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Theorem numclwwlk2lem1 24917
Description: In a friendship graph, for each walk of length n starting with a fixed vertex and ending not at this vertex, there is a unique vertex so that the walk extended by an edge to this vertex and an edge from this vertex to the first vertex of the walk is a value of operation H. If the walk is represented as a word, it is sufficient to add one vertex to the word to obtain the closed walk contained in the value of operation H, since in a word representing a closed walk the starting vertex is not repeated at the end. This theorem only generally holds for Friendship Graphs, because these guarantee that for the first and last vertex there is a third vertex "in between". (Contributed by Alexander van der Vekens, 3-Oct-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 ) ) } )
numclwwlk.q  |-  Q  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  (
( V WWalksN  E ) `  n )  |  ( ( w `  0
)  =  v  /\  ( lastS  `  w )  =/=  v ) } )
numclwwlk.h  |-  H  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( ( w `
 0 )  =  v  /\  ( w `
 ( n  - 
2 ) )  =/=  ( w `  0
) ) } )
Assertion
Ref Expression
numclwwlk2lem1  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  ( W  e.  ( X Q N )  ->  E! v  e.  V  ( W concat  <" v "> )  e.  ( X H ( N  +  2 ) ) ) )
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    w, Q    w, G    v, E    v, W, w
Allowed substitution hints:    Q( v, n)    F( v, n)    G( v, n)    H( w, v, n)    W( n)

Proof of Theorem numclwwlk2lem1
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 nnnn0 10814 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  NN0 )
21anim2i 569 . . . . . 6  |-  ( ( X  e.  V  /\  N  e.  NN )  ->  ( X  e.  V  /\  N  e.  NN0 ) )
323adant1 1014 . . . . 5  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  ( X  e.  V  /\  N  e.  NN0 ) )
4 numclwwlk.c . . . . . 6  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
5 numclwwlk.f . . . . . 6  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
6 numclwwlk.g . . . . . 6  |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>=
`  2 )  |->  { w  e.  ( C `
 n )  |  ( ( w ` 
0 )  =  v  /\  ( w `  ( n  -  2
) )  =  ( w `  0 ) ) } )
7 numclwwlk.q . . . . . 6  |-  Q  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  (
( V WWalksN  E ) `  n )  |  ( ( w `  0
)  =  v  /\  ( lastS  `  w )  =/=  v ) } )
84, 5, 6, 7numclwwlkovq 24914 . . . . 5  |-  ( ( X  e.  V  /\  N  e.  NN0 )  -> 
( X Q N )  =  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( ( w `
 0 )  =  X  /\  ( lastS  `  w
)  =/=  X ) } )
93, 8syl 16 . . . 4  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  ( X Q N )  =  { w  e.  ( ( V WWalksN  E ) `  N )  |  ( ( w `  0
)  =  X  /\  ( lastS  `  w )  =/= 
X ) } )
109eleq2d 2537 . . 3  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  ( W  e.  ( X Q N )  <->  W  e.  { w  e.  ( ( V WWalksN  E ) `  N
)  |  ( ( w `  0 )  =  X  /\  ( lastS  `  w )  =/=  X
) } ) )
11 fveq1 5871 . . . . . 6  |-  ( w  =  W  ->  (
w `  0 )  =  ( W ` 
0 ) )
1211eqeq1d 2469 . . . . 5  |-  ( w  =  W  ->  (
( w `  0
)  =  X  <->  ( W `  0 )  =  X ) )
13 fveq2 5872 . . . . . 6  |-  ( w  =  W  ->  ( lastS  `  w )  =  ( lastS  `  W ) )
1413neeq1d 2744 . . . . 5  |-  ( w  =  W  ->  (
( lastS  `  w )  =/= 
X  <->  ( lastS  `  W )  =/=  X ) )
1512, 14anbi12d 710 . . . 4  |-  ( w  =  W  ->  (
( ( w ` 
0 )  =  X  /\  ( lastS  `  w
)  =/=  X )  <-> 
( ( W ` 
0 )  =  X  /\  ( lastS  `  W
)  =/=  X ) ) )
