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Theorem numclwlk1lem2f1 25870
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 25868 . 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 25869 . . . . . . 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 435 . . . . . 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 728 . . . . 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 25869 . . . . . . 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 435 . . . . . 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 727 . . . . 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 2476 . . . 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 6342 . . . . . 6  |-  ( p substr  <. 0 ,  ( N  -  2 ) >.
)  e.  _V
14 fvex 5897 . . . . . 6  |-  ( p `
 ( N  - 
1 ) )  e. 
_V
1513, 14opth 4689 . . . . 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 11227 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  3
)  ->  N  e.  ( ZZ>= `  2 )
)
171, 2, 3numclwwlkovgelim 25865 . . . . . . . 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 1311 . . . . . . 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 25865 . . . . . . . 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 1311 . . . . . . 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 765 . . . . . . . . . . 11  |-  ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) ) )  ->  p  e. Word  V )
2221ad2antrl 739 . . . . . . . . . 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 777 . . . . . . . . . . 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 472 . . . . . . . . . 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 2527 . . . . . . . . . . . . . . . . 17  |-  ( N  =  ( # `  p
)  ->  ( N  e.  ( ZZ>= `  3 )  <->  (
# `  p )  e.  ( ZZ>= `  3 )
) )
2625eqcoms 2469 . . . . . . . . . . . . . . . 16  |-  ( (
# `  p )  =  N  ->  ( N  e.  ( ZZ>= `  3
)  <->  ( # `  p
)  e.  ( ZZ>= ` 
3 ) ) )
27 eluz2 11193 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  p )  e.  ( ZZ>= `  3 )  <->  ( 3  e.  ZZ  /\  ( # `  p )  e.  ZZ  /\  3  <_  ( # `  p
) ) )
28 1red 9683 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  p )  e.  ZZ  ->  1  e.  RR )
29 3re 10710 . . . . . . . . . . . . . . . . . . . . . 22  |-  3  e.  RR
3029a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  p )  e.  ZZ  ->  3  e.  RR )
31 zre 10969 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  p )  e.  ZZ  ->  ( # `  p
)  e.  RR )
3228, 30, 313jca 1194 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  p )  e.  ZZ  ->  ( 1  e.  RR  /\  3  e.  RR  /\  ( # `  p )  e.  RR ) )
33 1lt3 10806 . . . . . . . . . . . . . . . . . . . 20  |-  1  <  3
34 ltletr 9750 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 1  e.  RR  /\  3  e.  RR  /\  ( # `
 p )  e.  RR )  ->  (
( 1  <  3  /\  3  <_  ( # `  p ) )  -> 
1  <  ( # `  p
) ) )
3534expd 442 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1  e.  RR  /\  3  e.  RR  /\  ( # `
 p )  e.  RR )  ->  (
1  <  3  ->  ( 3  <_  ( # `  p
)  ->  1  <  (
# `  p )
) ) )
3632, 33, 35mpisyl 21 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  p )  e.  ZZ  ->  ( 3  <_  ( # `  p
)  ->  1  <  (
# `  p )
) )
3736imp 435 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( # `  p
)  e.  ZZ  /\  3  <_  ( # `  p
) )  ->  1  <  ( # `  p
) )
38373adant1 1032 . . . . . . . . . . . . . . . . 17  |-  ( ( 3  e.  ZZ  /\  ( # `  p )  e.  ZZ  /\  3  <_  ( # `  p
