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i  =  1  ->  ( P `  i )  =  ( P ` 
1 ) )
79 oveq1 6110 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( i  =  1  ->  (
i  +  1 )  =  ( 1  +  1 ) )
8079fveq2d 5707 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( i  =  1  ->  ( P `  ( i  +  1 ) )  =  ( P `  ( 1  +  1 ) ) )
81 1p1e2 10447 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 1  +  1 )  =  2
8281fveq2i 5706 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( P `
 ( 1  +  1 ) )  =  ( P `  2
)
8380, 82syl6eq 2491 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i  =  1  ->  ( P `  ( i  +  1 ) )  =  ( P ` 
2 ) )
8478, 83preq12d 3974 . . . . . . . . . . . . . . . . . . . . 21  |-  ( i  =  1  ->  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  =  { ( P ` 
1 ) ,  ( P `  2 ) } )
8577, 84eqeq12d 2457 . . . . . . . . . . . . . . . . . . . 20  |-  ( i  =  1  ->  (
( E `  ( F `  i )
)  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  <->  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } ) )
8664, 65, 75, 85ralpr 3941 . . . . . . . . . . . . . . . . . . 19  |-  ( A. i  e.  { 0 ,  1 }  ( E `  ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  <->  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) } ) )
8763, 86syl6bb 261 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  F )  =  2  ->  ( A. i  e.  (
0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  <->  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) } ) ) )
8861, 87anbi12d 710 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  F )  =  2  ->  (
( ( ( P `
 0 )  =  A  /\  ( P `
 ( # `  F
) )  =  B  /\  A  =/=  B
)  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } )  <->  ( (
( P `  0
)  =  A  /\  ( P `  2 )  =  B  /\  A  =/=  B )  /\  (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) ) ) )
8988imbi1d 317 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  =  2  ->  (
( ( ( ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B  /\  A  =/=  B
)  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } )  -> 
( F `  0
)  =/=  ( F `
 1 ) )  <-> 
( ( ( ( P `  0 )  =  A  /\  ( P `  2 )  =  B  /\  A  =/= 
B )  /\  (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) )  ->  ( F ` 
0 )  =/=  ( F `  1 )
) ) )
9058, 89imbi12d 320 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  2  ->  (
( F : ( 0..^ ( # `  F
) ) --> dom  E  ->  ( ( ( ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B  /\  A  =/=  B
)  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } )  -> 
( F `  0
)  =/=  ( F `
 1 ) ) )  <->  ( F :
( 0..^ 2 ) --> dom  E  ->  (
( ( ( P `
 0 )  =  A  /\  ( P `
 2 )  =  B  /\  A  =/= 
B )  /\  (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) )  ->  ( F ` 
0 )  =/=  ( F `  1 )
) ) ) )
9190adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E )  ->  (
( F : ( 0..^ ( # `  F
) ) --> dom  E  ->  ( ( ( ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B  /\  A  =/=  B
)  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } )  -> 
( F `  0
)  =/=  ( F `
 1 ) ) )  <->  ( F :
( 0..^ 2 ) --> dom  E  ->  (
( ( ( P `
 0 )  =  A  /\  ( P `
 2 )  =  B  /\  A  =/= 
B )  /\  (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) )  ->  ( F ` 
0 )  =/=  ( F `  1 )
) ) ) )
9257, 91mpbird 232 . . . . . . . . . . . . 13  |-  ( ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E )  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  ( ( ( ( P `
 0 )  =  A  /\  ( P `
 ( # `  F
) )  =  B  /\  A  =/=  B
)  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } )  -> 
( F `  0
)  =/=  ( F `
 1 ) ) ) )
9392impcom 430 . . . . . . . . . . . 12  |-  ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E ) )  -> 
( ( ( ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B  /\  A  =/=  B
)  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } )  -> 
( F `  0
)  =/=  ( F `
 1 ) ) )
9493imp 429 . . . . . . . . . . 11  |-  ( ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E ) )  /\  ( ( ( P `
 0 )  =  A  /\  ( P `
 ( # `  F
) )  =  B  /\  A  =/=  B
)  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } ) )  ->  ( F ` 
0 )  =/=  ( F `  1 )
)
9511, 94jca 532 . . . . . . . . . 10  |-  ( ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E ) )  /\  ( ( ( P `