1615elrab 3266 . . 3  |-  ( W  e.  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( ( w `
 0 )  =  X  /\  ( lastS  `  w
)  =/=  X ) }  <->  ( W  e.  ( ( V WWalksN  E
) `  N )  /\  ( ( W ` 
0 )  =  X  /\  ( lastS  `  W
)  =/=  X ) ) )
1710, 16syl6bb 261 . 2  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  ( W  e.  ( X Q N )  <->  ( W  e.  ( ( V WWalksN  E
) `  N )  /\  ( ( W ` 
0 )  =  X  /\  ( lastS  `  W
)  =/=  X ) ) ) )
18 simpl1 999 . . . . 5  |-  ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  V FriendGrph  E )
19 wwlknimp 24501 . . . . . . . . . . . 12  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( W  e. Word  V  /\  ( # `  W )  =  ( N  +  1 )  /\  A. i  e.  ( 0..^ N ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E
) )
20 peano2nn 10560 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  NN )
2120adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) )  /\  N  e.  NN )  ->  ( N  + 
1 )  e.  NN )
22 simpl 457 . . . . . . . . . . . . . . 15  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) )  /\  N  e.  NN )  ->  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )
2321, 22jca 532 . . . . . . . . . . . . . 14  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) )  /\  N  e.  NN )  ->  ( ( N  +  1 )  e.  NN  /\  ( W  e. Word  V  /\  ( # `
 W )  =  ( N  +  1 ) ) ) )
2423ex 434 . . . . . . . . . . . . 13  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( N  + 
1 ) )  -> 
( N  e.  NN  ->  ( ( N  + 
1 )  e.  NN  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) ) ) )
25243adant3 1016 . . . . . . . . . . . 12  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( N  + 
1 )  /\  A. i  e.  ( 0..^ N ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E )  ->  ( N  e.  NN  ->  ( ( N  +  1 )  e.  NN  /\  ( W  e. Word  V  /\  ( # `  W )  =  ( N  + 
1 ) ) ) ) )
2619, 25syl 16 . . . . . . . . . . 11  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( N  e.  NN  ->  ( ( N  +  1 )  e.  NN  /\  ( W  e. Word  V  /\  ( # `
 W )  =  ( N  +  1 ) ) ) ) )
27 lswlgt0cl 12570 . . . . . . . . . . 11  |-  ( ( ( N  +  1 )  e.  NN  /\  ( W  e. Word  V  /\  ( # `  W )  =  ( N  + 
1 ) ) )  ->  ( lastS  `  W )  e.  V )
2826, 27syl6 33 . . . . . . . . . 10  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( N  e.  NN  ->  ( lastS  `  W
)  e.  V ) )
2928adantr 465 . . . . . . . . 9  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( ( W `  0 )  =  X  /\  ( lastS  `  W )  =/=  X
) )  ->  ( N  e.  NN  ->  ( lastS  `  W )  e.  V
) )
3029com12 31 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) )  -> 
( lastS  `  W )  e.  V ) )
31303ad2ant3 1019 . . . . . . 7  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  (
( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) )  -> 
( lastS  `  W )  e.  V ) )
3231imp 429 . . . . . 6  |-  ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  ( lastS  `  W )  e.  V )
33 eleq1 2539 . . . . . . . . . . 11  |-  ( ( W `  0 )  =  X  ->  (
( W `  0
)  e.  V  <->  X  e.  V ) )
3433biimprd 223 . . . . . . . . . 10  |-  ( ( W `  0 )  =  X  ->  ( X  e.  V  ->  ( W `  0 )  e.  V ) )
3534ad2antrl 727 . . . . . . . . 9  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( ( W `  0 )  =  X  /\  ( lastS  `  W )  =/=  X
) )  ->  ( X  e.  V  ->  ( W `  0 )  e.  V ) )
3635com12 31 . . . . . . . 8  |-  ( X  e.  V  ->  (
( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) )  -> 
( W `  0
)  e.  V ) )
37363ad2ant2 1018 . . . . . . 7  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  (
( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) )  -> 
( W `  0
)  e.  V ) )
