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Theorem extwwlkfab 30832
Description: The set of closed walks (having a fixed length greater than 1 and starting at a fixed vertex) with the last but 2 vertex is identical with the first (and therefore last) vertex can be constructed from the set of closed walks with length smaller by 2 than the fixed length appending a neighbor of the last vertex and afterwards the last vertex (which is the first vertex) itself ("walking forth and back" from the last vertex).  3  <_  N is required since for  N  =  2:  ( X F ( N  -  2 ) )  =  ( X F 0 )  =  (/), see clwwlkgt0 30583 stating that a walk of length 0 is not represented as word, at least not for an undirected simple graph.) (Contributed by Alexander van der Vekens, 18-Sep-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 ) ) } )
Assertion
Ref Expression
extwwlkfab  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( X G N )  =  {
w  e.  ( C `
 N )  |  ( ( w substr  <. 0 ,  ( N  - 
2 ) >. )  e.  ( X F ( N  -  2 ) )  /\  ( w `
 ( N  - 
1 ) )  e.  ( <. V ,  E >. Neighbors  X )  /\  (
w `  ( N  -  2 ) )  =  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
Allowed substitution hints:    E( v)    F( v, n)    G( w, v, n)

Proof of Theorem extwwlkfab
Dummy variables  i  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uzuzle23 30342 . . . 4  |-  ( N  e.  ( ZZ>= `  3
)  ->  N  e.  ( ZZ>= `  2 )
)
2 numclwwlk.c . . . . 5  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
3 numclwwlk.f . . . . 5  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
4 numclwwlk.g . . . . 5  |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>=
`  2 )  |->  { w  e.  ( C `
 n )  |  ( ( w ` 
0 )  =  v  /\  ( w `  ( n  -  2
) )  =  ( w `  0 ) ) } )
52, 3, 4numclwwlkovg 30829 . . . 4  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( X G N )  =  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) } )
61, 5sylan2 474 . . 3  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  -> 
( X G N )  =  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) } )
763adant1 1006 . 2  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( X G N )  =  {
w  e.  ( C `
 N )  |  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) } )
8 eluzge2nn0 30341 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  NN0 )
92numclwwlkfvc 30819 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( C `
 N )  =  ( ( V ClWWalksN  E ) `
 N ) )
101, 8, 93syl 20 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  3
)  ->  ( C `  N )  =  ( ( V ClWWalksN  E ) `  N ) )
11103ad2ant3 1011 . . . . . . 7  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( C `  N )  =  ( ( V ClWWalksN  E ) `  N ) )
1211eleq2d 2524 . . . . . 6  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( w  e.  ( C `  N
)  <->  w  e.  (
( V ClWWalksN  E ) `  N ) ) )
132extwwlkfablem2 30820 . . . . . . . . . . . 12  |-  ( ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  3 ) )  /\  w  e.  ( ( V ClWWalksN  E ) `  N ) )  /\  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) )  ->  (
w substr  <. 0 ,  ( N  -  2 )
>. )  e.  ( C `  ( N  -  2 ) ) )
14 simpl 457 . . . . . . . . . . . . 13  |-  ( ( ( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =  ( w `
 0 ) )  ->  ( w ` 
0 )  =  X )
1514adantl 466 . . . . . . . . . . . 12  |-  ( ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  3 ) )  /\  w  e.  ( ( V ClWWalksN  E ) `  N ) )  /\  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) )  ->  (
w `  0 )  =  X )
