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Theorem extwwlkfablem1 24898
Description: Lemma 1 for extwwlkfab 24914. (Contributed by Alexander van der Vekens, 15-Sep-2018.)
Assertion
Ref Expression
extwwlkfablem1  |-  ( ( ( ( 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 ) )

Proof of Theorem extwwlkfablem1
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 clwwlknimp 24599 . . . 4  |-  ( 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
) )
2 simp2 997 . . . . . . . 8  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  2 )
)  ->  X  e.  V )
32adantl 466 . . . . . . 7  |-  ( ( ( ( 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 )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  2 ) ) )  ->  X  e.  V )
43adantr 465 . . . . . 6  |-  ( ( ( ( ( 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
)  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  2 ) ) )  /\  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) )  ->  X  e.  V
)
5 simplr1 1038 . . . . . . 7  |-  ( ( ( ( ( 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
)  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  2 ) ) )  /\  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) )  ->  V USGrph  E )
6 eluzelcn 11105 . . . . . . . . . . . . . 14  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  CC )
7 1e2m1 10663 . . . . . . . . . . . . . . . . 17  |-  1  =  ( 2  -  1 )
87a1i 11 . . . . . . . . . . . . . . . 16  |-  ( N  e.  CC  ->  1  =  ( 2  -  1 ) )
98oveq2d 6311 . . . . . . . . . . . . . . 15  |-  ( N  e.  CC  ->  ( N  -  1 )  =  ( N  -  ( 2  -  1 ) ) )
10 id 22 . . . . . . . . . . . . . . . 16  |-  ( N  e.  CC  ->  N  e.  CC )
11 2cnd 10620 . . . . . . . . . . . . . . . 16  |-  ( N  e.  CC  ->  2  e.  CC )
12 ax-1cn 9562 . . . . . . . . . . . . . . . . 17  |-  1  e.  CC
1312a1i 11 . . . . . . . . . . . . . . . 16  |-  ( N  e.  CC  ->  1  e.  CC )
1410, 11, 13subsubd 9970 . . . . . . . . . . . . . . 15  |-  ( N  e.  CC  ->  ( N  -  ( 2  -  1 ) )  =  ( ( N  -  2 )  +  1 ) )
159, 14eqtr2d 2509 . . . . . . . . . . . . . 14  |-  ( N  e.  CC  ->  (
( N  -  2 )  +  1 )  =  ( N  - 
1 ) )
166, 15syl 16 . . . . . . . . . . . . 13  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( ( N  -  2 )  +  1 )  =  ( N  -  1 ) )
1716fveq2d 5876 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( w `  ( ( N  - 
2 )  +  1 ) )  =  ( w `  ( N  -  1 ) ) )
1817preq2d 4119 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  2
)  ->  { (
w `  ( N  -  2 ) ) ,  ( w `  ( ( N  - 
2 )  +  1 ) ) }  =  { ( w `  ( N  -  2
) ) ,  ( w `  ( N  -  1 ) ) } )
19183ad2ant3 1019 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  2 )
)  ->  { (
w `  ( N  -  2 ) ) ,  ( w `  ( ( N  - 
2 )  +  1 ) ) }  =  { ( w `  ( N  -  2
) ) ,  ( w `  ( N  -  1 ) ) } )
2019adantl 466 . . . . . . . . 9  |-  ( ( ( ( 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 )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  2 ) ) )  ->  { (
w `  ( N  -  2 ) ) ,  ( w `  ( ( N  - 
2 )  +  1 ) ) }  =  { ( w `  ( N  -  2
) ) ,  ( w `  ( N  -  1 ) ) } )
21 uznn0sub 11125 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  -  2 )  e. 
NN0 )
22 uz2m1nn 11168 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  -  1 )  e.  NN )
23 1re 9607 . . . . . . . . . . . . . 14  |-  1  e.  RR
2423a1i 11 . . . . . . . . . . . . 13  |-  ( N  e.  ( ZZ>= `  2
)  ->  1  e.  RR )
25 2re 10617 . . . . . . . . . . . . . 14  |-  2  e.  RR
2625a1i 11 . . . . . . . . . . . . 13  |-  ( N  e.  ( ZZ>= `  2
)  ->  2  e.  RR )
27 eluzelre 11104 . . . . . . . . . . . . 13  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  RR )
28 1lt2 10714 . . . . . . . . . . . . . 14  |-  1  <  2
2928a1i 11 . . . . . . . . . . . . 13  |-  ( N  e.  ( ZZ>= `  2
)  ->  1  <  2 )
3024, 26, 27, 29ltsub2dd 10177 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  -  2 )  < 
( N  -  1 ) )
31 elfzo0 11843 . . . . . . . . . . . 12  |-  ( ( N  -  2 )  e.  ( 0..^ ( N  -  1 ) )  <->  ( ( N  -  2 )  e. 
