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Theorem numclwlk1lem2f1 25498
Description: T is a 1-1 function. (Contributed by AV, 26-Sep-2018.) (Proof shortened by AV, 23-Oct-2018.)
Hypotheses
Ref Expression
numclwwlk.c  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
numclwwlk.f  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
numclwwlk.g  |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>=
`  2 )  |->  { w  e.  ( C `
 n )  |  ( ( w ` 
0 )  =  v  /\  ( w `  ( n  -  2
) )  =  ( w `  0 ) ) } )
numclwwlk.t  |-  T  =  ( w  e.  ( X G N ) 
|->  <. ( w substr  <. 0 ,  ( N  - 
2 ) >. ) ,  ( w `  ( N  -  1
) ) >. )
Assertion
Ref Expression
numclwlk1lem2f1  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  T :
( X G N ) -1-1-> ( ( X F ( N  - 
2 ) )  X.  ( <. V ,  E >. Neighbors  X ) ) )
Distinct variable groups:    n, E    n, N    n, V    w, C    w, N    C, n, v, w    v, N    n, X, v, w    v, V   
w, E    w, V    w, F    w, G
Allowed substitution hints:    T( w, v, n)    E( v)    F( v, n)    G( v, n)

Proof of Theorem numclwlk1lem2f1
Dummy variables  u  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 numclwwlk.c . . 3  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
2 numclwwlk.f . . 3  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
3 numclwwlk.g . . 3  |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>=
`  2 )  |->  { w  e.  ( C `
 n )  |  ( ( w ` 
0 )  =  v  /\  ( w `  ( n  -  2
) )  =  ( w `  0 ) ) } )
4 numclwwlk.t . . 3  |-  T  =  ( w  e.  ( X G N ) 
|->  <. ( w substr  <. 0 ,  ( N  - 
2 ) >. ) ,  ( w `  ( N  -  1
) ) >. )
51, 2, 3, 4numclwlk1lem2f 25496 . 2  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  T :
( X G N ) --> ( ( X F ( N  - 
2 ) )  X.  ( <. V ,  E >. Neighbors  X ) ) )
61, 2, 3, 4numclwlk1lem2fv 25497 . . . . . . 7  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( p  e.  ( X G N )  ->  ( T `  p )  =  <. ( p substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( p `
 ( N  - 
1 ) ) >.
) )
76imp 427 . . . . . 6  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  p  e.  ( X G N ) )  -> 
( T `  p
)  =  <. (
p substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( p `
 ( N  - 
1 ) ) >.
)
87adantrr 715 . . . . 5  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  ( p  e.  ( X G N )  /\  u  e.  ( X G N ) ) )  ->  ( T `  p )  =  <. ( p substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( p `
 ( N  - 
1 ) ) >.
)
91, 2, 3, 4numclwlk1lem2fv 25497 . . . . . . 7  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( u  e.  ( X G N )  ->  ( T `  u )  =  <. ( u substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( u `
 ( N  - 
1 ) ) >.
) )
109imp 427 . . . . . 6  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  u  e.  ( X G N ) )  -> 
( T `  u
)  =  <. (
u substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( u `
 ( N  - 
1 ) ) >.
)
1110adantrl 714 . . . . 5  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  ( p  e.  ( X G N )  /\  u  e.  ( X G N ) ) )  ->  ( T `  u )  =  <. ( u substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( u `
 ( N  - 
1 ) ) >.
)
128, 11eqeq12d 2424 . . . 4  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  ( p  e.  ( X G N )  /\  u  e.  ( X G N ) ) )  ->  ( ( T `
 p )  =  ( T `  u
)  <->  <. ( p substr  <. 0 ,  ( N  - 
2 ) >. ) ,  ( p `  ( N  -  1
) ) >.  =  <. ( u substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( u `
 ( N  - 
1 ) ) >.
) )
13 ovex 6305 . . . . . 6  |-  ( p substr  <. 0 ,  ( N  -  2 ) >.
)  e.  _V
14 fvex 5858 . . . . . 6  |-  ( p `
 ( N  - 
1 ) )  e. 
_V
1513, 14opth 4664 . . . . 5  |-  ( <.
( p substr  <. 0 ,  ( N  -  2 ) >. ) ,  ( p `  ( N  -  1 ) )
>.  =  <. ( u substr  <. 0 ,  ( N  -  2 ) >.
