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Theorem usgra2wlkspthlem1 24746
Description: Lemma 1 for usgra2wlkspth 24748. (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 12558 . . 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 6304 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  ( 0..^ 2 ) )
5 fzo0to2pr 11902 . . . . . . . . . . . . . . . 16  |-  ( 0..^ 2 )  =  {
0 ,  1 }
64, 5syl6req 2515 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  2  ->  { 0 ,  1 }  =  ( 0..^ ( # `  F
) ) )
76adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( # `  F
)  =  2  /\  E : dom  E -1-1-> ran 
E )  ->  { 0 ,  1 }  =  ( 0..^ ( # `  F
) ) )
87ad2antlr 726 . . . . . . . . . . . . 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
) ) )
98feq2d 5724 . . . . . . . . . . . 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 ) )
103, 9mpbird 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 )
11 preq1 4111 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( P `  0 )  =  A  ->  { ( P `  0 ) ,  ( P ` 
1 ) }  =  { A ,  ( P `
 1 ) } )
1211eqeq2d 2471 . . . . . . . . . . . . . . . . . . 19  |-  ( ( P `  0 )  =  A  ->  (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  <->  ( E `  ( F `  0
) )  =  { A ,  ( P `  1 ) } ) )
13 preq2 4112 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( P `  2 )  =  B  ->  { ( P `  1 ) ,  ( P ` 
2 ) }  =  { ( P ` 
1 ) ,  B } )
1413eqeq2d 2471 . . . . . . . . . . . . . . . . . . 19  |-  ( ( P `  2 )  =  B  ->  (
( E `  ( F `  1 )
)  =  { ( P `  1 ) ,  ( P ` 
2 ) }  <->  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  B }
) )
1512, 14bi2anan9 873 . . . . . . . . . . . . . . . . . 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 } ) ) )
16153adant3 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 } ) ) )
17 fveq2 5872 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( F `  0 )  =  ( F ` 
1 )  ->  ( E `  ( F `  0 ) )  =  ( E `  ( F `  1 ) ) )
1817eqeq1d 2459 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( F `  0 )  =  ( F ` 
1 )  ->  (
( E `  ( F `  0 )
)  =  { A ,  ( P ` 
1 ) }  <->  ( E `  ( F `  1
) )  =  { A ,  ( P `  1 ) } ) )
1918anbi1d 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 } ) ) )
2019anbi2d 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 } ) ) ) )
21 eqtr2 2484 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( E `  ( F `  1 )
)  =  { A ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  B } )  ->  { A ,  ( P `
 1 ) }  =  { ( P `
 1 ) ,  B } )
22 fvex 5882 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( P `
 0 )  e. 
_V
23 eleq1 2529 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( A  =  ( P ` 
0 )  ->  ( A  e.  _V  <->  ( P `  0 )  e. 
_V ) )
2423eqcoms 2469 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( P `  0 )  =  A  ->  ( A  e.  _V  <->  ( P `  0 )  e. 
_V ) )
2522, 24mpbiri 233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( P `  0 )  =  A  ->  A  e.  _V )
26 fvex 5882 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( P `
 1 )  e. 
_V
2726a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( P `  0 )  =  A  ->  ( P `  1 )  e.  _V )
2825, 27jca 532 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( P `  0 )  =  A  ->  ( A  e.  _V  /\  ( P `  1 )  e.  _V ) )
2926a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( P `  2 )  =  B  ->  ( P `  1 )  e.  _V )
30 fvex 5882 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( P `
 2 )  e. 
_V
31 eleq1 2529 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( B  =  ( P ` 
2 )  ->  ( B  e.  _V  <->  ( P `  2 )  e. 
_V ) )
3231eqcoms 2469 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( P `  2 )  =  B  ->  ( B  e.  _V  <->  ( P `  2 )  e. 
_V ) )
3330, 32mpbiri 233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( P `  2 )  =  B  ->  B  e.  _V )
3429, 33jca 532 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( P `  2 )  =  B  ->  (
( P `  1
)  e.  _V  /\  B  e.  _V )
)
3528, 34anim12i 566 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( P `  0
)  =  A  /\  ( P `  2 )  =  B )  -> 
( ( A  e. 
_V  /\  ( P `  1 )  e. 
