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Theorem extwwlkfablem1 25195
Description: Lemma 1 for extwwlkfab 25211. (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 24897 . . . 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 1037 . . . . . 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 1036 . . . . . . 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 11012 . . . . . . . . . . . . . 14  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  CC )
5 1e2m1 10568 . . . . . . . . . . . . . . . . 17  |-  1  =  ( 2  -  1 )
65a1i 11 . . . . . . . . . . . . . . . 16  |-  ( N  e.  CC  ->  1  =  ( 2  -  1 ) )
76oveq2d 6212 . . . . . . . . . . . . . . 15  |-  ( N  e.  CC  ->  ( N  -  1 )  =  ( N  -  ( 2  -  1 ) ) )
8 id 22 . . . . . . . . . . . . . . . 16  |-  ( N  e.  CC  ->  N  e.  CC )
9 2cnd 10525 . . . . . . . . . . . . . . . 16  |-  ( N  e.  CC  ->  2  e.  CC )
10 1cnd 9523 . . . . . . . . . . . . . . . 16  |-  ( N  e.  CC  ->  1  e.  CC )
118, 9, 10subsubd 9872 . . . . . . . . . . . . . . 15  |-  ( N  e.  CC  ->  ( N  -  ( 2  -  1 ) )  =  ( ( N  -  2 )  +  1 ) )
127, 11eqtr2d 2424 . . . . . . . . . . . . . 14  |-  ( N  e.  CC  ->  (
( N  -  2 )  +  1 )  =  ( N  - 
1 ) )
134, 12syl 16 . . . . . . . . . . . . 13  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( ( N  -  2 )  +  1 )  =  ( N  -  1 ) )
1413fveq2d 5778 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( w `  ( ( N  - 
2 )  +  1 ) )  =  ( w `  ( N  -  1 ) ) )
1514preq2d 4030 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  2
)  ->  { (
w `  ( N  -  2 ) ) ,  ( w `  ( ( N  - 
2 )  +  1 ) ) }  =  { ( w `  ( N  -  2
) ) ,  ( w `  ( N  -  1 ) ) } )
16153ad2ant3 1017 . . . . . . . . . 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 464 . . . . . . . . 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 11778 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  -  2 )  e.  ( 0..^ ( N  -  1 ) ) )
19183ad2ant3 1017 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( N  -  2 )  e.  ( 0..^ ( N  -  1 ) ) )
20 simp2 995 . . . . . . . . . 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 5774 . . . . . . . . . . . . 13  |-  ( i  =  ( N  - 
2 )  ->  (
w `  i )  =  ( w `  ( N  -  2
) ) )
22 oveq1 6203 . . . . . . . . . . . . . 14  |-  ( i  =  ( N  - 
2 )  ->  (
i  +  1 )  =  ( ( N  -  2 )  +  1 ) )
2322fveq2d 5778 . . . . . . . . . . . . 13  |-  ( i  =  ( N  - 
2 )  ->  (
w `  ( i  +  1 ) )  =  ( w `  ( ( N  - 
2 )  +  1 ) ) )
2421, 23preq12d 4031 . . . . . . . . . . . 12  |-  ( i  =  ( N  - 
2 )  ->  { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  =  { ( w `  ( N  -  2
) ) ,  ( w `  ( ( N  -  2 )  +  1 ) ) } )
2524eleq1d 2451 . . . . . . . . . . 11  |-  ( i  =  ( N  - 
2 )  ->  ( { ( w `  i ) ,  ( w `  ( i  +  1 ) ) }  e.  ran  E  <->  { ( w `  ( N  -  2 ) ) ,  ( w `
 ( ( N  -  2 )  +  1 ) ) }  e.  ran  E ) )
2625rspcva 3133 . . . . . . . . . 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 476 . . . . . . . . 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 2471 . . . . . . . 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 463 . . . . . . 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 24498 . . . . . . . 8  |-  ( ( V USGrph  E  /\  { ( w `  ( N  -  2 ) ) ,  ( w `  ( N  -  1
) ) }  e.  ran  E )  ->  (
( w `  ( N  -  2 ) )  e.  V  /\  ( w `  ( N  -  1 ) )  e.  V ) )
3130simprd 461 . . . . . . 7  |-  ( ( V USGrph  E  /\  { ( w `  ( N  -  2 ) ) ,  ( w `  ( N  -  1
) ) }  e.  ran  E )  ->  (
w `  ( N  -  1 ) )  e.  V )
323, 29, 31syl2anc 659 . . . . . 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 4023 . . . . . . . . . . 11  |-  ( X  =  ( w ` 
0 )  ->  { X ,  ( w `  ( N  -  1
) ) }  =  { ( w ` 
0 ) ,  ( w `  ( N  -  1 ) ) } )
3433eqcoms 2394 . . . . . . . . . 10  |-  ( ( w `  0 )  =  X  ->  { X ,  ( w `  ( N  -  1
) ) }  =  { ( w ` 
0 ) ,  ( w `  ( N  -  1 ) ) } )
35 preq1 4023 . . . . . . . . . . 11  |-  ( ( w `  0 )  =  ( w `  ( N  -  2
) )  ->  { ( w `  0 ) ,  ( w `  ( N  -  1
) ) }  =  { ( w `  ( N  -  2
) ) ,  ( w `  ( N  -  1 ) ) } )
3635eqcoms 2394 . . . . . . . . . 10  |-  ( ( w `  ( N  -  2 ) )  =  ( w ` 
0 )  ->  { ( w `  0 ) ,  ( w `  ( N  -  1
) ) }  =  { ( w `  ( N  -  2
) ) ,  ( w `  ( N  -  1 ) ) } )
3734, 36sylan9eq 2443 . . . . . . . . 9  |-  ( ( ( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =  ( w `
 0 ) )  ->  { X , 
( w `  ( N  -  1 ) ) }  =  {
( w `  ( N  -  2 ) ) ,  ( w `
 ( N  - 
1 ) ) } )
3837eleq1d 2451 . . . . . . . 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 464 . . . . . . 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 24459 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
42413ad2ant1 1015 . . . . . . . . 9  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( V  e.  _V  /\  E  e. 
_V ) )
4342adantl 464 . . . . . . . 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 463 . . . . . . 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 24547 . . . . . . 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 1177 . . . . 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 602 . . . 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 428 . 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 427 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826   A.wral 2732   _Vcvv 3034   {cpr 3946   <.cop 3950   class class class wbr 4367   ran crn 4914   ` cfv 5496  (class class class)co 6196   CCcc 9401   0cc0 9403   1c1 9404    + caddc 9406    - cmin 9718   2c2 10502   ZZ>=cuz 11001  ..^cfzo 11717   #chash 12307  Word cword 12438   lastS clsw 12439   USGrph cusg 24451   Neighbors cnbgra 24538   ClWWalksN cclwwlkn 24870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-pm 7341  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-fzo 11718  df-hash 12308  df-word 12446  df-usgra 24454  df-nbgra 24541  df-clwwlk 24872  df-clwwlkn 24873
This theorem is referenced by:  extwwlkfab  25211
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