) ,  ( u `
 ( N  - 
1 ) ) >.  <->  ( ( p substr  <. 0 ,  ( N  - 
2 ) >. )  =  ( u substr  <. 0 ,  ( N  - 
2 ) >. )  /\  ( p `  ( N  -  1 ) )  =  ( u `
 ( N  - 
1 ) ) ) )
16 uzuzle23 11166 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  3
)  ->  N  e.  ( ZZ>= `  2 )
)
171, 2, 3numclwwlkovgelim 25493 . . . . . . . 8  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( p  e.  ( X G N )  ->  ( (
p  e. Word  V  /\  ( # `  p )  =  N )  /\  ( ( p ` 
0 )  =  X  /\  ( p `  ( N  -  2
) )  =  ( p `  0 ) ) ) ) )
1816, 17syl3an3 1265 . . . . . . 7  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( p  e.  ( X G N )  ->  ( (
p  e. Word  V  /\  ( # `  p )  =  N )  /\  ( ( p ` 
0 )  =  X  /\  ( p `  ( N  -  2
) )  =  ( p `  0 ) ) ) ) )
191, 2, 3numclwwlkovgelim 25493 . . . . . . . 8  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( u  e.  ( X G N )  ->  ( (
u  e. Word  V  /\  ( # `  u )  =  N )  /\  ( ( u ` 
0 )  =  X  /\  ( u `  ( N  -  2
) )  =  ( u `  0 ) ) ) ) )
2016, 19syl3an3 1265 . . . . . . 7  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( u  e.  ( X G N )  ->  ( (
u  e. Word  V  /\  ( # `  u )  =  N )  /\  ( ( u ` 
0 )  =  X  /\  ( u `  ( N  -  2
) )  =  ( u `  0 ) ) ) ) )
21 simpll 752 . . . . . . . . . . 11  |-  ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) ) )  ->  p  e. Word  V )
2221ad2antrl 726 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  ( ( ( p  e. Word  V  /\  ( # `
 p )  =  N )  /\  (
( p `  0
)  =  X  /\  ( p `  ( N  -  2 ) )  =  ( p `
 0 ) ) )  /\  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  /\  ( ( u ` 
0 )  =  X  /\  ( u `  ( N  -  2
) )  =  ( u `  0 ) ) ) ) )  ->  p  e. Word  V
)
23 simprll 764 . . . . . . . . . . 11  |-  ( ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) ) )  /\  ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) ) )  ->  u  e. Word  V
)
2423adantl 464 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  ( ( ( p  e. Word  V  /\  ( # `
 p )  =  N )  /\  (
( p `  0
)  =  X  /\  ( p `  ( N  -  2 ) )  =  ( p `
 0 ) ) )  /\  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  /\  ( ( u ` 
0 )  =  X  /\  ( u `  ( N  -  2
) )  =  ( u `  0 ) ) ) ) )  ->  u  e. Word  V
)
25 eleq1 2474 . . . . . . . . . . . . . . . . 17  |-  ( N  =  ( # `  p
)  ->  ( N  e.  ( ZZ>= `  3 )  <->  (
# `  p )  e.  ( ZZ>= `  3 )
) )
2625eqcoms 2414 . . . . . . . . . . . . . . . 16  |-  ( (
# `  p )  =  N  ->  ( N  e.  ( ZZ>= `  3
)  <->  ( # `  p
)  e.  ( ZZ>= ` 
3 ) ) )
27 eluz2 11132 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  p )  e.  ( ZZ>= `  3 )  <->  ( 3  e.  ZZ  /\  ( # `  p )  e.  ZZ  /\  3  <_  ( # `  p
) ) )
28 1red 9640 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  p )  e.  ZZ  ->  1  e.  RR )
29 3re 10649 . . . . . . . . . . . . . . . . . . . . . 22  |-  3  e.  RR
3029a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  p )  e.  ZZ  ->  3  e.  RR )
31 zre 10908 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  p )  e.  ZZ  ->  ( # `  p
)  e.  RR )
3228, 30, 313jca 1177 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  p )  e.  ZZ  ->  ( 1  e.  RR  /\  3  e.  RR  /\  ( # `  p )  e.  RR ) )
33 1lt3 10744 . . . . . . . . . . . . . . . . . . . 20  |-  1  <  3
34 ltletr 9706 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 1  e.  RR  /\  3  e.  RR  /\  ( # `
