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Theorem numclwlk1lem2f1 30699
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 30697 . 2  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  T :
( X G N ) --> ( ( X F ( N  - 
2 ) )  X.  ( <. V ,  E >. Neighbors  X ) ) )
6 simpl 457 . . . . . 6  |-  ( ( p  e.  ( X G N )  /\  u  e.  ( X G N ) )  ->  p  e.  ( X G N ) )
71, 2, 3, 4numclwlk1lem2fv 30698 . . . . . . 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 ) ) >.
) )
87imp 429 . . . . . 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 ) ) >.
)
96, 8sylan2 474 . . . . 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 ) ) >.
)
101, 2, 3, 4numclwlk1lem2fv 30698 . . . . . . 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 ) ) >.
) )
1110imp 429 . . . . . 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 ) ) >.
)
1211adantrl 715 . . . . 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 ) ) >.
)
139, 12eqeq12d 2457 . . . 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 ) ) >.
) )
14 ovex 6128 . . . . . 6  |-  ( p substr  <. 0 ,  ( N  -  2 ) >.
)  e.  _V
15 fvex 5713 . . . . . 6  |-  ( p `
 ( N  - 
1 ) )  e. 
_V
1614, 15opth 4578 . . . . 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 ) ) ) )
17 uzuzle23 30205 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  3
)  ->  N  e.  ( ZZ>= `  2 )
)
181, 2, 3numclwwlkovgelim 30694 . . . . . . . 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 ) ) ) ) )
1917, 18syl3an3 1253 . . . . . . 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 ) ) ) ) )
201, 2, 3numclwwlkovgelim 30694 . . . . . . . 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 ) ) ) ) )
2117, 20syl3an3 1253 . . . . . . 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 ) ) ) ) )
22 simpl 457 . . . . . . . . . . . 12  |-  ( ( p  e. Word  V  /\  ( # `  p )  =  N )  ->  p  e. Word  V )
2322adantr 465 . . . . . . . . . . 11  |-  ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) ) )  ->  p  e. Word  V )
2423ad2antrl 727 . . . . . . . . . 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
)
25 simprll 761 . . . . . . . . . . 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
)
2625adantl 466 . . . . . . . . . 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
)
27 eleq1 2503 . . . . . . . . . . . . . . . . 17  |-  ( N  =  ( # `  p
)  ->  ( N  e.  ( ZZ>= `  3 )  <->  (
# `  p )  e.  ( ZZ>= `  3 )
) )
2827eqcoms 2446 . . . . . . . . . . . . . . . 16  |-  ( (
# `  p )  =  N  ->  ( N  e.  ( ZZ>= `  3
)  <->  ( # `  p
)  e.  ( ZZ>= ` 
3 ) ) )
29 eluz2 10879 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  p )  e.  ( ZZ>= `  3 )  <->  ( 3  e.  ZZ  /\  ( # `  p )  e.  ZZ  /\  3  <_  ( # `  p
) ) )
30 1re 9397 . . . . . . . . . . . . . . . . . . . . . 22  |-  1  e.  RR
3130a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  p )  e.  ZZ  ->  1  e.  RR )
32 3re 10407 . . . . . . . . . . . . . . . . . . . . . 22  |-  3  e.  RR
3332a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  p )  e.  ZZ  ->  3  e.  RR )
34 zre 10662 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  p )  e.  ZZ  ->  ( # `  p
)  e.  RR )
3531, 33, 343jca 1168 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  p )  e.  ZZ  ->  ( 1  e.  RR  /\  3  e.  RR  /\  ( # `  p )  e.  RR ) )
36 1lt3 10502 . . . . . . . . . . . . . . . . . . . 20  |-  1  <  3
37 ltletr 9478 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 1  e.  RR  /\  3  e.  RR  /\  ( # `
 p )  e.  RR )  ->  (
( 1  <  3  /\  3  <_  ( # `  p ) )  -> 
1  <  ( # `  p
) ) )
3837expd 436 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1  e.  RR  /\  3  e.  RR  /\  ( # `
 p )  e.  RR )  ->  (
1  <  3  ->  ( 3  <_  ( # `  p
