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Theorem numclwlk1lem2f1 25814
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 25812 . 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 25813 . . . . . . 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 431 . . . . . 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 722 . . . . 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 25813 . . . . . . 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 431 . . . . . 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 721 . . . . 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 2445 . . . 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 6331 . . . . . 6  |-  ( p substr  <. 0 ,  ( N  -  2 ) >.
)  e.  _V
14 fvex 5889 . . . . . 6  |-  ( p `
 ( N  - 
1 ) )  e. 
_V
1513, 14opth 4693 . . . . 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 11201 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  3
)  ->  N  e.  ( ZZ>= `  2 )
)
171, 2, 3numclwwlkovgelim 25809 . . . . . . . 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 1300 . . . . . . 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 25809 . . . . . . . 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 1300 . . . . . . 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 759 . . . . . . . . . . 11  |-  ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) ) )  ->  p  e. Word  V )
2221ad2antrl 733 . . . . . . . . . 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 771 . . . . . . . . . . 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 468 . . . . . . . . . 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 2495 . . . . . . . . . . . . . . . . 17  |-  ( N  =  ( # `  p
)  ->  ( N  e.  ( ZZ>= `  3 )  <->  (
# `  p )  e.  ( ZZ>= `  3 )
) )
2625eqcoms 2435 . . . . . . . . . . . . . . . 16  |-  ( (
# `  p )  =  N  ->  ( N  e.  ( ZZ>= `  3
)  <->  ( # `  p
)  e.  ( ZZ>= ` 
3 ) ) )
27 eluz2 11167 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  p )  e.  ( ZZ>= `  3 )  <->  ( 3  e.  ZZ  /\  ( # `  p )  e.  ZZ  /\  3  <_  ( # `  p
) ) )
28 1red 9660 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  p )  e.  ZZ  ->  1  e.  RR )
29 3re 10685 . . . . . . . . . . . . . . . . . . . . . 22  |-  3  e.  RR
3029a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  p )  e.  ZZ  ->  3  e.  RR )
31 zre 10943 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  p )  e.  ZZ  ->  ( # `  p
)  e.  RR )
3228, 30, 313jca 1186 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  p )  e.  ZZ  ->  ( 1  e.  RR  /\  3  e.  RR  /\  ( # `  p )  e.  RR ) )
33 1lt3 10780 . . . . . . . . . . . . . . . . . . . 20  |-  1  <  3
34 ltletr 9727 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 1  e.  RR  /\  3  e.  RR  /\  ( # `
 p )  e.  RR )  ->  (
( 1  <  3  /\  3  <_  ( # `  p ) )  -> 
1  <  ( # `  p
) ) )
3534expd 438 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1  e.  RR  /\  3  e.  RR  /\  ( # `
 p )  e.  RR )  ->  (
1  <  3  ->  ( 3  <_  ( # `  p
)  ->  1  <  (
# `  p )
) ) )
3632, 33, 35mpisyl 22 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  p )  e.  ZZ  ->  ( 3  <_  ( # `  p
)  ->  1  <  (
# `  p )
) )
3736imp 431 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( # `  p
)  e.  ZZ  /\  3  <_  ( # `  p
) )  ->  1  <  ( # `  p
) )
38373adant1 1024 . . . . . . . . . . . . . . . . 17  |-  ( ( 3  e.  ZZ  /\  ( # `  p )  e.  ZZ  /\  3  <_  ( # `  p
