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Theorem extwwlkfablem1 24943
Description: Lemma 1 for extwwlkfab 24959. (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 24645 . . . 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 simplr2 1038 . . . . . 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
)
3 simplr1 1037 . . . . . . 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 )
4 eluzelcn 11098 . . . . . . . . . . . . . 14  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  CC )
5 1e2m1 10654 . . . . . . . . . . . . . . . . 17  |-  1  =  ( 2  -  1 )
65a1i 11 . . . . . . . . . . . . . . . 16  |-  ( N  e.  CC  ->  1  =  ( 2  -  1 ) )
76oveq2d 6294 . . . . . . . . . . . . . . 15  |-  ( N  e.  CC  ->  ( N  -  1 )  =  ( N  -  ( 2  -  1 ) ) )
8 id 22 . . . . . . . . . . . . . . . 16  |-  ( N  e.  CC  ->  N  e.  CC )
9 2cnd 10611 . . . . . . . . . . . . . . . 16  |-  ( N  e.  CC  ->  2  e.  CC )
10 1cnd 9612 . . . . . . . . . . . . . . . 16  |-  ( N  e.  CC  ->  1  e.  CC )
118, 9, 10subsubd 9961 . . . . . . . . . . . . . . 15  |-  ( N  e.  CC  ->  ( N  -  ( 2  -  1 ) )  =  ( ( N  -  2 )  +  1 ) )
127, 11eqtr2d 2483 . . . . . . . . . . . . . 14  |-  ( N  e.  CC  ->  (
( N  -  2 )  +  1 )  =  ( N  - 
1 ) )
134, 12syl 16 . . . . . . . . . . . . 13  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( ( N  -  2 )  +  1 )  =  ( N  -  1 ) )
1413fveq2d 5857 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( w `  ( ( N  - 
2 )  +  1 ) )  =  ( w `  ( N  -  1 ) ) )
1514preq2d 4098 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  2
)  ->  { (
w `  ( N  -  2 ) ) ,  ( w `  ( ( N  - 
2 )  +  1 ) ) }  =  { ( w `  ( N  -  2
) ) ,  ( w `  ( N  -  1 ) ) } )
16153ad2ant3 1018 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  2 )
)  ->  { (
w `  ( N  -  2 ) ) ,  ( w `  ( ( N  - 
2 )  +  1 ) ) }  =  { ( w `  ( N  -  2
) ) ,  ( w `  ( N  -  1 ) ) } )
1716adantl 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 ) ) } )
18 ige2m2fzo 11855 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  -  2 )  e.  ( 0..^ ( N  -  1 ) ) )
19183ad2ant3 1018 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( N  -  2 )  e.  ( 0..^ ( N  -  1 ) ) )
20 simp2 996 . . . . . . . . . 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 )
21 fveq2 5853 . . . . . . . . . . . . 13  |-  ( i  =  ( N  - 
2 )  ->  (
w `  i )  =  ( w `  ( N  -  2
) ) )
22 oveq1 6285 . . . . . . . . . . . . . 14  |-  ( i  =  ( N  - 
2 )  ->  (
i  +  1 )  =  ( ( N  -  2 )  +  1 ) )
2322fveq2d 5857 . . . . . . . . . . . . 13  |-  ( i  =  ( N  - 
2 )  ->  (
w `  ( i  +  1 ) )  =  ( w `  ( ( N  - 
2 )  +  1 ) ) )
2421, 23preq12d 4099 . . . . . . . . . . . 12  |-  ( i  =  ( N  - 
2 )  ->  { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  =  { ( w `  ( N  -  2
) ) ,  ( w `  ( ( N  -  2 )  +  1 ) ) } )
2524eleq1d 2510 . . . . . . . . . . 11  |-  ( i  =  ( N  - 
2 )  ->  ( { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( w `  ( N  -  2 ) ) ,  ( w `
 ( ( N  -  2 )  +  1 ) ) }  e.  ran  E ) )
2625rspcva 3192 . . . . . . . . . 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 )
2719, 20, 26syl2anr 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 )
2817, 27eqeltrrd 2530 . . . . . . . 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 )
2928adantr 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 )
