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Theorem usgra2wlkspthlem1 30205
Description: Lemma 1 for usgra2wlkspth 30207. (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 12236 . . 3  |-  ( F  e. Word  dom  E  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
2 simpl 454 . . . . . . . . . . . . 13  |-  ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E ) )  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
32adantr 462 . . . . . . . . . . . 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 6098 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  ( 0..^ 2 ) )
5 fzo0to2pr 11610 . . . . . . . . . . . . . . . . 17  |-  ( 0..^ 2 )  =  {
0 ,  1 }
64, 5syl6req 2490 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  =  2  ->  { 0 ,  1 }  =  ( 0..^ ( # `  F
) ) )
76adantr 462 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E )  ->  { 0 ,  1 }  =  ( 0..^ ( # `  F
) ) )
87adantl 463 . . . . . . . . . . . . . 14  |-  ( ( F : ( 0..^ ( # `  F
) ) --> dom  E  /\  ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E ) )  ->  { 0 ,  1 }  =  ( 0..^ ( # `  F
) ) )
98adantr 462 . . . . . . . . . . . . 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 5544 . . . . . . . . . . . 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 3951 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( P `  0 )  =  A  ->  { ( P `  0 ) ,  ( P ` 
1 ) }  =  { A ,  ( P `
 1 ) } )
1312eqeq2d 2452 . . . . . . . . . . . . . . . . . . 19  |-  ( ( P `  0 )  =  A  ->  (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  <->  ( E `  ( F `  0
) )  =  { A ,  ( P `  1 ) } ) )
14 preq2 3952 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( P `  2 )  =  B  ->  { ( P `  1 ) ,  ( P ` 
2 ) }  =  { ( P ` 
1 ) ,  B } )
1514eqeq2d 2452 . . . . . . . . . . . . . . . . . . 19  |-  ( ( P `  2 )  =  B  ->  (
( E `  ( F `  1 )
)  =  { ( P `  1 ) ,  ( P ` 
2 ) }  <->  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  B }
) )
1613, 15bi2anan9 863 . . . . . . . . . . . . . . . . . 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 1003 . . . . . . . . . . . . . . . . 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 5688 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( F `  0 )  =  ( F ` 
1 )  ->  ( E `  ( F `  0 ) )  =  ( E `  ( F `  1 ) ) )
1918eqeq1d 2449 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( F `  0 )  =  ( F ` 
1 )  ->  (
( E `  ( F `  0 )
)  =  { A ,  ( P ` 
1 ) }  <->  ( E `  ( F `  1
) )  =  { A ,  ( P `  1 ) } ) )
2019anbi1d 699 . . . . . . . . . . . . . . . . . . . . . . . . 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 698 . . . . . . . . . . . . . . . . . . . . . . . 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 2459 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( E `  ( F `  1 )
)  =  { A ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  B } )  ->  { A ,  ( P `
 1 ) }  =  { ( P `
 1 ) ,  B } )
23 fvex 5698 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( P `
 0 )  e. 
_V
2423a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( P `  0 )  =  A  ->  ( P `  0 )  e.  _V )
25 eleq1 2501 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( A  =  ( P ` 
0 )  ->  ( A  e.  _V  <->  ( P `  0 )  e. 
_V ) )
2625eqcoms 2444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( P `  0 )  =  A  ->  ( A  e.  _V  <->  ( P `  0 )  e. 
_V ) )
2724, 26mpbird 232 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( P `  0 )  =  A  ->  A  e.  _V )
28 fvex 5698 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( P `
 1 )  e. 
_V
2928a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( P `  0 )  =  A  ->  ( P `  1 )  e.  _V )
3027, 29jca 529 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( P `  0 )  =  A  ->  ( A  e.  _V  /\  ( P `  1 )  e.  _V ) )
3128a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( P `  2 )  =  B  ->  ( P `  1 )  e.  _V )
32 fvex 5698 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( P `
 2 )  e. 
_V
3332a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( P `  2 )  =  B  ->  ( P `  2 )  e.  _V )
34 eleq1 2501 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( B  =  ( P ` 
2 )  ->  ( B  e.  _V  <->  ( P `  2 )  e. 
_V ) )
3534eqcoms 2444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( P `  2 )  =  B  ->  ( B  e.  _V  <->  ( P `  2 )  e. 
_V ) )
3633, 35mpbird 232 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( P `  2 )  =  B  ->  B  e.  _V )
3731, 36jca 529 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( P `  2 )  =  B  ->  (
( P `  1
)  e.  _V  /\  B  e.  _V )
)
3830, 37anim12i 563 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( P `  0
)  =  A  /\  ( P `  2 )  =  B )  -> 
( ( A  e. 
_V  /\  ( P `  1 )  e. 