) )  ->  1  <  ( # `  p
) )
3927, 38sylbi 199 . . . . . . . . . . . . . . . 16  |-  ( (
# `  p )  e.  ( ZZ>= `  3 )  ->  1  <  ( # `  p ) )
4026, 39syl6bi 232 . . . . . . . . . . . . . . 15  |-  ( (
# `  p )  =  N  ->  ( N  e.  ( ZZ>= `  3
)  ->  1  <  (
# `  p )
) )
4140com12 33 . . . . . . . . . . . . . 14  |-  ( N  e.  ( ZZ>= `  3
)  ->  ( ( # `
 p )  =  N  ->  1  <  (
# `  p )
) )
42413ad2ant3 1029 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( ( # `
 p )  =  N  ->  1  <  (
# `  p )
) )
4342com12 33 . . . . . . . . . . . 12  |-  ( (
# `  p )  =  N  ->  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  1  <  (
# `  p )
) )
4443ad3antlr 736 . . . . . . . . . . 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 432 . . . . . . . . . 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 13022 . . . . . . . . . 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 1265 . . . . . . . . 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 2451 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( # `  p
)  =  N  /\  ( # `  u )  =  N )  -> 
( # `  p )  =  ( # `  u
) )
4948expcom 437 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  u )  =  N  ->  ( (
# `  p )  =  N  ->  ( # `  p )  =  (
# `  u )
) )
5049adantl 468 . . . . . . . . . . . . . . . 16  |-  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  -> 
( ( # `  p
)  =  N  -> 
( # `  p )  =  ( # `  u
) ) )
5150adantr 467 . . . . . . . . . . . . . . 15  |-  ( ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) )  -> 
( ( # `  p
)  =  N  -> 
( # `  p )  =  ( # `  u
) ) )
5251com12 33 . . . . . . . . . . . . . 14  |-  ( (
# `  p )  =  N  ->  ( ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) )  -> 
( # `  p )  =  ( # `  u
) ) )
5352adantl 468 . . . . . . . . . . . . 13  |-  ( ( p  e. Word  V  /\  ( # `  p )  =  N )  -> 
( ( ( u  e. Word  V  /\  ( # `
 u )  =  N )  /\  (
( u `  0
)  =  X  /\  ( u `  ( N  -  2 ) )  =  ( u `
 0 ) ) )  ->  ( # `  p
)  =  ( # `  u ) ) )
5453adantr 467 . . . . . . . . . . . 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 431 . . . . . . . . . . 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 468 . . . . . . . . . 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 511 . . . . . . . . 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 996 . . . . . . . . . . 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 2438 . . . . . . . . . . . . . . . . . . 19  |-  ( ( p `  0 )  =  X  ->  (
( p `  ( N  -  2 ) )  =  ( p `
 0 )  <->  ( p `  ( N  -  2 ) )  =  X ) )
61 oveq1 6310 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  =  ( # `  p
)  ->  ( N  -  2 )  =  ( ( # `  p
)  -  2 ) )
6261eqcoms 2435 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
# `  p )  =  N  ->  ( N  -  2 )  =  ( ( # `  p
)  -  2 ) )
6362fveq2d 5883 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
# `  p )  =  N  ->  ( p `
 ( N  - 
2 ) )  =  ( p `  (
( # `  p )  -  2 ) ) )
6463eqeq1d 2425 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  p )  =  N  ->  ( ( p `  ( N  -  2 ) )  =  X  <->  ( p `  ( ( # `  p
)  -  2 ) )  =  X ) )
6564biimpd 211 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  p )  =  N  ->  ( ( p `  ( N  -  2 ) )  =  X  ->  (
p `  ( ( # `
 p )  - 
2 ) )  =  X ) )
6665com12 33 . . . . . . . . . . . . . . . . . . 19  |-  ( ( p `  ( N  -  2 ) )  =  X  ->  (
