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Theorem extwwlkfablem1 30835
Description: Lemma 1 for extwwlkfab 30851. (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 30607 . . . 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 989 . . . . . . . 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 1030 . . . . . . 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 10986 . . . . . . . . . . . . . 14  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  CC )
7 1e2m1 10551 . . . . . . . . . . . . . . . . 17  |-  1  =  ( 2  -  1 )
87a1i 11 . . . . . . . . . . . . . . . 16  |-  ( N  e.  CC  ->  1  =  ( 2  -  1 ) )
98oveq2d 6219 . . . . . . . . . . . . . . 15  |-  ( N  e.  CC  ->  ( N  -  1 )  =  ( N  -  ( 2  -  1 ) ) )
10 id 22 . . . . . . . . . . . . . . . 16  |-  ( N  e.  CC  ->  N  e.  CC )
11 2cnd 10508 . . . . . . . . . . . . . . . 16  |-  ( N  e.  CC  ->  2  e.  CC )
12 ax-1cn 9454 . . . . . . . . . . . . . . . . 17  |-  1  e.  CC
1312a1i 11 . . . . . . . . . . . . . . . 16  |-  ( N  e.  CC  ->  1  e.  CC )
1410, 11, 13subsubd 9861 . . . . . . . . . . . . . . 15  |-  ( N  e.  CC  ->  ( N  -  ( 2  -  1 ) )  =  ( ( N  -  2 )  +  1 ) )
159, 14eqtr2d 2496 . . . . . . . . . . . . . 14  |-  ( N  e.  CC  ->  (
( N  -  2 )  +  1 )  =  ( N  - 
1 ) )
166, 15syl 16 . . . . . . . . . . . . 13  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( ( N  -  2 )  +  1 )  =  ( N  -  1 ) )
1716fveq2d 5806 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( w `  ( ( N  - 
2 )  +  1 ) )  =  ( w `  ( N  -  1 ) ) )
1817preq2d 4072 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  2
)  ->  { (
w `  ( N  -  2 ) ) ,  ( w `  ( ( N  - 
2 )  +  1 ) ) }  =  { ( w `  ( N  -  2
) ) ,  ( w `  ( N  -  1 ) ) } )
19183ad2ant3 1011 . . . . . . . . . 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 11006 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  -  2 )  e. 
NN0 )
22 uz2m1nn 11043 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  -  1 )  e.  NN )
23 1re 9499 . . . . . . . . . . . . . 14  |-  1  e.  RR
2423a1i 11 . . . . . . . . . . . . 13  |-  ( N  e.  ( ZZ>= `  2
)  ->  1  e.  RR )
25 2re 10505 . . . . . . . . . . . . . 14  |-  2  e.  RR
2625a1i 11 . . . . . . . . . . . . 13  |-  ( N  e.  ( ZZ>= `  2
)  ->  2  e.  RR )
27 eluzelre 10985 . . . . . . . . . . . . 13  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  RR )
28 1lt2 10602 . . . . . . . . . . . . . 14  |-  1  <  2
2928a1i 11 . . . . . . . . . . . . 13  |-  ( N  e.  ( ZZ>= `  2
)  ->  1  <  2 )
3024, 26, 27, 29ltsub2dd 10066 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  -  2 )  < 
( N  -  1 ) )
31 elfzo0 11707 . . . . . . . . . . . 12  |-  ( ( N  -  2 )  e.  ( 0..^ ( N  -  1 ) )  <->  ( ( N  -  2 )  e. 