) )  ->  1  <  ( # `  p
) )
3927, 38sylbi 200 . . . . . . . . . . . . . . . 16  |-  ( (
# `  p )  e.  ( ZZ>= `  3 )  ->  1  <  ( # `  p ) )
4026, 39syl6bi 236 . . . . . . . . . . . . . . 15  |-  ( (
# `  p )  =  N  ->  ( N  e.  ( ZZ>= `  3
)  ->  1  <  (
# `  p )
) )
4140com12 32 . . . . . . . . . . . . . 14  |-  ( N  e.  ( ZZ>= `  3
)  ->  ( ( # `
 p )  =  N  ->  1  <  (
# `  p )
) )
42413ad2ant3 1037 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( ( # `
 p )  =  N  ->  1  <  (
# `  p )
) )
4342com12 32 . . . . . . . . . . . 12  |-  ( (
# `  p )  =  N  ->  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  1  <  (
# `  p )
) )
4443ad3antlr 742 . . . . . . . . . . 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 436 . . . . . . . . . 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 13076 . . . . . . . . . 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 1276 . . . . . . . . 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 2482 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( # `  p
)  =  N  /\  ( # `  u )  =  N )  -> 
( # `  p )  =  ( # `  u
) )
4948expcom 441 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  u )  =  N  ->  ( (
# `  p )  =  N  ->  ( # `  p )  =  (
# `  u )
) )
5049adantl 472 . . . . . . . . . . . . . . . 16  |-  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  -> 
( ( # `  p
)  =  N  -> 
( # `  p )  =  ( # `  u
) ) )
5150adantr 471 . . . . . . . . . . . . . . 15  |-  ( ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) )  -> 
( ( # `  p
)  =  N  -> 
( # `  p )  =  ( # `  u
) ) )
5251com12 32 . . . . . . . . . . . . . 14  |-  ( (
# `  p )  =  N  ->  ( ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) )  -> 
( # `  p )  =  ( # `  u
) ) )
5352adantl 472 . . . . . . . . . . . . 13  |-  ( ( p  e. Word  V  /\  ( # `  p )  =  N )  -> 
( ( ( u  e. Word  V  /\  ( # `
 u )  =  N )  /\  (
( u `  0
)  =  X  /\  ( u `  ( N  -  2 ) )  =  ( u `
 0 ) ) )  ->  ( # `  p
)  =  ( # `  u ) ) )
5453adantr 471 . . . . . . . . . . . 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 435 . . . . . . . . . . 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 472 . . . . . . . . . 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 515 . . . . . . . . 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 1004 . . . . . . . . . . 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 2472 . . . . . . . . . . . . . . . . . . 19  |-  ( ( p `  0 )  =  X  ->  (
( p `  ( N  -  2 ) )  =  ( p `
 0 )  <->  ( p `  ( N  -  2 ) )  =  X ) )
61 oveq1 6321 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  =  ( # `  p
)  ->  ( N  -  2 )  =  ( ( # `  p
)  -  2 ) )
6261eqcoms 2469 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
# `  p )  =  N  ->  ( N  -  2 )  =  ( ( # `  p
)  -  2 ) )
6362fveq2d 5891 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
# `  p )  =  N  ->  ( p `
 ( N  - 
2 ) )  =  ( p `  (
( # `  p )  -  2 ) ) )
6463eqeq1d 2463 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  p )  =  N  ->  ( ( p `  ( N  -  2 ) )  =  X  <->  ( p `  ( ( # `  p
)  -  2 ) )  =  X ) )
6564biimpd 212 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  p )  =  N  ->  ( ( p `  ( N  -  2 ) )  =  X  ->  (
p `  ( ( # `
 p )  - 
2 ) )  =  X ) )
6665com12 32 . . . . . . . . . . . . . . . . . . 19  |-  ( ( p `  ( N  -  2 ) )  =  X  ->  (