NN0  /\  ( N  -  1 )  e.  NN  /\  ( N  -  2 )  < 
( N  -  1 ) ) )
3221, 22, 30, 31syl3anbrc 1180 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  -  2 )  e.  ( 0..^ ( N  -  1 ) ) )
33323ad2ant3 1019 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( N  -  2 )  e.  ( 0..^ ( N  -  1 ) ) )
34 simp2 997 . . . . . . . . . 10  |-  ( ( ( 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 )  ->  A. i  e.  ( 0..^ ( N  - 
1 ) ) { ( w `  i
) ,  ( w `
 ( i  +  1 ) ) }  e.  ran  E )
35 fveq2 5872 . . . . . . . . . . . . 13  |-  ( i  =  ( N  - 
2 )  ->  (
w `  i )  =  ( w `  ( N  -  2
) ) )
36 oveq1 6302 . . . . . . . . . . . . . 14  |-  ( i  =  ( N  - 
2 )  ->  (
i  +  1 )  =  ( ( N  -  2 )  +  1 ) )
3736fveq2d 5876 . . . . . . . . . . . . 13  |-  ( i  =  ( N  - 
2 )  ->  (
w `  ( i  +  1 ) )  =  ( w `  ( ( N  - 
2 )  +  1 ) ) )
3835, 37preq12d 4120 . . . . . . . . . . . 12  |-  ( i  =  ( N  - 
2 )  ->  { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  =  { ( w `  ( N  -  2
) ) ,  ( w `  ( ( N  -  2 )  +  1 ) ) } )
3938eleq1d 2536 . . . . . . . . . . 11  |-  ( i  =  ( N  - 
2 )  ->  ( { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( w `  ( N  -  2 ) ) ,  ( w `
 ( ( N  -  2 )  +  1 ) ) }  e.  ran  E ) )
4039rspcva 3217 . . . . . . . . . 10  |-  ( ( ( N  -  2 )  e.  ( 0..^ ( N  -  1 ) )  /\  A. i  e.  ( 0..^ ( N  -  1 ) ) { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E )  ->  { ( w `  ( N  -  2 ) ) ,  ( w `  ( ( N  - 
2 )  +  1 ) ) }  e.  ran  E )
4133, 34, 40syl2anr 478 . . . . . . . . 9  |-  ( ( ( ( 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 )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  2 ) ) )  ->  { (
w `  ( N  -  2 ) ) ,  ( w `  ( ( N  - 
2 )  +  1 ) ) }  e.  ran  E )
4220, 41eqeltrrd 2556 . . . . . . . 8  |-  ( ( ( ( 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 )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  2 ) ) )  ->  { (
w `  ( N  -  2 ) ) ,  ( w `  ( N  -  1
) ) }  e.  ran  E )
4342adantr 465 . . . . . . 7  |-  ( ( ( ( ( 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
)  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  2 ) ) )  /\  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) )  ->  { ( w `
 ( N  - 
2 ) ) ,  ( w `  ( N  -  1 ) ) }  e.  ran  E )
44 usgraedgrnv 24200 . . . . . . . 8  |-  ( ( V USGrph  E  /\  { ( w `  ( N  -  2 ) ) ,  ( w `  ( N  -  1
) ) }  e.  ran  E )  ->  (
( w `  ( N  -  2 ) )  e.  V  /\  ( w `  ( N  -  1 ) )  e.  V ) )
4544simprd 463 . . . . . . 7  |-  ( ( V USGrph  E  /\  { ( w `  ( N  -  2 ) ) ,  ( w `  ( N  -  1
) ) }  e.  ran  E )  ->  (
w `  ( N  -  1 ) )  e.  V )
465, 43, 45syl2anc 661 . . . . . 6  |-  ( ( ( ( ( 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
)  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  2 ) ) )  /\  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) )  ->  ( w `  ( N  -  1
) )  e.  V
)
47 preq1 4112 . . . . . . . . . . 11  |-  ( X  =  ( w ` 
0 )  ->  { X ,  ( w `  ( N  -  1
) ) }  =  { ( w ` 
0 ) ,  ( w `  ( N  -  1 ) ) } )
4847eqcoms 2479 . . . . . . . . . 10  |-  ( ( w `  0 )  =  X  ->  { X ,  ( w `  ( N  -  1
) ) }  =  { ( w ` 
0 ) ,  ( w `  ( N  -  1 ) ) } )
49 preq1 4112 . . . . . . . . . . 11  |-  ( ( w `  0 )  =  ( w `  ( N  -  2
) )  ->  { ( w `  0 ) ,  ( w `  ( N  -  1
) ) }  =  { ( w `  ( N  -  2
) ) ,  ( w `  ( N  -  1 ) ) } )
5049eqcoms 2479 . . . . . . . . . 10  |-  ( ( w `  ( N  -  2 ) )  =  ( w ` 
0 )  ->  { ( w `  0 ) ,  ( w `  ( N  -  1
) ) }  =  { ( w `  ( N  -  2
) ) ,  ( w `  ( N  -  1 ) ) } )
5148, 50sylan9eq 2528 . . . . . . . . 9  |-  ( ( ( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =  ( w `
 0 ) )  ->  { X , 
( w `  ( N  -  1 ) ) }  =  {
( w `  ( N  -  2 ) ) ,  ( w `
 ( N  - 
1 ) ) } )