NN0  /\  ( N  -  1 )  e.  NN  /\  ( N  -  2 )  < 
( N  -  1 ) ) )
3221, 22, 30, 31syl3anbrc 1172 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  -  2 )  e.  ( 0..^ ( N  -  1 ) ) )
33323ad2ant3 1011 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( N  -  2 )  e.  ( 0..^ ( N  -  1 ) ) )
34 simp2 989 . . . . . . . . . 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 5802 . . . . . . . . . . . . 13  |-  ( i  =  ( N  - 
2 )  ->  (
w `  i )  =  ( w `  ( N  -  2
) ) )
36 oveq1 6210 . . . . . . . . . . . . . 14  |-  ( i  =  ( N  - 
2 )  ->  (
i  +  1 )  =  ( ( N  -  2 )  +  1 ) )
3736fveq2d 5806 . . . . . . . . . . . . 13  |-  ( i  =  ( N  - 
2 )  ->  (
w `  ( i  +  1 ) )  =  ( w `  ( ( N  - 
2 )  +  1 ) ) )
3835, 37preq12d 4073 . . . . . . . . . . . 12  |-  ( i  =  ( N  - 
2 )  ->  { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  =  { ( w `  ( N  -  2
) ) ,  ( w `  ( ( N  -  2 )  +  1 ) ) } )
3938eleq1d 2523 . . . . . . . . . . 11  |-  ( i  =  ( N  - 
2 )  ->  ( { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( w `  ( N  -  2 ) ) ,  ( w `
 ( ( N  -  2 )  +  1 ) ) }  e.  ran  E ) )
4039rspcva 3177 . . . . . . . . . 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 2543 . . . . . . . 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 23468 . . . . . . . 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 4065 . . . . . . . . . . 11  |-  ( X  =  ( w ` 
0 )  ->  { X ,  ( w `  ( N  -  1
) ) }  =  { ( w ` 
0 ) ,  ( w `  ( N  -  1 ) ) } )
4847eqcoms 2466 . . . . . . . . . 10  |-  ( ( w `  0 )  =  X  ->  { X ,  ( w `  ( N  -  1
) ) }  =  { ( w ` 
0 ) ,  ( w `  ( N  -  1 ) ) } )
49 preq1 4065 . . . . . . . . . . 11  |-  ( ( w `  0 )  =  ( w `  ( N  -  2
) )  ->  { ( w `  0 ) ,  ( w `  ( N  -  1
) ) }  =  { ( w `  ( N  -  2
) ) ,  ( w `  ( N  -  1 ) ) } )
5049eqcoms 2466 . . . . . . . . . 10  |-  ( ( w `  ( N  -  2 ) )  =  ( w ` 
0 )  ->  { ( w `  0 ) ,  ( w `  ( N  -  1
) ) }  =  { ( w `  ( N  -  2
) ) ,  ( w `  ( N  -  1 ) ) } )
5148, 50sylan9eq 2515 . . . . . . . . 9  |-  ( ( ( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =  ( w `
 0 ) )  ->  { X , 
( w `  ( N  -  1 ) ) }  =  {
( w `  ( N  -  2 ) ) ,  ( w `
 ( N  - 
1 ) ) } )
5251eleq1d 2523 . . . . . . . 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 23442 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
56553ad2ant1 1009 . . . . . . . . 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 23509 . . . . . . 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 1171 . . . . 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 965    = wceq 1370    e. wcel 1758   A.wral 2799   _Vcvv 3078   {cpr 3990   <.cop 3994   class class class wbr 4403   ran crn 4952   ` cfv 5529  (class class class)co 6203   CCcc 9394   RRcr 9395   0cc0 9396   1c1 9397    + caddc 9399    < clt 9532    - cmin 9709   NNcn 10436   2c2 10485   NN0cn0 10693   ZZ>=cuz 10975  ..^cfzo 11668   #chash 12223  Word cword 12342   lastS clsw 12343   USGrph cusg 23436   Neighbors cnbgra 23501   ClWWalksN cclwwlkn 30582
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 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
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 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-fzo 11669  df-hash 12224  df-word 12350  df-usgra 23438  df-nbgra 23504  df-clwwlk 30584  df-clwwlkn 30585
This theorem is referenced by:  extwwlkfab  30851
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