 p )  e.  RR )  ->  (
( 1  <  3  /\  3  <_  ( # `  p ) )  -> 
1  <  ( # `  p
) ) )
3534expd 434 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1  e.  RR  /\  3  e.  RR  /\  ( # `
 p )  e.  RR )  ->  (
1  <  3  ->  ( 3  <_  ( # `  p
)  ->  1  <  (
# `  p )
) ) )
3632, 33, 35mpisyl 19 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  p )  e.  ZZ  ->  ( 3  <_  ( # `  p
)  ->  1  <  (
# `  p )
) )
3736imp 427 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( # `  p
)  e.  ZZ  /\  3  <_  ( # `  p
) )  ->  1  <  ( # `  p
) )
38373adant1 1015 . . . . . . . . . . . . . . . . 17  |-  ( ( 3  e.  ZZ  /\  ( # `  p )  e.  ZZ  /\  3  <_  ( # `  p
) )  ->  1  <  ( # `  p
) )
3927, 38sylbi 195 . . . . . . . . . . . . . . . 16  |-  ( (
# `  p )  e.  ( ZZ>= `  3 )  ->  1  <  ( # `  p ) )
4026, 39syl6bi 228 . . . . . . . . . . . . . . 15  |-  ( (
# `  p )  =  N  ->  ( N  e.  ( ZZ>= `  3
)  ->  1  <  (
# `  p )
) )
4140com12 29 . . . . . . . . . . . . . 14  |-  ( N  e.  ( ZZ>= `  3
)  ->  ( ( # `
 p )  =  N  ->  1  <  (
# `  p )
) )
42413ad2ant3 1020 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( ( # `
 p )  =  N  ->  1  <  (
# `  p )
) )
4342com12 29 . . . . . . . . . . . 12  |-  ( (
# `  p )  =  N  ->  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  1  <  (
# `  p )
) )
4443ad3antlr 729 . . . . . . . . . . 11  |-  ( ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) ) )  /\  ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) ) )  ->  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  3 ) )  ->  1  <  ( # `
 p ) ) )
4544impcom 428 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  ( ( ( p  e. Word  V  /\  ( # `
 p )  =  N )  /\  (
( p `  0
)  =  X  /\  ( p `  ( N  -  2 ) )  =  ( p `
 0 ) ) )  /\  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  /\  ( ( u ` 
0 )  =  X  /\  ( u `  ( N  -  2
) )  =  ( u `  0 ) ) ) ) )  ->  1  <  ( # `
 p ) )
46 2swrd2eqwrdeq 12945 . . . . . . . . . 10  |-  ( ( p  e. Word  V  /\  u  e. Word  V  /\  1  <  ( # `  p
) )  ->  (
p  =  u  <->  ( ( # `
 p )  =  ( # `  u
)  /\  ( (
p substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  =  (
u substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  /\  (
p `  ( ( # `
 p )  - 
2 ) )  =  ( u `  (
( # `  p )  -  2 ) )  /\  ( lastS  `  p
)  =  ( lastS  `  u
) ) ) ) )
4722, 24, 45, 46syl3anc 1230 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  ( ( ( p  e. Word  V  /\  ( # `
 p )  =  N )  /\  (
( p `  0
)  =  X  /\  ( p `  ( N  -  2 ) )  =  ( p `
 0 ) ) )  /\  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  /\  ( ( u ` 
0 )  =  X  /\  ( u `  ( N  -  2
) )  =  ( u `  0 ) ) ) ) )  ->  ( p  =  u  <->  ( ( # `  p )  =  (
# `  u )  /\  ( ( p substr  <. 0 ,  ( ( # `  p )  -  2 ) >. )  =  ( u substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  /\  (
p `  ( ( # `
 p )  - 
2 ) )  =  ( u `  (
( # `  p )  -  2 ) )  /\  ( lastS  `  p
)  =  ( lastS  `  u
) ) ) ) )
48 eqtr3 2430 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( # `  p
)  =  N  /\  ( # `  u )  =  N )  -> 
( # `  p )  =  ( # `  u
) )
4948expcom 433 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  u )  =  N  ->  ( (
# `  p )  =  N  ->  ( # `  p )  =  (
# `  u )
) )