( # `  p )  =  N  ->  (
p `  ( ( # `
 p )  - 
2 ) )  =  X ) )
6760, 66syl6bi 236 . . . . . . . . . . . . . . . . . 18  |-  ( ( p `  0 )  =  X  ->  (
( p `  ( N  -  2 ) )  =  ( p `
 0 )  -> 
( ( # `  p
)  =  N  -> 
( p `  (
( # `  p )  -  2 ) )  =  X ) ) )
6867imp 435 . . . . . . . . . . . . . . . . 17  |-  ( ( ( p `  0
)  =  X  /\  ( p `  ( N  -  2 ) )  =  ( p `
 0 ) )  ->  ( ( # `  p )  =  N  ->  ( p `  ( ( # `  p
)  -  2 ) )  =  X ) )
6968com12 32 . . . . . . . . . . . . . . . 16  |-  ( (
# `  p )  =  N  ->  ( ( ( p `  0
)  =  X  /\  ( p `  ( N  -  2 ) )  =  ( p `
 0 ) )  ->  ( p `  ( ( # `  p
)  -  2 ) )  =  X ) )
7069adantl 472 . . . . . . . . . . . . . . 15  |-  ( ( p  e. Word  V  /\  ( # `  p )  =  N )  -> 
( ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) )  ->  (
p `  ( ( # `
 p )  - 
2 ) )  =  X ) )
7170imp 435 . . . . . . . . . . . . . 14  |-  ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) ) )  -> 
( p `  (
( # `  p )  -  2 ) )  =  X )
7271adantr 471 . . . . . . . . . . . . 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 6321 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  p )  =  N  ->  ( (
# `  p )  -  2 )  =  ( N  -  2 ) )
7473fveq2d 5891 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  p )  =  N  ->  ( u `
 ( ( # `  p )  -  2 ) )  =  ( u `  ( N  -  2 ) ) )
75 eqeq1 2465 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( u `  0 )  =  ( u `  ( N  -  2
) )  ->  (
( u `  0
)  =  X  <->  ( u `  ( N  -  2 ) )  =  X ) )
7675eqcoms 2469 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( u `  ( N  -  2 ) )  =  ( u ` 
0 )  ->  (
( u `  0
)  =  X  <->  ( u `  ( N  -  2 ) )  =  X ) )
7776biimpd 212 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( u `  ( N  -  2 ) )  =  ( u ` 
0 )  ->  (
( u `  0
)  =  X  -> 
( u `  ( N  -  2 ) )  =  X ) )
7877impcom 436 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( u `  0
)  =  X  /\  ( u `  ( N  -  2 ) )  =  ( u `
 0 ) )  ->  ( u `  ( N  -  2
) )  =  X )
7978adantl 472 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) )  -> 
( u `  ( N  -  2 ) )  =  X )
8074, 79sylan9eq 2515 . . . . . . . . . . . . . . . . 17  |-  ( ( ( # `  p
)  =  N  /\  ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) ) )  ->  ( u `  ( ( # `  p
)  -  2 ) )  =  X )
8180ex 440 . . . . . . . . . . . . . . . 16  |-  ( (
# `  p )  =  N  ->  ( ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) )  -> 
( u `  (
( # `  p )  -  2 ) )  =  X ) )
8281adantl 472 . . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . . 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 435 . . . . . . . . . . . . 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 2498 . . . . . . . . . . . 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 472 . . . . . . . . . . 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 515 . . . . . . . . . 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 4186 . . . . . . . . . . . . . . 15  |-  ( (
# `  p )  =  N  ->  <. 0 ,  ( ( # `  p )  -  2 ) >.  =  <. 0 ,  ( N  -  2 ) >.
)
8988oveq2d 6330 . . . . . . . . . . . . . 14  |-  ( (
# `  p )  =  N  ->  ( p substr  <. 0 ,  ( (
# `  p )  -  2 ) >.
)  =  ( p substr  <. 0 ,  ( N  -  2 ) >.
) )
9088oveq2d 6330 . . . . . . . . . . . . . 14  |-  ( (
# `  p )  =  N  ->  ( u substr  <. 0 ,  ( (
# `  p )  -  2 ) >.
)  =  ( u substr  <. 0 ,  ( N  -  2 ) >.