)  ->  1  <  (
# `  p )
) ) )
3935, 36, 38mpisyl 18 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  p )  e.  ZZ  ->  ( 3  <_  ( # `  p
)  ->  1  <  (
# `  p )
) )
4039imp 429 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( # `  p
)  e.  ZZ  /\  3  <_  ( # `  p
) )  ->  1  <  ( # `  p
) )
41403adant1 1006 . . . . . . . . . . . . . . . . 17  |-  ( ( 3  e.  ZZ  /\  ( # `  p )  e.  ZZ  /\  3  <_  ( # `  p
) )  ->  1  <  ( # `  p
) )
4229, 41sylbi 195 . . . . . . . . . . . . . . . 16  |-  ( (
# `  p )  e.  ( ZZ>= `  3 )  ->  1  <  ( # `  p ) )
4328, 42syl6bi 228 . . . . . . . . . . . . . . 15  |-  ( (
# `  p )  =  N  ->  ( N  e.  ( ZZ>= `  3
)  ->  1  <  (
# `  p )
) )
4443com12 31 . . . . . . . . . . . . . 14  |-  ( N  e.  ( ZZ>= `  3
)  ->  ( ( # `
 p )  =  N  ->  1  <  (
# `  p )
) )
45443ad2ant3 1011 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( ( # `
 p )  =  N  ->  1  <  (
# `  p )
) )
4645com12 31 . . . . . . . . . . . 12  |-  ( (
# `  p )  =  N  ->  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  1  <  (
# `  p )
) )
4746ad3antlr 730 . . . . . . . . . . 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 ) ) )
4847impcom 430 . . . . . . . . . 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 ) )
49 2swrd2eqwrdeq 12565 . . . . . . . . . 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
) ) ) ) )
5024, 26, 48, 49syl3anc 1218 . . . . . . . . 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
) ) ) ) )
51 eqtr3 2462 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( # `  p
)  =  N  /\  ( # `  u )  =  N )  -> 
( # `  p )  =  ( # `  u
) )
5251expcom 435 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  u )  =  N  ->  ( (
# `  p )  =  N  ->  ( # `  p )  =  (
# `  u )
) )
5352adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  -> 
( ( # `  p
)  =  N  -> 
( # `  p )  =  ( # `  u
) ) )
5453adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) )  -> 
( ( # `  p
)  =  N  -> 
( # `  p )  =  ( # `  u
) ) )
5554com12 31 . . . . . . . . . . . . . 14  |-  ( (
# `  p )  =  N  ->  ( ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) )  -> 
( # `  p )  =  ( # `  u
) ) )
5655adantl 466 . . . . . . . . . . . . 13  |-  ( ( p  e. Word  V  /\  ( # `  p )  =  N )  -> 
( ( ( u  e. Word  V  /\  ( # `
 u )  =  N )  /\  (
( u `  0
)  =  X  /\  ( u `  ( N  -  2 ) )  =  ( u `
 0 ) ) )  ->  ( # `  p
)  =  ( # `  u ) ) )
5756adantr 465 . . . . . . . . . . . 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 ) ) )
5857imp 429 . . . . . . . . . . 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 ) )
5958adantl 466 . . . . . . . . . 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 ) )
6059biantrurd 508 . . . . . . . . 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
) ) ) ) )
61 3anan12 978 . . . . . . . . . . 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
) ) ) )
6261a1i 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
) ) ) ) )
63 eqeq2 2452 . . . . . . . . . . . . . . . . . . 19  |-  ( ( p `  0 )  =  X  ->  (
( p `  ( N  -  2 ) )  =  ( p `
 0 )  <->  ( p `  ( N  -  2 ) )  =  X ) )
64 oveq1 6110 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  =  ( # `  p
)  ->  ( N  -  2 )  =  ( ( # `  p
)  -  2 ) )
6564eqcoms 2446 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
# `  p )  =  N  ->  ( N  -  2 )  =  ( ( # `  p
)  -  2 ) )
6665fveq2d 5707 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
# `  p )  =  N  ->  ( p `
 ( N  - 
2 ) )  =  ( p `  (
( # `  p )  -  2 ) ) )
6766eqeq1d 2451 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  p )  =  N  ->  ( ( p `  ( N  -  2 ) )  =  X  <->  ( p `  ( ( # `  p
)  -  2 ) )  =  X ) )