5251eleq1d 2536 . . . . . . . 8  |-  ( ( ( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =  ( w `
 0 ) )  ->  ( { X ,  ( w `  ( N  -  1
) ) }  e.  ran  E  <->  { ( w `  ( N  -  2
) ) ,  ( w `  ( N  -  1 ) ) }  e.  ran  E
) )
5352adantl 466 . . . . . . 7  |-  ( ( ( ( ( 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
)  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  2 ) ) )  /\  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) )  ->  ( { X ,  ( w `  ( N  -  1
) ) }  e.  ran  E  <->  { ( w `  ( N  -  2
) ) ,  ( w `  ( N  -  1 ) ) }  e.  ran  E
) )
5443, 53mpbird 232 . . . . . 6  |-  ( ( ( ( ( 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
)  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  2 ) ) )  /\  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) )  ->  { X , 
( w `  ( N  -  1 ) ) }  e.  ran  E )
55 usgrav 24161 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
56553ad2ant1 1017 . . . . . . . . 9  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( V  e.  _V  /\  E  e. 
_V ) )
5756adantl 466 . . . . . . . 8  |-  ( ( ( ( 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 )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  2 ) ) )  ->  ( V  e.  _V  /\  E  e. 
_V ) )
5857adantr 465 . . . . . . 7  |-  ( ( ( ( ( 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
)  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  2 ) ) )  /\  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) )  ->  ( V  e. 
_V  /\  E  e.  _V ) )
59 nbgrael 24249 . . . . . . 7  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( ( w `  ( N  -  1
) )  e.  (
<. V ,  E >. Neighbors  X
)  <->  ( X  e.  V  /\  ( w `
 ( N  - 
1 ) )  e.  V  /\  { X ,  ( w `  ( N  -  1
) ) }  e.  ran  E ) ) )
6058, 59syl 16 . . . . . 6  |-  ( ( ( ( ( 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
)  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  2 ) ) )  /\  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) )  ->  ( ( w `
 ( N  - 
1 ) )  e.  ( <. V ,  E >. Neighbors  X )  <->  ( X  e.  V  /\  (
w `  ( N  -  1 ) )  e.  V  /\  { X ,  ( w `  ( N  -  1 ) ) }  e.  ran  E ) ) )
614, 46, 54, 60mpbir3and 1179 . . . . 5  |-  ( ( ( ( ( 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
)  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  2 ) ) )  /\  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) )  ->  ( w `  ( N  -  1
) )  e.  (
<. V ,  E >. Neighbors  X
) )
6261exp31 604 . . . 4  |-  ( ( ( 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 )  ->  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  2 ) )  ->  ( ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) )  -> 
( w `  ( N  -  1 ) )  e.  ( <. V ,  E >. Neighbors  X
) ) ) )
631, 62syl 16 . . 3  |-  ( w  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( (
( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =  ( w `
 0 ) )  ->  ( w `  ( N  -  1
) )  e.  (
<. V ,  E >. Neighbors  X
) ) ) )
6463impcom 430 . 2  |-  ( ( ( 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
) ) )
6564imp 429 1  |-  ( ( ( ( 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 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118   {cpr 4035   <.cop 4039   class class class wbr 4453   ran crn 5006   ` cfv 5594  (class class class)co 6295   CCcc 9502   RRcr 9503   0cc0 9504   1c1 9505    + caddc 9507    < clt 9640    - cmin 9817   NNcn 10548   2c2 10597   NN0cn0 10807   ZZ>=cuz 11094  ..^cfzo 11804   #chash 12385  Word cword 12515   lastS clsw 12516   USGrph cusg 24153   Neighbors cnbgra 24240   ClWWalksN cclwwlkn 24572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-hash 12386  df-word 12523  df-usgra 24156  df-nbgra 24243  df-clwwlk 24574  df-clwwlkn 24575
This theorem is referenced by:  extwwlkfab  24914
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