 0 )  =  A  /\  ( P `
 ( # `  F
) )  =  B  /\  A  =/=  B
)  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } ) )  ->  ( F : { 0 ,  1 } --> dom  E  /\  ( F `  0 )  =/=  ( F ` 
1 ) ) )
9664, 65pm3.2i 455 . . . . . . . . . . . 12  |-  ( 0  e.  _V  /\  1  e.  _V )
97 0ne1 10401 . . . . . . . . . . . 12  |-  0  =/=  1
9896, 97pm3.2i 455 . . . . . . . . . . 11  |-  ( ( 0  e.  _V  /\  1  e.  _V )  /\  0  =/=  1
)
99 eqid 2443 . . . . . . . . . . . 12  |-  { 0 ,  1 }  =  { 0 ,  1 }
10099f12dfv 30158 . . . . . . . . . . 11  |-  ( ( ( 0  e.  _V  /\  1  e.  _V )  /\  0  =/=  1
)  ->  ( F : { 0 ,  1 } -1-1-> dom  E  <->  ( F : { 0 ,  1 } --> dom  E  /\  ( F `  0 )  =/=  ( F ` 
1 ) ) ) )
10198, 100mp1i 12 . . . . . . . . . 10  |-  ( ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E ) )  /\  ( ( ( P `
 0 )  =  A  /\  ( P `
 ( # `  F
) )  =  B  /\  A  =/=  B
)  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } ) )  ->  ( F : { 0 ,  1 } -1-1-> dom  E  <->  ( F : { 0 ,  1 } --> dom  E  /\  ( F `  0 )  =/=  ( F ` 
1 ) ) ) )
10295, 101mpbird 232 . . . . . . . . 9  |-  ( ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E ) )  /\  ( ( ( P `
 0 )  =  A  /\  ( P `
 ( # `  F
) )  =  B  /\  A  =/=  B
)  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } ) )  ->  F : {
0 ,  1 }
-1-1-> dom  E )
103 df-f1 5435 . . . . . . . . 9  |-  ( F : { 0 ,  1 } -1-1-> dom  E  <->  ( F : { 0 ,  1 } --> dom  E  /\  Fun  `' F ) )
104102, 103sylib 196 . . . . . . . 8  |-  ( ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E ) )  /\  ( ( ( P `
 0 )  =  A  /\  ( P `
 ( # `  F
) )  =  B  /\  A  =/=  B
)  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } ) )  ->  ( F : { 0 ,  1 } --> dom  E  /\  Fun  `' F ) )
105104simprd 463 . . . . . . 7  |-  ( ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E ) )  /\  ( ( ( P `
 0 )  =  A  /\  ( P `
 ( # `  F
) )  =  B  /\  A  =/=  B
)  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } ) )  ->  Fun  `' F
)
106105ex 434 . . . . . 6  |-  ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E ) )  -> 
( ( ( ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B  /\  A  =/=  B
)  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } )  ->  Fun  `' F ) )
107106expcom 435 . . . . 5  |-  ( ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E )  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  ( ( ( ( P `
 0 )  =  A  /\  ( P `
 ( # `  F
) )  =  B  /\  A  =/=  B
)  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } )  ->  Fun  `' F ) ) )
108107ex 434 . . . 4  |-  ( (
# `  F )  =  2  ->  ( E : dom  E -1-1-> ran  E  ->  ( F :
( 0..^ ( # `  F ) ) --> dom 
E  ->  ( (
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/=  B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )  ->  Fun  `' F ) ) ) )
109108com13 80 . . 3  |-  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  ( E : dom  E -1-1-> ran 
E  ->  ( ( # `
 F )  =  2  ->  ( (
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/=  B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )  ->  Fun  `' F ) ) ) )
1101, 109syl 16 . 2  |-  ( F  e. Word  dom  E  ->  ( E : dom  E -1-1-> ran 
E  ->  ( ( # `
 F )  =  2  ->  ( (
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/=  B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )  ->  Fun  `' F ) ) ) )
1111103imp 1181 1  |-  ( ( F  e. Word  dom  E  /\  E : dom  E -1-1-> ran 
E  /\  ( # `  F
)  =  2 )  ->  ( ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/= 
B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )  ->  Fun  `' F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2618   A.wral 2727   _Vcvv 2984   {cpr 3891   `'ccnv 4851   dom cdm 4852   ran crn 4853   Fun wfun 5424   -->wf 5426   -1-1->wf1 5427   ` cfv 5430  (class class class)co 6103   0cc0 9294   1c1 9295    + caddc 9297   2c2 10383  ..^cfzo 11560   #chash 12115  Word cword 12233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-card 8121  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-n0 10592  df-z 10659  df-uz 10874  df-fz 11450  df-fzo 11561  df-hash 12116  df-word 12241
This theorem is referenced by:  usgra2wlkspth  30310
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