6867biimpd 207 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  p )  =  N  ->  ( ( p `  ( N  -  2 ) )  =  X  ->  (
p `  ( ( # `
 p )  - 
2 ) )  =  X ) )
6968com12 31 . . . . . . . . . . . . . . . . . . 19  |-  ( ( p `  ( N  -  2 ) )  =  X  ->  (
( # `  p )  =  N  ->  (
p `  ( ( # `
 p )  - 
2 ) )  =  X ) )
7063, 69syl6bi 228 . . . . . . . . . . . . . . . . . 18  |-  ( ( p `  0 )  =  X  ->  (
( p `  ( N  -  2 ) )  =  ( p `
 0 )  -> 
( ( # `  p
)  =  N  -> 
( p `  (
( # `  p )  -  2 ) )  =  X ) ) )
7170imp 429 . . . . . . . . . . . . . . . . 17  |-  ( ( ( p `  0
)  =  X  /\  ( p `  ( N  -  2 ) )  =  ( p `
 0 ) )  ->  ( ( # `  p )  =  N  ->  ( p `  ( ( # `  p
)  -  2 ) )  =  X ) )
7271com12 31 . . . . . . . . . . . . . . . 16  |-  ( (
# `  p )  =  N  ->  ( ( ( p `  0
)  =  X  /\  ( p `  ( N  -  2 ) )  =  ( p `
 0 ) )  ->  ( p `  ( ( # `  p
)  -  2 ) )  =  X ) )
7372adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( p  e. Word  V  /\  ( # `  p )  =  N )  -> 
( ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) )  ->  (
p `  ( ( # `
 p )  - 
2 ) )  =  X ) )
7473imp 429 . . . . . . . . . . . . . 14  |-  ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) ) )  -> 
( p `  (
( # `  p )  -  2 ) )  =  X )
7574adantr 465 . . . . . . . . . . . . 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 )
76 oveq1 6110 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  p )  =  N  ->  ( (
# `  p )  -  2 )  =  ( N  -  2 ) )
7776fveq2d 5707 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  p )  =  N  ->  ( u `
 ( ( # `  p )  -  2 ) )  =  ( u `  ( N  -  2 ) ) )
78 eqeq1 2449 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( u `  0 )  =  ( u `  ( N  -  2
) )  ->  (
( u `  0
)  =  X  <->  ( u `  ( N  -  2 ) )  =  X ) )
7978eqcoms 2446 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( u `  ( N  -  2 ) )  =  ( u ` 
0 )  ->  (
( u `  0
)  =  X  <->  ( u `  ( N  -  2 ) )  =  X ) )
8079biimpd 207 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( u `  ( N  -  2 ) )  =  ( u ` 
0 )  ->  (
( u `  0
)  =  X  -> 
( u `  ( N  -  2 ) )  =  X ) )
8180impcom 430 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( u `  0
)  =  X  /\  ( u `  ( N  -  2 ) )  =  ( u `
 0 ) )  ->  ( u `  ( N  -  2
) )  =  X )
8281adantl 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) )  -> 
( u `  ( N  -  2 ) )  =  X )
8377, 82sylan9eq 2495 . . . . . . . . . . . . . . . . 17  |-  ( ( ( # `  p
)  =  N  /\  ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) ) )  ->  ( u `  ( ( # `  p
)  -  2 ) )  =  X )
8483ex 434 . . . . . . . . . . . . . . . 16  |-  ( (
# `  p )  =  N  ->  ( ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) )  -> 
( u `  (
( # `  p )  -  2 ) )  =  X ) )
8584adantl 466 . . . . . . . . . . . . . . 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 ) )
8685adantr 465 . . . . . . . . . . . . . 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 ) )
8786imp 429 . . . . . . . . . . . . 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 )
8875, 87eqtr4d 2478 . . . . . . . . . . . 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 ) ) )
8988adantl 466 . . . . . . . . . . 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 ) ) )
9089biantrurd 508 . . . . . . . . . 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
) ) ) ) )
9176opeq2d 4078 . . . . . . . . . . . . . . 15  |-  ( (
# `  p )  =  N  ->  <. 0 ,  ( ( # `  p )  -  2 ) >.  =  <. 0 ,  ( N  -  2 ) >.
)
9291oveq2d 6119 . . . . . . . . . . . . . 14  |-  ( (
# `  p )  =  N  ->  ( p substr  <. 0 ,  ( (
# `  p )  -  2 ) >.
)  =  ( p substr  <. 0 ,  ( N  -  2 ) >.
) )
9391oveq2d 6119 . . . . . . . . . . . . . 14  |-  ( (
# `  p )  =  N  ->  ( u substr  <. 0 ,  ( (
# `  p )  -  2 ) >.
)  =  ( u substr  <. 0 ,  ( N  -  2 ) >.