30 usgraedgrnv 24246 . . . . . . . 8  |-  ( ( V USGrph  E  /\  { ( w `  ( N  -  2 ) ) ,  ( w `  ( N  -  1
) ) }  e.  ran  E )  ->  (
( w `  ( N  -  2 ) )  e.  V  /\  ( w `  ( N  -  1 ) )  e.  V ) )
3130simprd 463 . . . . . . 7  |-  ( ( V USGrph  E  /\  { ( w `  ( N  -  2 ) ) ,  ( w `  ( N  -  1
) ) }  e.  ran  E )  ->  (
w `  ( N  -  1 ) )  e.  V )
323, 29, 31syl2anc 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
)
33 preq1 4091 . . . . . . . . . . 11  |-  ( X  =  ( w ` 
0 )  ->  { X ,  ( w `  ( N  -  1
) ) }  =  { ( w ` 
0 ) ,  ( w `  ( N  -  1 ) ) } )
3433eqcoms 2453 . . . . . . . . . 10  |-  ( ( w `  0 )  =  X  ->  { X ,  ( w `  ( N  -  1
) ) }  =  { ( w ` 
0 ) ,  ( w `  ( N  -  1 ) ) } )
35 preq1 4091 . . . . . . . . . . 11  |-  ( ( w `  0 )  =  ( w `  ( N  -  2
) )  ->  { ( w `  0 ) ,  ( w `  ( N  -  1
) ) }  =  { ( w `  ( N  -  2
) ) ,  ( w `  ( N  -  1 ) ) } )
3635eqcoms 2453 . . . . . . . . . 10  |-  ( ( w `  ( N  -  2 ) )  =  ( w ` 
0 )  ->  { ( w `  0 ) ,  ( w `  ( N  -  1
) ) }  =  { ( w `  ( N  -  2
) ) ,  ( w `  ( N  -  1 ) ) } )
3734, 36sylan9eq 2502 . . . . . . . . 9  |-  ( ( ( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =  ( w `
 0 ) )  ->  { X , 
( w `  ( N  -  1 ) ) }  =  {
( w `  ( N  -  2 ) ) ,  ( w `
 ( N  - 
1 ) ) } )
3837eleq1d 2510 . . . . . . . 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
) )
3938adantl 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
) )
4029, 39mpbird 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 )
41 usgrav 24207 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
42413ad2ant1 1016 . . . . . . . . 9  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( V  e.  _V  /\  E  e. 
_V ) )
4342adantl 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 ) )
4443adantr 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 ) )
45 nbgrael 24295 . . . . . . 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 ) ) )
4644, 45syl 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 ) ) )
472, 32, 40, 46mpbir3and 1178 . . . . 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
) )
4847exp31 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
) ) ) )
491, 48syl 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
) ) ) )
5049impcom 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
) ) )
5150imp 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 972    = wceq 1381    e. wcel 1802   A.wral 2791   _Vcvv 3093   {cpr 4013   <.cop 4017   class class class wbr 4434   ran crn 4987   ` cfv 5575  (class class class)co 6278   CCcc 9490   0cc0 9492   1c1 9493    + caddc 9495    - cmin 9807   2c2 10588   ZZ>=cuz 11087  ..^cfzo 11800   #chash 12381  Word cword 12510   lastS clsw 12511   USGrph cusg 24199   Neighbors cnbgra 24286   ClWWalksN cclwwlkn 24618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-int 4269  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6683  df-1st 6782  df-2nd 6783  df-recs 7041  df-rdg 7075  df-1o 7129  df-oadd 7133  df-er 7310  df-map 7421  df-pm 7422  df-en 7516  df-dom 7517  df-sdom 7518  df-fin 7519  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-nn 10540  df-2 10597  df-n0 10799  df-z 10868  df-uz 11088  df-fz 11679  df-fzo 11801  df-hash 12382  df-word 12518  df-usgra 24202  df-nbgra 24289  df-clwwlk 24620  df-clwwlkn 24621
This theorem is referenced by:  extwwlkfab  24959
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