5049adantl 464 . . . . . . . . . . . . . . . 16  |-  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  -> 
( ( # `  p
)  =  N  -> 
( # `  p )  =  ( # `  u
) ) )
5150adantr 463 . . . . . . . . . . . . . . 15  |-  ( ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) )  -> 
( ( # `  p
)  =  N  -> 
( # `  p )  =  ( # `  u
) ) )
5251com12 29 . . . . . . . . . . . . . 14  |-  ( (
# `  p )  =  N  ->  ( ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) )  -> 
( # `  p )  =  ( # `  u
) ) )
5352adantl 464 . . . . . . . . . . . . 13  |-  ( ( p  e. Word  V  /\  ( # `  p )  =  N )  -> 
( ( ( u  e. Word  V  /\  ( # `
 u )  =  N )  /\  (
( u `  0
)  =  X  /\  ( u `  ( N  -  2 ) )  =  ( u `
 0 ) ) )  ->  ( # `  p
)  =  ( # `  u ) ) )
5453adantr 463 . . . . . . . . . . . 12  |-  ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) ) )  -> 
( ( ( u  e. Word  V  /\  ( # `
 u )  =  N )  /\  (
( u `  0
)  =  X  /\  ( u `  ( N  -  2 ) )  =  ( u `
 0 ) ) )  ->  ( # `  p
)  =  ( # `  u ) ) )
5554imp 427 . . . . . . . . . . 11  |-  ( ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) ) )  /\  ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) ) )  ->  ( # `  p
)  =  ( # `  u ) )
5655adantl 464 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  ( ( ( p  e. Word  V  /\  ( # `
 p )  =  N )  /\  (
( p `  0
)  =  X  /\  ( p `  ( N  -  2 ) )  =  ( p `
 0 ) ) )  /\  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  /\  ( ( u ` 
0 )  =  X  /\  ( u `  ( N  -  2
) )  =  ( u `  0 ) ) ) ) )  ->  ( # `  p
)  =  ( # `  u ) )
5756biantrurd 506 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  ( ( ( p  e. Word  V  /\  ( # `
 p )  =  N )  /\  (
( p `  0
)  =  X  /\  ( p `  ( N  -  2 ) )  =  ( p `
 0 ) ) )  /\  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  /\  ( ( u ` 
0 )  =  X  /\  ( u `  ( N  -  2
) )  =  ( u `  0 ) ) ) ) )  ->  ( ( ( p substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  =  (
u substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  /\  (
p `  ( ( # `
 p )  - 
2 ) )  =  ( u `  (
( # `  p )  -  2 ) )  /\  ( lastS  `  p
)  =  ( lastS  `  u
) )  <->  ( ( # `
 p )  =  ( # `  u
)  /\  ( (
p substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  =  (
u substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  /\  (
p `  ( ( # `
 p )  - 
2 ) )  =  ( u `  (
( # `  p )  -  2 ) )  /\  ( lastS  `  p
)  =  ( lastS  `  u
) ) ) ) )
58 3anan12 987 . . . . . . . . . . 11  |-  ( ( ( p substr  <. 0 ,  ( ( # `  p )  -  2 ) >. )  =  ( u substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  /\  (
p `  ( ( # `
 p )  - 
2 ) )  =  ( u `  (
( # `  p )  -  2 ) )  /\  ( lastS  `  p
)  =  ( lastS  `  u
) )  <->  ( (
p `  ( ( # `
 p )  - 
2 ) )  =  ( u `  (
( # `  p )  -  2 ) )  /\  ( ( p substr  <. 0 ,  ( (
# `  p )  -  2 ) >.
)  =  ( u substr  <. 0 ,  ( (
# `  p )  -  2 ) >.
)  /\  ( lastS  `  p
)  =  ( lastS  `  u
) ) ) )
5958a1i 11 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  ( ( ( p  e. Word  V  /\  ( # `
 p )  =  N )  /\  (
( p `  0
)  =  X  /\  ( p `  ( N  -  2 ) )  =  ( p `
 0 ) ) )  /\  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  /\  ( ( u ` 
0 )  =  X  /\  ( u `  ( N  -  2
) )  =  ( u `  0 ) ) ) ) )  ->  ( ( ( p substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  =  (
u substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  /\  (
p `  ( ( # `
 p )  - 
2 ) )  =  ( u `  (
( # `  p )  -  2 ) )  /\  ( lastS  `  p
)  =  ( lastS  `  u
) )  <->  ( (
p `  ( ( # `
 p )  - 
2 ) )  =  ( u `  (
( # `  p )  -  2 ) )  /\  ( ( p substr  <. 0 ,  ( (
# `  p )  -  2 ) >.
)  =  ( u substr  <. 0 ,  ( (
# `  p )  -  2 ) >.
)  /\  ( lastS  `  p
)  =  ( lastS  `  u
) ) ) ) )
60 eqeq2 2417 . . . . . . . . . . . . . . . . . . 19  |-  ( ( p `  0 )  =  X  ->  (
( p `  ( N  -  2 ) )  =  ( p `
 0 )  <->  ( p `  ( N  -  2 ) )  =  X ) )
61 oveq1 6284 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  =  ( # `  p
)  ->  ( N  -  2 )  =  ( ( # `  p
)  -  2 ) )
6261eqcoms 2414 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
# `  p )  =  N  ->  ( N  -  2 )  =  ( ( # `  p
)  -  2 ) )
6362fveq2d 5852 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
# `  p )  =  N  ->  ( p `
 ( N  - 
2 ) )  =  ( p `  (
( # `  p )  -  2 ) ) )
6463eqeq1d 2404 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  p )  =  N  ->  ( ( p `  ( N  -  2 ) )  =  X  <->  ( p `  ( ( # `  p
)  -  2 ) )  =  X ) )
6564biimpd 207 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  p )  =  N  ->  ( ( p `  ( N  -  2 ) )  =  X  ->  (
p `  ( ( # `
 p )  - 
2 ) )  =  X ) )
6665com12 29 . . . . . . . . . . . . . . . . . . 19  |-  ( ( p `  ( N  -  2 ) )  =  X  ->  (
( # `  p )  =  N  ->  (
p `  ( ( # `
 p )  - 
2 ) )  =  X ) )
6760, 66syl6bi 228 . . . . . . . . . . . . . . . . . 18  |-  ( ( p `  0 )  =  X  ->  (
( p `  ( N  -  2 ) )  =  ( p `
 0 )  -> 
( ( # `  p
)  =  N  -> 
( p `  (
( # `  p )  -  2 ) )  =  X ) ) )
6867imp 427 . . . . . . . . . . . . . . . . 17  |-  ( ( ( p `  0
)  =  X  /\  ( p `  ( N  -  2 ) )  =  ( p `
 0 ) )  ->  ( ( # `  p )  =  N  ->  ( p `  ( ( # `  p
)  -  2 ) )  =  X ) )
6968com12 29 . . . . . . . . . . . . . . . 16  |-  ( (
# `  p )  =  N  ->  ( ( ( p `  0
)  =  X  /\  ( p `  ( N  -  2 ) )  =  ( p `
 0 ) )  ->  ( p `  ( ( # `  p
)  -  2 ) )  =  X ) )
7069adantl 464 . . . . . . . . . . . . . . 15  |-  ( ( p  e. Word  V  /\  ( # `  p )  =  N )  -> 
( ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) )  ->  (
p `  ( ( # `
 p )  - 
2 ) )  =  X ) )
7170imp 427 . . . . . . . . . . . . . 14  |-  ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) ) )  -> 
( p `  (
( # `  p )  -  2 ) )  =  X )
7271adantr 463 . . . . . . . . . . . . 13  |-  ( ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) ) )  /\  ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) ) )  ->  ( p `  ( ( # `  p
)  -  2 ) )  =  X )
73 oveq1 6284 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  p )  =  N  ->  ( (
# `  p )  -  2 )  =  ( N  -  2 ) )
7473fveq2d 5852 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  p )  =  N  ->  ( u `
 ( ( # `  p )  -  2 ) )  =  ( u `  ( N  -  2 ) ) )
75 eqeq1 2406 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( u `  0 )  =  ( u `  ( N  -  2
) )  ->  (
( u `  0
)  =  X  <->  ( u `  ( N  -  2 ) )  =  X ) )
7675eqcoms 2414 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( u `  ( N  -  2 ) )  =  ( u ` 
0 )  ->  (
( u `  0
)  =  X  <->  ( u `  ( N  -  2 ) )  =  X ) )
7776biimpd 207 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( u `  ( N  -  2 ) )  =  ( u ` 
0 )  ->  (
( u `  0
)  =  X  -> 
( u `  ( N  -  2 ) )  =  X ) )
7877impcom 428 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( u `  0
)  =  X  /\  ( u `  ( N  -  2 ) )  =  ( u `
 0 ) )  ->  ( u `  ( N  -  2
) )  =  X )
7978adantl 464 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) )  -> 
( u `  ( N  -  2 ) )  =  X )
8074, 79sylan9eq 2463 . . . . . . . . . . . . . . . . 17  |-  ( ( ( # `  p