1613, 15jca 532 . . . . . . . . . . 11  |-  ( ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  3 ) )  /\  w  e.  ( ( V ClWWalksN  E ) `  N ) )  /\  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) )  ->  (
( w substr  <. 0 ,  ( N  -  2 ) >. )  e.  ( C `  ( N  -  2 ) )  /\  ( w ` 
0 )  =  X ) )
1713anim3i 1176 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  2 ) ) )
18 extwwlkfablem1 30816 . . . . . . . . . . . 12  |-  ( ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  2 ) )  /\  w  e.  ( ( V ClWWalksN  E ) `  N ) )  /\  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) )  ->  (
w `  ( N  -  1 ) )  e.  ( <. V ,  E >. Neighbors  X ) )
1917, 18sylanl1 650 . . . . . . . . . . 11  |-  ( ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  3 ) )  /\  w  e.  ( ( V ClWWalksN  E ) `  N ) )  /\  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) )  ->  (
w `  ( N  -  1 ) )  e.  ( <. V ,  E >. Neighbors  X ) )
20 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =  ( w `
 0 ) )  ->  ( w `  ( N  -  2
) )  =  ( w `  0 ) )
2120, 14eqtrd 2495 . . . . . . . . . . . 12  |-  ( ( ( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =  ( w `
 0 ) )  ->  ( w `  ( N  -  2
) )  =  X )
2221adantl 466 . . . . . . . . . . 11  |-  ( ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  3 ) )  /\  w  e.  ( ( V ClWWalksN  E ) `  N ) )  /\  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) )  ->  (
w `  ( N  -  2 ) )  =  X )
2316, 19, 223jca 1168 . . . . . . . . . 10  |-  ( ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  3 ) )  /\  w  e.  ( ( V ClWWalksN  E ) `  N ) )  /\  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) )  ->  (
( ( w substr  <. 0 ,  ( N  - 
2 ) >. )  e.  ( C `  ( N  -  2 ) )  /\  ( w `
 0 )  =  X )  /\  (
w `  ( N  -  1 ) )  e.  ( <. V ,  E >. Neighbors  X )  /\  (
w `  ( N  -  2 ) )  =  X ) )
2423ex 434 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  w  e.  ( ( V ClWWalksN  E ) `  N
) )  ->  (
( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) )  ->  ( (
( w substr  <. 0 ,  ( N  -  2 ) >. )  e.  ( C `  ( N  -  2 ) )  /\  ( w ` 
0 )  =  X )  /\  ( w `
 ( N  - 
1 ) )  e.  ( <. V ,  E >. Neighbors  X )  /\  (
w `  ( N  -  2 ) )  =  X ) ) )
25 simpl 457 . . . . . . . . . . . . . 14  |-  ( ( ( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =  X )  ->  ( w ` 
0 )  =  X )
26 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( ( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =  X )  ->  ( w `  ( N  -  2
) )  =  X )
2725eqcomd 2462 . . . . . . . . . . . . . . 15  |-  ( ( ( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =  X )  ->  X  =  ( w `  0 ) )
2826, 27eqtrd 2495 . . . . . . . . . . . . . 14  |-  ( ( ( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =  X )  ->  ( w `  ( N  -  2
) )  =  ( w `  0 ) )
2925, 28jca 532 . . . . . . . . . . . . 13  |-  ( ( ( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =  X )  ->  ( ( w `
 0 )  =  X  /\  ( w `
 ( N  - 
2 ) )  =  ( w `  0
) ) )
3029ex 434 . . . . . . . . . . . 12  |-  ( ( w `  0 )  =  X  ->  (
( w `  ( N  -  2 ) )  =  X  -> 
( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) ) )
3130a1d 25 . . . . . . . . . . 11  |-  ( ( w `  0 )  =  X  ->  (
( w `  ( N  -  1 ) )  e.  ( <. V ,  E >. Neighbors  X
)  ->  ( (
w `  ( N  -  2 ) )  =  X  ->  (
( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =  ( w `
 0 ) ) ) ) )
3231adantl 466 . . . . . . . . . 10  |-  ( ( ( w substr  <. 0 ,  ( N  - 
2 ) >. )  e.  ( C `  ( N  -  2 ) )  /\  ( w `
 0 )  =  X )  ->  (
( w `  ( N  -  1 ) )  e.  ( <. V ,  E >. Neighbors  X
)  ->  ( (
w `  ( N  -  2 ) )  =  X  ->  (
( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =  ( w `