( # `  p )  =  N  ->  (
p `  ( ( # `
 p )  - 
2 ) )  =  X ) )
6760, 66syl6bi 232 . . . . . . . . . . . . . . . . . 18  |-  ( ( p `  0 )  =  X  ->  (
( p `  ( N  -  2 ) )  =  ( p `
 0 )  -> 
( ( # `  p
)  =  N  -> 
( p `  (
( # `  p )  -  2 ) )  =  X ) ) )
6867imp 431 . . . . . . . . . . . . . . . . 17  |-  ( ( ( p `  0
)  =  X  /\  ( p `  ( N  -  2 ) )  =  ( p `
 0 ) )  ->  ( ( # `  p )  =  N  ->  ( p `  ( ( # `  p
)  -  2 ) )  =  X ) )
6968com12 33 . . . . . . . . . . . . . . . 16  |-  ( (
# `  p )  =  N  ->  ( ( ( p `  0
)  =  X  /\  ( p `  ( N  -  2 ) )  =  ( p `
 0 ) )  ->  ( p `  ( ( # `  p
)  -  2 ) )  =  X ) )
7069adantl 468 . . . . . . . . . . . . . . 15  |-  ( ( p  e. Word  V  /\  ( # `  p )  =  N )  -> 
( ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) )  ->  (
p `  ( ( # `
 p )  - 
2 ) )  =  X ) )
7170imp 431 . . . . . . . . . . . . . 14  |-  ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) ) )  -> 
( p `  (
( # `  p )  -  2 ) )  =  X )
7271adantr 467 . . . . . . . . . . . . 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 6310 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  p )  =  N  ->  ( (
# `  p )  -  2 )  =  ( N  -  2 ) )
7473fveq2d 5883 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  p )  =  N  ->  ( u `
 ( ( # `  p )  -  2 ) )  =  ( u `  ( N  -  2 ) ) )
75 eqeq1 2427 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( u `  0 )  =  ( u `  ( N  -  2
) )  ->  (
( u `  0
)  =  X  <->  ( u `  ( N  -  2 ) )  =  X ) )
7675eqcoms 2435 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( u `  ( N  -  2 ) )  =  ( u ` 
0 )  ->  (
( u `  0
)  =  X  <->  ( u `  ( N  -  2 ) )  =  X ) )
7776biimpd 211 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( u `  ( N  -  2 ) )  =  ( u ` 
0 )  ->  (
( u `  0
)  =  X  -> 
( u `  ( N  -  2 ) )  =  X ) )
7877impcom 432 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( u `  0
)  =  X  /\  ( u `  ( N  -  2 ) )  =  ( u `
 0 ) )  ->  ( u `  ( N  -  2
) )  =  X )
7978adantl 468 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) )  -> 
( u `  ( N  -  2 ) )  =  X )
8074, 79sylan9eq 2484 . . . . . . . . . . . . . . . . 17  |-  ( ( ( # `  p
)  =  N  /\  ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) ) )  ->  ( u `  ( ( # `  p
)  -  2 ) )  =  X )
8180ex 436 . . . . . . . . . . . . . . . 16  |-  ( (
# `  p )  =  N  ->  ( ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) )  -> 
( u `  (
( # `  p )  -  2 ) )  =  X ) )
8281adantl 468 . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . 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 431 . . . . . . . . . . . . 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 2467 . . . . . . . . . . . 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 468 . . . . . . . . . . 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 511 . . . . . . . . . 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 4192 . . . . . . . . . . . . . . 15  |-  ( (
# `  p )  =  N  ->  <. 0 ,  ( ( # `  p )  -  2 ) >.  =  <. 0 ,  ( N  -  2 ) >.
)
8988oveq2d 6319 . . . . . . . . . . . . . 14  |-  ( (
# `  p )  =  N  ->  ( p substr  <. 0 ,  ( (
# `  p )  -  2 ) >.
)  =  ( p substr  <. 0 ,  ( N  -  2 ) >.
) )
9088oveq2d 6319 . . . . . . . . . . . . . 14  |-  ( (
# `  p )  =  N  ->  ( u substr  <. 0 ,  ( (
# `  p )  -  2 ) >.
)  =  ( u substr  <. 0 ,  ( N  -  2 ) >.