) )
9189, 90eqeq12d 2476 . . . . . . . . . . . . 13  |-  ( (
# `  p )  =  N  ->  ( ( p substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  =  (
u substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  <->  ( p substr  <. 0 ,  ( N  - 
2 ) >. )  =  ( u substr  <. 0 ,  ( N  - 
2 ) >. )
) )
9291ad3antlr 742 . . . . . . . . . . . 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 472 . . . . . . . . . . 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 12746 . . . . . . . . . . . . . . 15  |-  ( p  e. Word  V  ->  ( lastS  `  p )  =  ( p `  ( (
# `  p )  -  1 ) ) )
95 oveq1 6321 . . . . . . . . . . . . . . . 16  |-  ( (
# `  p )  =  N  ->  ( (
# `  p )  -  1 )  =  ( N  -  1 ) )
9695fveq2d 5891 . . . . . . . . . . . . . . 15  |-  ( (
# `  p )  =  N  ->  ( p `
 ( ( # `  p )  -  1 ) )  =  ( p `  ( N  -  1 ) ) )
9794, 96sylan9eq 2515 . . . . . . . . . . . . . 14  |-  ( ( p  e. Word  V  /\  ( # `  p )  =  N )  -> 
( lastS  `  p )  =  ( p `  ( N  -  1 ) ) )
9897adantr 471 . . . . . . . . . . . . 13  |-  ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) ) )  -> 
( lastS  `  p )  =  ( p `  ( N  -  1 ) ) )
99 lsw 12746 . . . . . . . . . . . . . . . 16  |-  ( u  e. Word  V  ->  ( lastS  `  u )  =  ( u `  ( (
# `  u )  -  1 ) ) )
10099adantr 471 . . . . . . . . . . . . . . 15  |-  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  -> 
( lastS  `  u )  =  ( u `  (
( # `  u )  -  1 ) ) )
101 oveq1 6321 . . . . . . . . . . . . . . . . . . 19  |-  ( N  =  ( # `  u
)  ->  ( N  -  1 )  =  ( ( # `  u
)  -  1 ) )
102101eqcoms 2469 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  u )  =  N  ->  ( N  -  1 )  =  ( ( # `  u
)  -  1 ) )
103102fveq2d 5891 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  u )  =  N  ->  ( u `
 ( N  - 
1 ) )  =  ( u `  (
( # `  u )  -  1 ) ) )
104103eqeq2d 2471 . . . . . . . . . . . . . . . 16  |-  ( (
# `  u )  =  N  ->  ( ( lastS  `  u )  =  ( u `  ( N  -  1 ) )  <-> 
( lastS  `  u )  =  ( u `  (
( # `  u )  -  1 ) ) ) )
105104adantl 472 . . . . . . . . . . . . . . 15  |-  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  -> 
( ( lastS  `  u
)  =  ( u `
 ( N  - 
1 ) )  <->  ( lastS  `  u
)  =  ( u `
 ( ( # `  u )  -  1 ) ) ) )
106100, 105mpbird 240 . . . . . . . . . . . . . 14  |-  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  -> 
( lastS  `  u )  =  ( u `  ( N  -  1 ) ) )
107106adantr 471 . . . . . . . . . . . . 13  |-  ( ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) )  -> 
( lastS  `  u )  =  ( u `  ( N  -  1 ) ) )
10898, 107eqeqan12d 2477 . . . . . . . . . . . 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 472 . . . . . . . . . . 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 722 . . . . . . . . . 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 289 . . . . . . . . 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 289 . . . . . . . 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 632 . . . . . . 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 490 . . . . . 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 435 . . . . 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 225 . . . 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 223 . . 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 2820 . 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 6183 . 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 675 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 189    /\ wa 375    /\ w3a 991    = wceq 1454    e. wcel 1897   A.wral 2748   {crab 2752   <.cop 3985   class class class wbr 4415    |-> cmpt 4474    X. cxp 4850   -->wf 5596   -1-1->wf1 5597   ` cfv 5600  (class class class)co 6314    |-> cmpt2 6316   RRcr 9563   0cc0 9564   1c1 9565    < clt 9700    <_ cle 9701    - cmin 9885   2c2 10686   3c3 10687   NN0cn0 10897   ZZcz 10965   ZZ>=cuz 11187   #chash 12546  Word cword 12688   lastS clsw 12689   substr csubstr 12692   USGrph cusg 25105   Neighbors cnbgra 25193   ClWWalksN cclwwlkn 25525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-map 7499  df-pm 7500  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-card 8398  df-cda 8623  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-2 10695  df-3 10696  df-n0 10898  df-z 10966  df-uz 11188  df-fz 11813  df-fzo 11946  df-hash 12547  df-word 12696  df-lsw 12697  df-concat 12698  df-s1 12699  df-substr 12700  df-s2 12980  df-usgra 25108  df-nbgra 25196  df-clwwlk 25527  df-clwwlkn 25528
This theorem is referenced by:  numclwlk1lem2f1o  25872
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