 0 ) ) ) ) )
33323imp 1182 . . . . . . . . 9  |-  ( ( ( ( w substr  <. 0 ,  ( N  - 
2 ) >. )  e.  ( C `  ( N  -  2 ) )  /\  ( w `
 0 )  =  X )  /\  (
w `  ( N  -  1 ) )  e.  ( <. V ,  E >. Neighbors  X )  /\  (
w `  ( N  -  2 ) )  =  X )  -> 
( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) )
3424, 33impbid1 203 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  w  e.  ( ( V ClWWalksN  E ) `  N
) )  ->  (
( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) )  <->  ( ( ( w substr  <. 0 ,  ( N  -  2 )
>. )  e.  ( C `  ( N  -  2 ) )  /\  ( w ` 
0 )  =  X )  /\  ( w `
 ( N  - 
1 ) )  e.  ( <. V ,  E >. Neighbors  X )  /\  (
w `  ( N  -  2 ) )  =  X ) ) )
35 clwwlknimp 30588 . . . . . . . . . . . . . . . . 17  |-  ( w  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( (
w  e. Word  V  /\  ( # `  w )  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  w ) ,  ( w `  0 ) }  e.  ran  E
) )
36 ige3m2fz 30356 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  ( ZZ>= `  3
)  ->  ( N  -  2 )  e.  ( 1 ... N
) )
37 oveq2 6209 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
# `  w )  =  N  ->  ( 1 ... ( # `  w
) )  =  ( 1 ... N ) )
3837eleq2d 2524 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  w )  =  N  ->  ( ( N  -  2 )  e.  ( 1 ... ( # `  w
) )  <->  ( N  -  2 )  e.  ( 1 ... N
) ) )
3936, 38syl5ibr 221 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  w )  =  N  ->  ( N  e.  ( ZZ>= `  3
)  ->  ( N  -  2 )  e.  ( 1 ... ( # `
 w ) ) ) )
4039adantl 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  N )  -> 
( N  e.  (
ZZ>= `  3 )  -> 
( N  -  2 )  e.  ( 1 ... ( # `  w
) ) ) )
41 simpl 457 . . . . . . . . . . . . . . . . . . 19  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  N )  ->  w  e. Word  V )
4240, 41jctild 543 . . . . . . . . . . . . . . . . . 18  |-  ( ( w  e. Word  V  /\  ( # `  w )  =  N )  -> 
( N  e.  (
ZZ>= `  3 )  -> 
( w  e. Word  V  /\  ( N  -  2 )  e.  ( 1 ... ( # `  w
) ) ) ) )
43423ad2ant1 1009 . . . . . . . . . . . . . . . . 17  |-  ( ( ( w  e. Word  V  /\  ( # `  w
)  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  w
) ,  ( w `
 0 ) }  e.  ran  E )  ->  ( N  e.  ( ZZ>= `  3 )  ->  ( w  e. Word  V  /\  ( N  -  2 )  e.  ( 1 ... ( # `  w
) ) ) ) )
4435, 43syl 16 . . . . . . . . . . . . . . . 16  |-  ( w  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( N  e.  ( ZZ>= `  3 )  ->  ( w  e. Word  V  /\  ( N  -  2 )  e.  ( 1 ... ( # `  w
) ) ) ) )
4544com12 31 . . . . . . . . . . . . . . 15  |-  ( N  e.  ( ZZ>= `  3
)  ->  ( w  e.  ( ( V ClWWalksN  E ) `
 N )  -> 
( w  e. Word  V  /\  ( N  -  2 )  e.  ( 1 ... ( # `  w
) ) ) ) )
46453ad2ant3 1011 . . . . . . . . . . . . . 14  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( w  e.  ( ( V ClWWalksN  E ) `
 N )  -> 
( w  e. Word  V  /\  ( N  -  2 )  e.  ( 1 ... ( # `  w
) ) ) ) )
4746imp 429 . . . . . . . . . . . . 13  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  w  e.  ( ( V ClWWalksN  E ) `  N
) )  ->  (
w  e. Word  V  /\  ( N  -  2
)  e.  ( 1 ... ( # `  w
) ) ) )
48 swrd0fv0 12455 . . . . . . . . . . . . 13  |-  ( ( w  e. Word  V  /\  ( N  -  2
)  e.  ( 1 ... ( # `  w
) ) )  -> 
( ( w substr  <. 0 ,  ( N  - 
2 ) >. ) `  0 )  =  ( w `  0
) )
4947, 48syl 16 . . . . . . . . . . . 12  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  w  e.  ( ( V ClWWalksN  E ) `  N
) )  ->  (
( w substr  <. 0 ,  ( N  -  2 ) >. ) `  0
)  =  ( w `
 0 ) )
5049eqcomd 2462 . . . . . . . . . . 11  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  w  e.  ( ( V ClWWalksN  E ) `  N
) )  ->  (
w `  0 )  =  ( ( w substr  <. 0 ,  ( N  -  2 ) >.
