MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  usgra2wlkspthlem1 Structured version   Unicode version

Theorem usgra2wlkspthlem1 24311
Description: Lemma 1 for usgra2wlkspth 24313. (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 12518 . . 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 6291 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  ( 0..^ 2 ) )
5 fzo0to2pr 11866 . . . . . . . . . . . . . . . . 17  |-  ( 0..^ 2 )  =  {
0 ,  1 }
64, 5syl6req 2525 . . . . . . . . . . . . . . . 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 5717 . . . . . . . . . . . 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 4106 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( P `  0 )  =  A  ->  { ( P `  0 ) ,  ( P ` 
1 ) }  =  { A ,  ( P `
 1 ) } )
1312eqeq2d 2481 . . . . . . . . . . . . . . . . . . 19  |-  ( ( P `  0 )  =  A  ->  (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  <->  ( E `  ( F `  0
) )  =  { A ,  ( P `  1 ) } ) )
14 preq2 4107 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( P `  2 )  =  B  ->  { ( P `  1 ) ,  ( P ` 
2 ) }  =  { ( P ` 
1 ) ,  B } )
1514eqeq2d 2481 . . . . . . . . . . . . . . . . . . 19  |-  ( ( P `  2 )  =  B  ->  (
( E `  ( F `  1 )
)  =  { ( P `  1 ) ,  ( P ` 
2 ) }  <->  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  B }
) )
1613, 15bi2anan9 871 . . . . . . . . . . . . . . . . . 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 1016 . . . . . . . . . . . . . . . . 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 5865 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( F `  0 )  =  ( F ` 
1 )  ->  ( E `  ( F `  0 ) )  =  ( E `  ( F `  1 ) ) )
1918eqeq1d 2469 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2494 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( E `  ( F `  1 )
)  =  { A ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  B } )  ->  { A ,  ( P `
 1 ) }  =  { ( P `
 1 ) ,  B } )
23 fvex 5875 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( P `
 0 )  e. 
_V
2423a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( P `  0 )  =  A  ->  ( P `  0 )  e.  _V )
25 eleq1 2539 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( A  =  ( P ` 
0 )  ->  ( A  e.  _V  <->  ( P `  0 )  e. 
_V ) )
2625eqcoms 2479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( P `  0 )  =  A  ->  ( A  e.  _V  <->  ( P `  0 )  e. 
_V ) )
2724, 26mpbird 232 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( P `  0 )  =  A  ->  A  e.  _V )
28 fvex 5875 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5875 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( P `
 2 )  e. 
_V
3332a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( P `  2 )  =  B  ->  ( P `  2 )  e.  _V )
34 eleq1 2539 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( B  =  ( P ` 
2 )  ->  ( B  e.  _V  <->  ( P `  2 )  e. 
_V ) )
3534eqcoms 2479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4205 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2493 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2691 . . . . . . . . . . . . . . . . . . . . 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 1193 . . . . . . . . . . . . . . . . 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 5717 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  =  2  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  <->  F :
( 0..^ 2 ) --> dom  E ) )
59 fveq2 5865 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  =  2  ->  ( P `  ( # `  F
) )  =  ( P `  2 ) )
6059eqeq1d 2469 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  F )  =  2  ->  (
( P `  ( # `
 F ) )  =  B  <->  ( P `  2 )  =  B ) )
61603anbi2d 1304 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  F )  =  2  ->  (
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/=  B )  <->  ( ( P `  0 )  =  A  /\  ( P `  2 )  =  B  /\  A  =/= 
B ) ) )
624, 5syl6eq 2524 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  {
0 ,  1 } )