) )
9492, 93eqeq12d 2457 . . . . . . . . . . . . 13  |-  ( (
# `  p )  =  N  ->  ( ( p substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  =  (
u substr  <. 0 ,  ( ( # `  p
)  -  2 )
>. )  <->  ( p substr  <. 0 ,  ( N  - 
2 ) >. )  =  ( u substr  <. 0 ,  ( N  - 
2 ) >. )
) )
9594ad3antlr 730 . . . . . . . . . . . 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 ) >. )
) )
9695adantl 466 . . . . . . . . . . 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 ) >. )
) )
97 lsw 12278 . . . . . . . . . . . . . . 15  |-  ( p  e. Word  V  ->  ( lastS  `  p )  =  ( p `  ( (
# `  p )  -  1 ) ) )
98 oveq1 6110 . . . . . . . . . . . . . . . 16  |-  ( (
# `  p )  =  N  ->  ( (
# `  p )  -  1 )  =  ( N  -  1 ) )
9998fveq2d 5707 . . . . . . . . . . . . . . 15  |-  ( (
# `  p )  =  N  ->  ( p `
 ( ( # `  p )  -  1 ) )  =  ( p `  ( N  -  1 ) ) )
10097, 99sylan9eq 2495 . . . . . . . . . . . . . 14  |-  ( ( p  e. Word  V  /\  ( # `  p )  =  N )  -> 
( lastS  `  p )  =  ( p `  ( N  -  1 ) ) )
101100adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( p  e. Word  V  /\  ( # `  p
)  =  N )  /\  ( ( p `
 0 )  =  X  /\  ( p `
 ( N  - 
2 ) )  =  ( p `  0
) ) )  -> 
( lastS  `  p )  =  ( p `  ( N  -  1 ) ) )
102 lsw 12278 . . . . . . . . . . . . . . . 16  |-  ( u  e. Word  V  ->  ( lastS  `  u )  =  ( u `  ( (
# `  u )  -  1 ) ) )
103102adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  -> 
( lastS  `  u )  =  ( u `  (
( # `  u )  -  1 ) ) )
104 oveq1 6110 . . . . . . . . . . . . . . . . . . 19  |-  ( N  =  ( # `  u
)  ->  ( N  -  1 )  =  ( ( # `  u
)  -  1 ) )
105104eqcoms 2446 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  u )  =  N  ->  ( N  -  1 )  =  ( ( # `  u
)  -  1 ) )
106105fveq2d 5707 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  u )  =  N  ->  ( u `
 ( N  - 
1 ) )  =  ( u `  (
( # `  u )  -  1 ) ) )
107106eqeq2d 2454 . . . . . . . . . . . . . . . 16  |-  ( (
# `  u )  =  N  ->  ( ( lastS  `  u )  =  ( u `  ( N  -  1 ) )  <-> 
( lastS  `  u )  =  ( u `  (
( # `  u )  -  1 ) ) ) )
108107adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  -> 
( ( lastS  `  u
)  =  ( u `
 ( N  - 
1 ) )  <->  ( lastS  `  u
)  =  ( u `
 ( ( # `  u )  -  1 ) ) ) )
109103, 108mpbird 232 . . . . . . . . . . . . . 14  |-  ( ( u  e. Word  V  /\  ( # `  u )  =  N )  -> 
( lastS  `  u )  =  ( u `  ( N  -  1 ) ) )
110109adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( u  e. Word  V  /\  ( # `  u
)  =  N )  /\  ( ( u `
 0 )  =  X  /\  ( u `
 ( N  - 
2 ) )  =  ( u `  0
) ) )  -> 
( lastS  `  u )  =  ( u `  ( N  -  1 ) ) )
111101, 110eqeqan12d 2458 . . . . . . . . . . . 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 ) ) ) )
112111adantl 466 . . . . . . . . . . 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 ) ) ) )
11396, 112anbi12d 710 . . . . . . . . . 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
) ) ) ) )
11462, 90, 1133bitr2d 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
) ) ) ) )
11550, 60, 1143bitr2d 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 ) ) ) ) )
116115exbiri 622 . . . . . . 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 )
) )
11719, 21, 116syl2and 483 . . . . . 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 ) ) )
118117imp 429 . . . . 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 )
)
11916, 118syl5bi 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 ) )
12013, 119sylbid 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 ) )
121120ralrimivva 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 ) )
122 dff13 5983 . 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 ) ) )
1235, 121, 122sylanbrc 664 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2727   {crab 2731   <.cop 3895   class class class wbr 4304    e. cmpt 4362    X. cxp 4850   -->wf 5426   -1-1->wf1 5427   ` cfv 5430  (class class class)co 6103    e. cmpt2 6105   RRcr 9293   0cc0 9294   1c1 9295    < clt 9430    <_ cle 9431    - cmin 9607   2c2 10383   3c3 10384   NN0cn0 10591   ZZcz 10658   ZZ>=cuz 10873   #chash 12115  Word cword 12233   lastS clsw 12234   substr csubstr 12237   USGrph cusg 23276   Neighbors cnbgra 23341   ClWWalksN cclwwlkn 30426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  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-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-pm 7229  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-card 8121  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-3 10393  df-n0 10592  df-z 10659  df-uz 10874  df-fz 11450  df-fzo 11561  df-hash 12116  df-word 12241  df-lsw 12242  df-concat 12243  df-s1 12244  df-substr 12245  df-s2 12487  df-usgra 23278  df-nbgra 23344  df-clwwlk 30428  df-clwwlkn 30429
This theorem is referenced by:  numclwlk1lem2f1o  30701
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