) `  0 )
)
5150eqeq1d 2456 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  w  e.  ( ( V ClWWalksN  E ) `  N
) )  ->  (
( w `  0
)  =  X  <->  ( (
w substr  <. 0 ,  ( N  -  2 )
>. ) `  0 )  =  X ) )
5251anbi2d 703 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  w  e.  ( ( V ClWWalksN  E ) `  N
) )  ->  (
( ( w substr  <. 0 ,  ( N  - 
2 ) >. )  e.  ( C `  ( N  -  2 ) )  /\  ( w `
 0 )  =  X )  <->  ( (
w substr  <. 0 ,  ( N  -  2 )
>. )  e.  ( C `  ( N  -  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  -  2 ) >.
) `  0 )  =  X ) ) )
53523anbi1d 1294 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  w  e.  ( ( V ClWWalksN  E ) `  N
) )  ->  (
( ( ( w substr  <. 0 ,  ( N  -  2 ) >.
)  e.  ( C `
 ( N  - 
2 ) )  /\  ( w `  0
)  =  X )  /\  ( w `  ( N  -  1
) )  e.  (
<. V ,  E >. Neighbors  X
)  /\  ( w `  ( N  -  2 ) )  =  X )  <->  ( ( ( w substr  <. 0 ,  ( N  -  2 )
>. )  e.  ( C `  ( N  -  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  -  2 ) >.
) `  0 )  =  X )  /\  (
w `  ( N  -  1 ) )  e.  ( <. V ,  E >. Neighbors  X )  /\  (
w `  ( N  -  2 ) )  =  X ) ) )
5434, 53bitrd 253 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  w  e.  ( ( V ClWWalksN  E ) `  N
) )  ->  (
( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) )  <->  ( ( ( w substr  <. 0 ,  ( N  -  2 )
>. )  e.  ( C `  ( N  -  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  -  2 ) >.
) `  0 )  =  X )  /\  (
w `  ( N  -  1 ) )  e.  ( <. V ,  E >. Neighbors  X )  /\  (
w `  ( N  -  2 ) )  =  X ) ) )
5554ex 434 . . . . . 6  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( w  e.  ( ( V ClWWalksN  E ) `
 N )  -> 
( ( ( w `
 0 )  =  X  /\  ( w `
 ( N  - 
2 ) )  =  ( w `  0
) )  <->  ( (
( w substr  <. 0 ,  ( N  -  2 ) >. )  e.  ( C `  ( N  -  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  -  2 ) >.
) `  0 )  =  X )  /\  (
w `  ( N  -  1 ) )  e.  ( <. V ,  E >. Neighbors  X )  /\  (
w `  ( N  -  2 ) )  =  X ) ) ) )
5612, 55sylbid 215 . . . . 5  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( w  e.  ( C `  N
)  ->  ( (
( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =  ( w `
 0 ) )  <-> 
( ( ( w substr  <. 0 ,  ( N  -  2 ) >.