6362raleqdv 3064 . . . . . . . . . . . . . . . . . . 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 9589 . . . . . . . . . . . . . . . . . . . 20  |-  0  e.  _V
65 1ex 9590 . . . . . . . . . . . . . . . . . . . 20  |-  1  e.  _V
66 fveq2 5865 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i  =  0  ->  ( F `  i )  =  ( F ` 
0 ) )
6766fveq2d 5869 . . . . . . . . . . . . . . . . . . . . 21  |-  ( i  =  0  ->  ( E `  ( F `  i ) )  =  ( E `  ( F `  0 )
) )
68 fveq2 5865 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i  =  0  ->  ( P `  i )  =  ( P ` 
0 ) )
69 oveq1 6290 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( i  =  0  ->  (
i  +  1 )  =  ( 0  +  1 ) )
7069fveq2d 5869 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( i  =  0  ->  ( P `  ( i  +  1 ) )  =  ( P `  ( 0  +  1 ) ) )
71 0p1e1 10646 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 0  +  1 )  =  1
7271fveq2i 5868 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( P `
 ( 0  +  1 ) )  =  ( P `  1
)
7370, 72syl6eq 2524 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i  =  0  ->  ( P `  ( i  +  1 ) )  =  ( P ` 
1 ) )
7468, 73preq12d 4114 . . . . . . . . . . . . . . . . . . . . 21  |-  ( i  =  0  ->  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  =  { ( P ` 
0 ) ,  ( P `  1 ) } )
7567, 74eqeq12d 2489 . . . . . . . . . . . . . . . . . . . 20  |-  ( i  =  0  ->  (
( E `  ( F `  i )
)  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  <->  ( E `  ( F `  0
) )  =  {
( P `  0
) ,  ( P `
 1 ) } ) )
76 fveq2 5865 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i  =  1  ->  ( F `  i )  =  ( F ` 
1 ) )
7776fveq2d 5869 . . . . . . . . . . . . . . . . . . . . 21  |-  ( i  =  1  ->  ( E `  ( F `  i ) )  =  ( E `  ( F `  1 )
) )
78 fveq2 5865 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i  =  1  ->  ( P `  i )  =  ( P ` 
1 ) )
79 oveq1 6290 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( i  =  1  ->  (
i  +  1 )  =  ( 1  +  1 ) )
8079fveq2d 5869 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( i  =  1  ->  ( P `  ( i  +  1 ) )  =  ( P `  ( 1  +  1 ) ) )
81 1p1e2 10648 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 1  +  1 )  =  2
8281fveq2i 5868 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( P `
 ( 1  +  1 ) )  =  ( P `  2
)
8380, 82syl6eq 2524 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( i  =  1  ->  ( P `  ( i  +  1 ) )  =  ( P ` 
2 ) )
8478, 83preq12d 4114 . . . . . . . . . . . . . . . . . . . . 21  |-  ( i  =  1  ->  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  =  { ( P ` 
1 ) ,  ( P `  2 ) } )
8577, 84eqeq12d 2489 . . . . . . . . . . . . . . . . . . . 20  |-  ( i  =  1  ->  (
( E `  ( F `  i )
)  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  <->  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } ) )
8664, 65, 75, 85ralpr 4080 . . . . . . . . . . . . . . . . . . 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 10602 . . . . . . . . . . . 12  |-  0  =/=  1
9896, 97pm3.2i 455 . . . . . . . . . . 11  |-  ( ( 0  e.  _V  /\  1  e.  _V )  /\  0  =/=  1
)
99 eqid 2467 . . . . . . . . . . . 12  |-  { 0 ,  1 }  =  { 0 ,  1 }
10099f12dfv 6166 . . . . . . . . . . 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 5592 . . . . . . . . 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 1190 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   _Vcvv 3113   {cpr 4029   `'ccnv 4998   dom cdm 4999   ran crn 5000   Fun wfun 5581   -->wf 5583   -1-1->wf1 5584   ` cfv 5587  (class class class)co 6283   0cc0 9491   1c1 9492    + caddc 9494   2c2 10584  ..^cfzo 11791   #chash 12372  Word cword 12499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8319  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-2 10593  df-n0 10795  df-z 10864  df-uz 11082  df-fz 11672  df-fzo 11792  df-hash 12373  df-word 12507
This theorem is referenced by:  usgra2wlkspth  24313
  Copyright terms: Public domain W3C validator