_V )  /\  (
( P `  1
)  e.  _V  /\  B  e.  _V )
) )
3635adantr 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 )
) )
37 preq12bg 4211 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
3836, 37syl 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 ) ) ) ) )
39 eqtr 2483 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( A  =  ( P `
 1 )  /\  ( P `  1 )  =  B )  ->  A  =  B )
40 simpl 457 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( A  =  B  /\  ( P `  1 )  =  ( P ` 
1 ) )  ->  A  =  B )
4139, 40jaoi 379 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( A  =  ( P `  1 )  /\  ( P ` 
1 )  =  B )  \/  ( A  =  B  /\  ( P `  1 )  =  ( P ` 
1 ) ) )  ->  A  =  B )
4238, 41syl6bi 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 ) )
4321, 42syl5com 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 ) )
4443impcom 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 )
4520, 44syl6bi 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 ) )
4645com12 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 ) )
4746necon3d 2681 . . . . . . . . . . . . . . . . . . . . 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
) ) )
4847a1d 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
) ) ) )
4948exp31 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 ) ) ) ) ) )
5049com25 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
) ) ) ) ) )
51503impia 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
) ) ) ) )
5216, 51sylbid 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
) ) ) ) )
5352imp 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
) ) ) )
5453com13 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 )
) ) )
554feq2d 5724 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  =  2  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  <->  F :
( 0..^ 2 ) --> dom  E ) )
56 fveq2 5872 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  =  2  ->  ( P `  ( # `  F
) )  =  ( P `  2 ) )
5756eqeq1d 2459 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  F )  =  2  ->  (
( P `  ( # `
 F ) )  =  B  <->  ( P `  2 )  =  B ) )
58573anbi2d 1304 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  F )  =  2  ->  (
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/=  B )  <->  ( ( P `  0 )  =  A  /\  ( P `  2 )  =  B  /\  A  =/= 
B ) ) )
594, 5syl6eq 2514 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  {
0 ,  1 } )
6059raleqdv 3060 . . . . . . . . . . . . . . . . . . 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 ) ) } ) )
61 2wlklem 24693 . . . . . . . . . . . . . . . . . . 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 ) } ) )
6260, 61syl6bb 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 ) } ) ) )
6358, 62anbi12d 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
) } ) ) ) )
6463imbi1d 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 )
) ) )
6555, 64imbi12d 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 )
) ) ) )
6665adantr 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 )
) ) ) )
6754, 66mpbird 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 ) ) ) )
6867impcom 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 ) ) )
6968imp 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 )
)
7010, 69jca 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 ) ) )
71 c0ex 9607 . . . . . . . . . . . . 13  |-  0  e.  _V
72 1ex 9608 . . . . . . . . . . . . 13  |-  1  e.  _V
7371, 72pm3.2i 455 . . . . . . . . . . . 12  |-  ( 0  e.  _V  /\  1  e.  _V )
74 0ne1 10624 . . . . . . . . . . . 12  |-  0  =/=  1
7573, 74pm3.2i 455 . . . . . . . . . . 11  |-  ( ( 0  e.  _V  /\  1  e.  _V )  /\  0  =/=  1
)
76 eqid 2457 . . . . . . . . . . . 12  |-  { 0 ,  1 }  =  { 0 ,  1 }
7776f12dfv 6180 . . . . . . . . . . 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 ) ) ) )
7875, 77mp1i 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 ) ) ) )
7970, 78mpbird 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 )
80 df-f1 5599 . . . . . . . . 9  |-  ( F : { 0 ,  1 } -1-1-> dom  E  <->  ( F : { 0 ,  1 } --> dom  E  /\  Fun  `' F ) )
8179, 80sylib 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 ) )
8281simprd 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
)
8382ex 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 ) )
8483expcom 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 ) ) )
8584ex 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 ) ) ) )
8685com13 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 ) ) ) )
871, 86syl 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 ) ) ) )
88873imp 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 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   _Vcvv 3109   {cpr 4034   `'ccnv 5007   dom cdm 5008   ran crn 5009   Fun wfun 5588   -->wf 5590   -1-1->wf1 5591   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510    + caddc 9512   2c2 10606  ..^cfzo 11821   #chash 12408  Word cword 12538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546
This theorem is referenced by:  usgra2wlkspth  24748
  Copyright terms: Public domain W3C validator