)  e.  ( C `
 ( N  - 
2 ) )  /\  ( ( w substr  <. 0 ,  ( N  - 
2 ) >. ) `  0 )  =  X )  /\  (
w `  ( N  -  1 ) )  e.  ( <. V ,  E >. Neighbors  X )  /\  (
w `  ( N  -  2 ) )  =  X ) ) ) )
5756imp 429 . . . 4  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  w  e.  ( C `  N ) )  -> 
( ( ( w `
 0 )  =  X  /\  ( w `
 ( N  - 
2 ) )  =  ( w `  0
) )  <->  ( (
( w substr  <. 0 ,  ( N  -  2 ) >. )  e.  ( C `  ( N  -  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  -  2 ) >.
) `  0 )  =  X )  /\  (
w `  ( N  -  1 ) )  e.  ( <. V ,  E >. Neighbors  X )  /\  (
w `  ( N  -  2 ) )  =  X ) ) )
58 uznn0sub 11004 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  -  2 )  e. 
NN0 )
591, 58syl 16 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  3
)  ->  ( N  -  2 )  e. 
NN0 )
602, 3numclwwlkovf 30823 . . . . . . . . . 10  |-  ( ( X  e.  V  /\  ( N  -  2
)  e.  NN0 )  ->  ( X F ( N  -  2 ) )  =  { w  e.  ( C `  ( N  -  2 ) )  |  ( w `
 0 )  =  X } )
6159, 60sylan2 474 . . . . . . . . 9  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  -> 
( X F ( N  -  2 ) )  =  { w  e.  ( C `  ( N  -  2 ) )  |  ( w `
 0 )  =  X } )
62613adant1 1006 . . . . . . . 8  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( X F ( N  - 
2 ) )  =  { w  e.  ( C `  ( N  -  2 ) )  |  ( w ` 
0 )  =  X } )
6362adantr 465 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  w  e.  ( C `  N ) )  -> 
( X F ( N  -  2 ) )  =  { w  e.  ( C `  ( N  -  2 ) )  |  ( w `
 0 )  =  X } )
6463eleq2d 2524 . . . . . 6  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  w  e.  ( C `  N ) )  -> 
( ( w substr  <. 0 ,  ( N  - 
2 ) >. )  e.  ( X F ( N  -  2 ) )  <->  ( w substr  <. 0 ,  ( N  - 
2 ) >. )  e.  { w  e.  ( C `  ( N  -  2 ) )  |  ( w ` 
0 )  =  X } ) )
65 fveq1 5799 . . . . . . . 8  |-  ( u  =  ( w substr  <. 0 ,  ( N  - 
2 ) >. )  ->  ( u `  0
)  =  ( ( w substr  <. 0 ,  ( N  -  2 )
>. ) `  0 ) )
6665eqeq1d 2456 . . . . . . 7  |-  ( u  =  ( w substr  <. 0 ,  ( N  - 
2 ) >. )  ->  ( ( u ` 
0 )  =  X  <-> 
( ( w substr  <. 0 ,  ( N  - 
2 ) >. ) `  0 )  =  X ) )
67 fveq1 5799 . . . . . . . . 9  |-  ( w  =  u  ->  (
w `  0 )  =  ( u ` 
0 ) )
6867eqeq1d 2456 . . . . . . . 8  |-  ( w  =  u  ->  (
( w `  0
)  =  X  <->  ( u `  0 )  =  X ) )
6968cbvrabv 3077 . . . . . . 7  |-  { w  e.  ( C `  ( N  -  2 ) )  |  ( w `
 0 )  =  X }  =  {
u  e.  ( C `
 ( N  - 
2 ) )  |  ( u `  0
)  =  X }
7066, 69elrab2 3226 . . . . . 6  |-  ( ( w substr  <. 0 ,  ( N  -  2 )
>. )  e.  { w  e.  ( C `  ( N  -  2 ) )  |  ( w `
 0 )  =  X }  <->  ( (
w substr  <. 0 ,  ( N  -  2 )
>. )  e.  ( C `  ( N  -  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  -  2 ) >.