)  =  N  /\  ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) ) )  ->  ( u `  ( ( # `  p
)  -  2 ) )  =  X )
8180ex 432 . . . . . . . . . . . . . . . 16  |-  ( (
# `  p )  =  N  ->  ( ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) )  -> 
( u `  (
( # `  p )  -  2 ) )  =  X ) )
8281adantl 464 . . . . . . . . . . . . . . 15  |-  ( ( p  e. Word  V  /\  ( # `  p )  =  N )  -> 
( ( ( u  e. Word  V  /\  ( # `
 u )  =  N )  /\  (
( u `  0
)  =  X  /\  ( u `  ( N  -  2 ) )  =  ( u `
 0 ) ) )  ->  ( u `  ( ( # `  p
)  -  2 ) )  =  X ) )
8382adantr 463 . . . . . . . . . . . . . 14  |-  ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) ) )  -> 
( ( ( u  e. Word  V  /\  ( # `
 u )  =  N )  /\  (
( u `  0
)  =  X  /\  ( u `  ( N  -  2 ) )  =  ( u `
 0 ) ) )  ->  ( u `  ( ( # `  p
)  -  2 ) )  =  X ) )
8483imp 427 . . . . . . . . . . . . 13  |-  ( ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) ) )  /\  ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) ) )  ->  ( u `  ( ( # `  p
)  -  2 ) )  =  X )
8572, 84eqtr4d 2446 . . . . . . . . . . . 12  |-  ( ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) ) )  /\  ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) ) )  ->  ( p `  ( ( # `  p
)  -  2 ) )  =  ( u `
 ( ( # `  p )  -  2 ) ) )
8685adantl 464 . . . . . . . . . . 11  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  ( ( ( p  e. Word  V  /\  ( # `
 p )  =  N )  /\  (
( p `  0
)  =  X  /\  ( p `  ( N  -  2 ) )  =  ( p `
 0 ) ) )  /\  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  /\  ( ( u ` 
0 )  =  X  /\  ( u `  ( N  -  2
) )  =  ( u `  0 ) ) ) ) )  ->  ( p `  ( ( # `  p
)  -  2 ) )  =  ( u `
 ( ( # `  p )  -  2 ) ) )
8786biantrurd 506 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  ( ( ( p  e. Word  V  /\  ( # `
 p )  =  N )  /\  (
( p `  0
)  =  X  /\  ( p `  ( N  -  2 ) )  =  ( p `
 0 ) ) )  /\  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  /\  ( ( u ` 
0 )  =  X  /\  ( u `  ( N  -  2
) )  =  ( u `  0 ) ) ) ) )  ->  ( ( ( p substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  =  (
u substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  /\  ( lastS  `  p )  =  ( lastS  `  u ) )  <->  ( (
p `  ( ( # `
 p )  - 
2 ) )  =  ( u `  (
( # `  p )  -  2 ) )  /\  ( ( p substr  <. 0 ,  ( (
# `  p )  -  2 ) >.
)  =  ( u substr  <. 0 ,  ( (
# `  p )  -  2 ) >.
)  /\  ( lastS  `  p
)  =  ( lastS  `  u
) ) ) ) )
8873opeq2d 4165 . . . . . . . . . . . . . . 15  |-  ( (
# `  p )  =  N  ->  <. 0 ,  ( ( # `  p )  -  2 ) >.  =  <. 0 ,  ( N  -  2 ) >.
)
8988oveq2d 6293 . . . . . . . . . . . . . 14  |-  ( (
# `  p )  =  N  ->  ( p substr  <. 0 ,  ( (
# `  p )  -  2 ) >.
)  =  ( p substr  <. 0 ,  ( N  -  2 ) >.
) )
9088oveq2d 6293 . . . . . . . . . . . . . 14  |-  ( (
# `  p )  =  N  ->  ( u substr  <. 0 ,  ( (
# `  p )  -  2 ) >.
)  =  ( u substr  <. 0 ,  ( N  -  2 ) >.
) )
9189, 90eqeq12d 2424 . . . . . . . . . . . . 13  |-  ( (
# `  p )  =  N  ->  ( ( p substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  =  (
u substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  <->  ( p substr  <. 0 ,  ( N  - 
2 ) >. )  =  ( u substr  <. 0 ,  ( N  - 
2 ) >. )
) )
9291ad3antlr 729 . . . . . . . . . . . 12  |-  ( ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) ) )  /\  ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) ) )  ->  ( ( p substr  <. 0 ,  ( (
# `  p )  -  2 ) >.
)  =  ( u substr  <. 0 ,  ( (
# `  p )  -  2 ) >.
)  <->  ( p substr  <. 0 ,  ( N  - 
2 ) >. )  =  ( u substr  <. 0 ,  ( N  - 
2 ) >. )
) )
9392adantl 464 . . . . . . . . . . 11  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  ( ( ( p  e. Word  V  /\  ( # `