) )
9189, 90eqeq12d 2445 . . . . . . . . . . . . 13  |-  ( (
# `  p )  =  N  ->  ( ( p substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  =  (
u substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  <->  ( p substr  <. 0 ,  ( N  - 
2 ) >. )  =  ( u substr  <. 0 ,  ( N  - 
2 ) >. )
) )
9291ad3antlr 736 . . . . . . . . . . . 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 468 . . . . . . . . . . 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 12709 . . . . . . . . . . . . . . 15  |-  ( p  e. Word  V  ->  ( lastS  `  p )  =  ( p `  ( (
# `  p )  -  1 ) ) )
95 oveq1 6310 . . . . . . . . . . . . . . . 16  |-  ( (
# `  p )  =  N  ->  ( (
# `  p )  -  1 )  =  ( N  -  1 ) )
9695fveq2d 5883 . . . . . . . . . . . . . . 15  |-  ( (
# `  p )  =  N  ->  ( p `
 ( ( # `  p )  -  1 ) )  =  ( p `  ( N  -  1 ) ) )
9794, 96sylan9eq 2484 . . . . . . . . . . . . . 14  |-  ( ( p  e. Word  V  /\  ( # `  p )  =  N )  -> 
( lastS  `  p )  =  ( p `  ( N  -  1 ) ) )
9897adantr 467 . . . . . . . . . . . . 13  |-  ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) ) )  -> 
( lastS  `  p )  =  ( p `  ( N  -  1 ) ) )
99 lsw 12709 . . . . . . . . . . . . . . . 16  |-  ( u  e. Word  V  ->  ( lastS  `  u )  =  ( u `  ( (
# `  u )  -  1 ) ) )
10099adantr 467 . . . . . . . . . . . . . . 15  |-  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  -> 
( lastS  `  u )  =  ( u `  (
( # `  u )  -  1 ) ) )
101 oveq1 6310 . . . . . . . . . . . . . . . . . . 19  |-  ( N  =  ( # `  u
)  ->  ( N  -  1 )  =  ( ( # `  u
)  -  1 ) )
102101eqcoms 2435 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  u )  =  N  ->  ( N  -  1 )  =  ( ( # `  u
)  -  1 ) )
103102fveq2d 5883 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  u )  =  N  ->  ( u `
 ( N  - 
1 ) )  =  ( u `  (
( # `  u )  -  1 ) ) )
104103eqeq2d 2437 . . . . . . . . . . . . . . . 16  |-  ( (
# `  u )  =  N  ->  ( ( lastS  `  u )  =  ( u `  ( N  -  1 ) )  <-> 
( lastS  `  u )  =  ( u `  (
( # `  u )  -  1 ) ) ) )
105104adantl 468 . . . . . . . . . . . . . . 15  |-  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  -> 
( ( lastS  `  u
)  =  ( u `
 ( N  - 
1 ) )  <->  ( lastS  `  u
)  =  ( u `
 ( ( # `  u )  -  1 ) ) ) )
106100, 105mpbird 236 . . . . . . . . . . . . . 14  |-  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  -> 
( lastS  `  u )  =  ( u `  ( N  -  1 ) ) )
107106adantr 467 . . . . . . . . . . . . 13  |-  ( ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) )  -> 
( lastS  `  u )  =  ( u `  ( N  -  1 ) ) )
10898, 107eqeqan12d 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
) ) ) )  ->  ( ( lastS  `  p
)  =  ( lastS  `  u
)  <->  ( p `  ( N  -  1
) )  =  ( u `  ( N  -  1 ) ) ) )
109108adantl 468 . . . . . . . . . . 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 716 . . . . . . . . . 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 285 . . . . . . . . 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 285 . . . . . . . 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 627 . . . . . . 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 486 . . . . . 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 431 . . . . 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 221 . . . 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 219 . . 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 2847 . 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 6172 . 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 669 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 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869   A.wral 2776   {crab 2780   <.cop 4003   class class class wbr 4421    |-> cmpt 4480    X. cxp 4849   -->wf 5595   -1-1->wf1 5596   ` cfv 5599  (class class class)co 6303    |-> cmpt2 6305   RRcr 9540   0cc0 9541   1c1 9542    < clt 9677    <_ cle 9678    - cmin 9862   2c2 10661   3c3 10662   NN0cn0 10871   ZZcz 10939   ZZ>=cuz 11161   #chash 12516  Word cword 12654   lastS clsw 12655   substr csubstr 12658   USGrph cusg 25049   Neighbors cnbgra 25137   ClWWalksN cclwwlkn 25469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-oadd 7192  df-er 7369  df-map 7480  df-pm 7481  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-card 8376  df-cda 8600  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-nn 10612  df-2 10670  df-3 10671  df-n0 10872  df-z 10940  df-uz 11162  df-fz 11787  df-fzo 11918  df-hash 12517  df-word 12662  df-lsw 12663  df-concat 12664  df-s1 12665  df-substr 12666  df-s2 12940  df-usgra 25052  df-nbgra 25140  df-clwwlk 25471  df-clwwlkn 25472
This theorem is referenced by:  numclwlk1lem2f1o  25816
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