) `  0 )  =  X ) )
7164, 70syl6rbb 262 . . . . 5  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  w  e.  ( C `  N ) )  -> 
( ( ( w substr  <. 0 ,  ( N  -  2 ) >.
)  e.  ( C `
 ( N  - 
2 ) )  /\  ( ( w substr  <. 0 ,  ( N  - 
2 ) >. ) `  0 )  =  X )  <->  ( w substr  <.
0 ,  ( N  -  2 ) >.
)  e.  ( X F ( N  - 
2 ) ) ) )
72713anbi1d 1294 . . . 4  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  w  e.  ( C `  N ) )  -> 
( ( ( ( w substr  <. 0 ,  ( N  -  2 )
>. )  e.  ( C `  ( N  -  2 ) )  /\  ( ( w substr  <. 0 ,  ( N  -  2 ) >.
) `  0 )  =  X )  /\  (
w `  ( N  -  1 ) )  e.  ( <. V ,  E >. Neighbors  X )  /\  (
w `  ( N  -  2 ) )  =  X )  <->  ( (
w substr  <. 0 ,  ( N  -  2 )
>. )  e.  ( X F ( N  - 
2 ) )  /\  ( w `  ( N  -  1 ) )  e.  ( <. V ,  E >. Neighbors  X
)  /\  ( w `  ( N  -  2 ) )  =  X ) ) )
7357, 72bitrd 253 . . 3  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  w  e.  ( C `  N ) )  -> 
( ( ( w `
 0 )  =  X  /\  ( w `
 ( N  - 
2 ) )  =  ( w `  0
) )  <->  ( (
w substr  <. 0 ,  ( N  -  2 )
>. )  e.  ( X F ( N  - 
2 ) )  /\  ( w `  ( N  -  1 ) )  e.  ( <. V ,  E >. Neighbors  X
)  /\  ( w `  ( N  -  2 ) )  =  X ) ) )
7473rabbidva 3069 . 2  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) }  =  { w  e.  ( C `  N
)  |  ( ( w substr  <. 0 ,  ( N  -  2 )
>. )  e.  ( X F ( N  - 
2 ) )  /\  ( w `  ( N  -  1 ) )  e.  ( <. V ,  E >. Neighbors  X
)  /\  ( w `  ( N  -  2 ) )  =  X ) } )
757, 74eqtrd 2495 1  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( X G N )  =  {
w  e.  ( C `
 N )  |  ( ( w substr  <. 0 ,  ( N  - 
2 ) >. )  e.  ( X F ( N  -  2 ) )  /\  ( w `
 ( N  - 
1 ) )  e.  ( <. V ,  E >. Neighbors  X )  /\  (
w `  ( N  -  2 ) )  =  X ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   {crab 2803   {cpr 3988   <.cop 3992   class class class wbr 4401    |-> cmpt 4459   ran crn 4950   ` cfv 5527  (class class class)co 6201    |-> cmpt2 6203   0cc0 9394   1c1 9395    + caddc 9397    - cmin 9707   2c2 10483   3c3 10484   NN0cn0 10691   ZZ>=cuz 10973   ...cfz 11555  ..^cfzo 11666   #chash 12221  Word cword 12340   lastS clsw 12341   substr csubstr 12344   USGrph cusg 23417   Neighbors cnbgra 23482   ClWWalksN cclwwlkn 30563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-pm 7328  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-hash 12222  df-word 12348  df-lsw 12349  df-substr 12352  df-usgra 23419  df-nbgra 23485  df-clwwlk 30565  df-clwwlkn 30566
This theorem is referenced by:  numclwlk1lem2foa  30833  numclwlk1lem2f  30834
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