 p )  =  N )  /\  (
( p `  0
)  =  X  /\  ( p `  ( N  -  2 ) )  =  ( p `
 0 ) ) )  /\  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  /\  ( ( u ` 
0 )  =  X  /\  ( u `  ( N  -  2
) )  =  ( u `  0 ) ) ) ) )  ->  ( ( p substr  <. 0 ,  ( (
# `  p )  -  2 ) >.
)  =  ( u substr  <. 0 ,  ( (
# `  p )  -  2 ) >.
)  <->  ( p substr  <. 0 ,  ( N  - 
2 ) >. )  =  ( u substr  <. 0 ,  ( N  - 
2 ) >. )
) )
94 lsw 12636 . . . . . . . . . . . . . . 15  |-  ( p  e. Word  V  ->  ( lastS  `  p )  =  ( p `  ( (
# `  p )  -  1 ) ) )
95 oveq1 6284 . . . . . . . . . . . . . . . 16  |-  ( (
# `  p )  =  N  ->  ( (
# `  p )  -  1 )  =  ( N  -  1 ) )
9695fveq2d 5852 . . . . . . . . . . . . . . 15  |-  ( (
# `  p )  =  N  ->  ( p `
 ( ( # `  p )  -  1 ) )  =  ( p `  ( N  -  1 ) ) )
9794, 96sylan9eq 2463 . . . . . . . . . . . . . 14  |-  ( ( p  e. Word  V  /\  ( # `  p )  =  N )  -> 
( lastS  `  p )  =  ( p `  ( N  -  1 ) ) )
9897adantr 463 . . . . . . . . . . . . 13  |-  ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) ) )  -> 
( lastS  `  p )  =  ( p `  ( N  -  1 ) ) )
99 lsw 12636 . . . . . . . . . . . . . . . 16  |-  ( u  e. Word  V  ->  ( lastS  `  u )  =  ( u `  ( (
# `  u )  -  1 ) ) )
10099adantr 463 . . . . . . . . . . . . . . 15  |-  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  -> 
( lastS  `  u )  =  ( u `  (
( # `  u )  -  1 ) ) )
101 oveq1 6284 . . . . . . . . . . . . . . . . . . 19  |-  ( N  =  ( # `  u
)  ->  ( N  -  1 )  =  ( ( # `  u
)  -  1 ) )
102101eqcoms 2414 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  u )  =  N  ->  ( N  -  1 )  =  ( ( # `  u
)  -  1 ) )
103102fveq2d 5852 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  u )  =  N  ->  ( u `
 ( N  - 
1 ) )  =  ( u `  (
( # `  u )  -  1 ) ) )
104103eqeq2d 2416 . . . . . . . . . . . . . . . 16  |-  ( (
# `  u )  =  N  ->  ( ( lastS  `  u )  =  ( u `  ( N  -  1 ) )  <-> 
( lastS  `  u )  =  ( u `  (
( # `  u )  -  1 ) ) ) )
105104adantl 464 . . . . . . . . . . . . . . 15  |-  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  -> 
( ( lastS  `  u
)  =  ( u `
 ( N  - 
1 ) )  <->  ( lastS  `  u
)  =  ( u `
 ( ( # `  u )  -  1 ) ) ) )
106100, 105mpbird 232 . . . . . . . . . . . . . 14  |-  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  -> 
( lastS  `  u )  =  ( u `  ( N  -  1 ) ) )
107106adantr 463 . . . . . . . . . . . . 13  |-  ( ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) )  -> 
( lastS  `  u )  =  ( u `  ( N  -  1 ) ) )
10898, 107eqeqan12d 2425 . . . . . . . . . . . 12  |-  ( ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) ) )  /\  ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) ) )  ->  ( ( lastS  `  p
)  =  ( lastS  `  u
)  <->  ( p `  ( N  -  1
) )  =  ( u `  ( N  -  1 ) ) ) )
109108adantl 464 . . . . . . . . . . 11  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  ( ( ( p  e. Word  V  /\  ( # `
 p )  =  N )  /\  (
( p `  0
)  =  X  /\  ( p `  ( N  -  2 ) )  =  ( p `
 0 ) ) )  /\  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  /\  ( ( u ` 
0 )  =  X  /\  ( u `  ( N  -  2
) )  =  ( u `  0 ) ) ) ) )  ->  ( ( lastS  `  p
)  =  ( lastS  `  u
)  <->  ( p `  ( N  -  1
) )  =  ( u `  ( N  -  1 ) ) ) )
11093, 109anbi12d 709 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  ( ( ( p  e. Word  V  /\  ( # `
 p )  =  N )  /\  (
( p `  0
)  =  X  /\  ( p `  ( N  -  2 ) )  =  ( p `
 0 ) ) )  /\  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  /\  ( ( u ` 
0 )  =  X  /\  ( u `  ( N  -  2
) )  =  ( u `  0 ) ) ) ) )  ->  ( ( ( p substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  =  (
u substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  /\  ( lastS  `  p )  =  ( lastS  `  u ) )  <->  ( (
p substr  <. 0 ,  ( N  -  2 )
>. )  =  (
u substr  <. 0 ,  ( N  -  2 )
>. )  /\  (
p `  ( N  -  1 ) )  =  ( u `  ( N  -  1
) ) ) ) )
11159, 87, 1103bitr2d 281 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  ( ( ( p  e. Word  V  /\  ( # `
 p )  =  N )  /\  (
( p `  0
)  =  X  /\  ( p `  ( N  -  2 ) )  =  ( p `
 0 ) ) )  /\  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  /\  ( ( u ` 
0 )  =  X  /\  ( u `  ( N  -  2
) )  =  ( u `  0 ) ) ) ) )  ->  ( ( ( p substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  =  (
u substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  /\  (
p `  ( ( # `
 p )  - 
2 ) )  =  ( u `  (
( # `  p )  -  2 ) )  /\  ( lastS  `  p
)  =  ( lastS  `  u
) )  <->  ( (
p substr  <. 0 ,  ( N  -  2 )
>. )  =  (
u substr  <. 0 ,  ( N  -  2 )
>. )  /\  (
p `  ( N  -  1 ) )  =  ( u `  ( N  -  1
) ) ) ) )
11247, 57, 1113bitr2d 281 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  ( ( ( p  e. Word  V  /\  ( # `
 p )  =  N )  /\  (
( p `  0
)  =  X  /\  ( p `  ( N  -  2 ) )  =  ( p `
 0 ) ) )  /\  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  /\  ( ( u ` 
0 )  =  X  /\  ( u `  ( N  -  2
) )  =  ( u `  0 ) ) ) ) )  ->  ( p  =  u  <->  ( ( p substr  <. 0 ,  ( N  -  2 ) >.