_V )  /\  (
( P `  1
)  e.  _V  /\  B  e.  _V )
) )
3938adantr 462 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4048 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2458 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( A  =  ( P `
 1 )  /\  ( P `  1 )  =  B )  ->  A  =  B )
43 simpl 454 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2644 . . . . . . . . . . . . . . . . . . . . 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 601 . . . . . . . . . . . . . . . . . . 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 1179 . . . . . . . . . . . . . . . . 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 5544 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  =  2  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  <->  F :
( 0..^ 2 ) --> dom  E ) )
59 fveq2 5688 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  =  2  ->  ( P `  ( # `  F
) )  =  ( P `  2 ) )
6059eqeq1d 2449 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  F )  =  2  ->  (
( P `  ( # `
 F ) )  =  B  <->  ( P `  2 )  =  B ) )
61603anbi2d 1289 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  F )  =  2  ->  (
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/=  B )  <->  ( ( P `  0 )  =  A  /\  ( P `  2 )  =  B  /\  A  =/= 
B ) ) )
624, 5syl6eq 2489 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  {
0 ,  1 } )
6362raleqdv 2921 . . . . . . . . . . . . . . . . . . 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 9376 . . . . . . . . . . . . . . . . . . . 20  |-  0  e.  _V
65 1ex 9377 . . . . . . . . . . . . . . . . . . . 20  |-  1  e.  _V
66 fveq2 5688 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i  =  0  ->  ( F `  i )  =  ( F ` 
0 ) )
6766fveq2d 5692 . . . . . . . . . . . . . . . . . . . . 21  |-  ( i  =  0  ->  ( E `  ( F `  i ) )  =  ( E `  ( F `  0 )
) )
68 fveq2 5688 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i  =  0  ->  ( P `  i )  =  ( P ` 
0 ) )
69 oveq1 6097 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( i  =  0  ->  (
i  +  1 )  =  ( 0  +  1 ) )
7069fveq2d 5692 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( i  =  0  ->  ( P `  ( i  +  1 ) )  =  ( P `  ( 0  +  1 ) ) )
71 0p1e1 10429 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 0  +  1 )  =  1
7271fveq2i 5691 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( P `
 ( 0  +  1 ) )  =  ( P `  1
)
7370, 72syl6eq 2489 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i  =  0  ->  ( P `  ( i  +  1 ) )  =  ( P ` 
1 ) )
7468, 73preq12d 3959 . . . . . . . . . . . . . . . . . . . . 21  |-  ( i  =  0  ->  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  =  { ( P ` 
0 ) ,  ( P `  1 ) } )
7567, 74eqeq12d 2455 . . . . . . . . . . . . . . . . . . . 20  |-  ( i  =  0  ->  (
( E `  ( F `  i )
)  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  <->  ( E `  ( F `  0
) )  =  {
( P `  0
) ,  ( P `
 1 ) } ) )
76 fveq2 5688 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i  =  1  ->  ( F `  i )  =  ( F ` 
1 ) )
7776fveq2d 5692 . . . . . . . . . . . . . . . . . . . . 21  |-  ( i  =  1  ->  ( E `  ( F `  i ) )  =  ( E `  ( F `  1 )
) )
78 fveq2 5688 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i  =  1  ->  ( P `  i )  =  ( P ` 
1 ) )
79 oveq1 6097 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( i  =  1  ->  (
i  +  1 )  =  ( 1  +  1 ) )
8079fveq2d 5692 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( i  =  1  ->  ( P `  ( i  +  1 ) )  =  ( P `  ( 1  +  1 ) ) )
81 1p1e2 10431 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 1  +  1 )  =  2
8281fveq2i 5691 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( P `
 ( 1  +  1 ) )  =  ( P `  2
)
8380, 82syl6eq 2489 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i  =  1  ->  ( P `  ( i  +  1 ) )  =  ( P ` 
2 ) )
8478, 83preq12d 3959 . . . . . . . . . . . . . . . . . . . . 21  |-  ( i  =  1  ->  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  =  { ( P ` 
1 ) ,  ( P `  2 ) } )
8577, 84eqeq12d 2455 . . . . . . . . . . . . . . . . . . . 20  |-  ( i  =  1  ->  (
( E `  ( F `  i )
)  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  <->  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } ) )
8664, 65, 75, 85ralpr 3926 . . . . . . . . . . . . . . . . . . 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 705 . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . 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 529 . . . . . . . . . 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 452 . . . . . . . . . . . 12  |-  ( 0  e.  _V  /\  1  e.  _V )
97 0ne1 10385 . . . . . . . . . . . 12  |-  0  =/=  1
9896, 97pm3.2i 452 . . . . . . . . . . 11  |-  ( ( 0  e.  _V  /\  1  e.  _V )  /\  0  =/=  1
)
99 eqid 2441 . . . . . . . . . . . 12  |-  { 0 ,  1 }  =  { 0 ,  1 }
10099f12dfv 30055 . . . . . . . . . . 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 5420 . . . . . . . . 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 460 . . . . . . 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 1176 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 960    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   _Vcvv 2970   {cpr 3876   `'ccnv 4835   dom cdm 4836   ran crn 4837   Fun wfun 5409   -->wf 5411   -1-1->wf1 5412   ` cfv 5415  (class class class)co 6090   0cc0 9278   1c1 9279    + caddc 9281   2c2 10367  ..^cfzo 11544   #chash 12099  Word cword 12217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-hash 12100  df-word 12225
This theorem is referenced by:  usgra2wlkspth  30207
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