)  =  ( u substr  <. 0 ,  ( N  -  2 ) >.
)  /\  ( p `  ( N  -  1 ) )  =  ( u `  ( N  -  1 ) ) ) ) )
113112exbiri 620 . . . . . . 7  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( (
( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) ) )  /\  ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) ) )  ->  ( ( ( p substr  <. 0 ,  ( N  -  2 )
>. )  =  (
u substr  <. 0 ,  ( N  -  2 )
>. )  /\  (
p `  ( N  -  1 ) )  =  ( u `  ( N  -  1
) ) )  ->  p  =  u )
) )
11418, 20, 113syl2and 481 . . . . . 6  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( (
p  e.  ( X G N )  /\  u  e.  ( X G N ) )  -> 
( ( ( p substr  <. 0 ,  ( N  -  2 ) >.
)  =  ( u substr  <. 0 ,  ( N  -  2 ) >.
)  /\  ( p `  ( N  -  1 ) )  =  ( u `  ( N  -  1 ) ) )  ->  p  =  u ) ) )
115114imp 427 . . . . 5  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  ( p  e.  ( X G N )  /\  u  e.  ( X G N ) ) )  ->  ( ( ( p substr  <. 0 ,  ( N  -  2 )
>. )  =  (
u substr  <. 0 ,  ( N  -  2 )
>. )  /\  (
p `  ( N  -  1 ) )  =  ( u `  ( N  -  1
) ) )  ->  p  =  u )
)
11615, 115syl5bi 217 . . . 4  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  ( p  e.  ( X G N )  /\  u  e.  ( X G N ) ) )  ->  ( <. (
p substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( p `
 ( N  - 
1 ) ) >.  =  <. ( u substr  <. 0 ,  ( N  - 
2 ) >. ) ,  ( u `  ( N  -  1
) ) >.  ->  p  =  u ) )
11712, 116sylbid 215 . . 3  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  ( p  e.  ( X G N )  /\  u  e.  ( X G N ) ) )  ->  ( ( T `
 p )  =  ( T `  u
)  ->  p  =  u ) )
118117ralrimivva 2824 . 2  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  A. p  e.  ( X G N ) A. u  e.  ( X G N ) ( ( T `
 p )  =  ( T `  u
)  ->  p  =  u ) )
119 dff13 6146 . 2  |-  ( T : ( X G N ) -1-1-> ( ( X F ( N  -  2 ) )  X.  ( <. V ,  E >. Neighbors  X ) )  <->  ( T : ( X G N ) --> ( ( X F ( N  -  2 ) )  X.  ( <. V ,  E >. Neighbors  X ) )  /\  A. p  e.  ( X G N ) A. u  e.  ( X G N ) ( ( T `  p )  =  ( T `  u )  ->  p  =  u ) ) )
1205, 118, 119sylanbrc 662 1  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  T :
( X G N ) -1-1-> ( ( X F ( N  - 
2 ) )  X.  ( <. V ,  E >. Neighbors  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2753   {crab 2757   <.cop 3977   class class class wbr 4394    |-> cmpt 4452    X. cxp 4820   -->wf 5564   -1-1->wf1 5565   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279   RRcr 9520   0cc0 9521   1c1 9522    < clt 9657    <_ cle 9658    - cmin 9840   2c2 10625   3c3 10626   NN0cn0 10835   ZZcz 10904   ZZ>=cuz 11126   #chash 12450  Word cword 12581   lastS clsw 12582   substr csubstr 12585   USGrph cusg 24734   Neighbors cnbgra 24821   ClWWalksN cclwwlkn 25153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-3 10635  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-fzo 11853  df-hash 12451  df-word 12589  df-lsw 12590  df-concat 12591  df-s1 12592  df-substr 12593  df-s2 12867  df-usgra 24737  df-nbgra 24824  df-clwwlk 25155  df-clwwlkn 25156
This theorem is referenced by:  numclwlk1lem2f1o  25500
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