3837imp 429 . . . . . 6  |-  ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  ( W ` 
0 )  e.  V
)
39 neeq2 2750 . . . . . . . . . 10  |-  ( X  =  ( W ` 
0 )  ->  (
( lastS  `  W )  =/= 
X  <->  ( lastS  `  W )  =/=  ( W ` 
0 ) ) )
4039eqcoms 2479 . . . . . . . . 9  |-  ( ( W `  0 )  =  X  ->  (
( lastS  `  W )  =/= 
X  <->  ( lastS  `  W )  =/=  ( W ` 
0 ) ) )
4140biimpa 484 . . . . . . . 8  |-  ( ( ( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X )  ->  ( lastS  `  W )  =/=  ( W `  0 )
)
4241adantl 466 . . . . . . 7  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( ( W `  0 )  =  X  /\  ( lastS  `  W )  =/=  X
) )  ->  ( lastS  `  W )  =/=  ( W `  0 )
)
4342adantl 466 . . . . . 6  |-  ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  ( lastS  `  W )  =/=  ( W ` 
0 ) )
4432, 38, 433jca 1176 . . . . 5  |-  ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  ( ( lastS  `  W
)  e.  V  /\  ( W `  0 )  e.  V  /\  ( lastS  `  W )  =/=  ( W `  0 )
) )
45 frgraun 24810 . . . . 5  |-  ( V FriendGrph  E  ->  ( ( ( lastS  `  W )  e.  V  /\  ( W `  0
)  e.  V  /\  ( lastS  `  W )  =/=  ( W `  0
) )  ->  E! v  e.  V  ( { ( lastS  `  W ) ,  v }  e.  ran  E  /\  { v ,  ( W ` 
0 ) }  e.  ran  E ) ) )
4618, 44, 45sylc 60 . . . 4  |-  ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  E! v  e.  V  ( { ( lastS  `  W ) ,  v }  e.  ran  E  /\  { v ,  ( W `  0 ) }  e.  ran  E
) )
47 simpl 457 . . . . . . . . . . 11  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( ( W `  0 )  =  X  /\  ( lastS  `  W )  =/=  X
) )  ->  W  e.  ( ( V WWalksN  E
) `  N )
)
4847ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  /\  v  e.  V
)  ->  W  e.  ( ( V WWalksN  E
) `  N )
)
49 simpr 461 . . . . . . . . . 10  |-  ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  /\  v  e.  V
)  ->  v  e.  V )
5013ad2ant3 1019 . . . . . . . . . . 11  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  N  e.  NN0 )
5150ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  /\  v  e.  V
)  ->  N  e.  NN0 )
5248, 49, 513jca 1176 . . . . . . . . 9  |-  ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  /\  v  e.  V
)  ->  ( W  e.  ( ( V WWalksN  E
) `  N )  /\  v  e.  V  /\  N  e.  NN0 ) )
53 wwlkext2clwwlk 24617 . . . . . . . . . 10  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  v  e.  V  /\  N  e.  NN0 )  ->  ( ( { ( lastS  `  W ) ,  v }  e.  ran  E  /\  { v ,  ( W ` 
0 ) }  e.  ran  E )  ->  ( W concat  <" v "> )  e.  ( ( V ClWWalksN  E ) `  ( N  +  2 ) ) ) )
5453imp 429 . . . . . . . . 9  |-  ( ( ( W  e.  ( ( V WWalksN  E ) `  N )  /\  v  e.  V  /\  N  e. 
NN0 )  /\  ( { ( lastS  `  W ) ,  v }  e.  ran  E  /\  { v ,  ( W ` 
0 ) }  e.  ran  E ) )  -> 
( W concat  <" v "> )  e.  ( ( V ClWWalksN  E ) `  ( N  +  2 ) ) )
5552, 54sylan 471 . . . . . . . 8  |-  ( ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E
) `  N )  /\  ( ( W ` 
0 )  =  X  /\  ( lastS  `  W
)  =/=  X ) ) )  /\  v  e.  V )  /\  ( { ( lastS  `  W ) ,  v }  e.  ran  E  /\  { v ,  ( W ` 
0 ) }  e.  ran  E ) )  -> 
( W concat  <" v "> )  e.  ( ( V ClWWalksN  E ) `  ( N  +  2 ) ) )
56 2nn0 10824 . . . . . . . . . . . . 13  |-  2  e.  NN0
5756a1i 11 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  2  e.  NN0 )
581, 57nn0addcld 10868 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  ( N  +  2 )  e.  NN0 )
594numclwwlkfvc 24892 . . . . . . . . . . 11  |-  ( ( N  +  2 )  e.  NN0  ->  ( C `
 ( N  + 
2 ) )  =  ( ( V ClWWalksN  E ) `
 ( N  + 
2 ) ) )
6058, 59syl 16 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( C `  ( N  +  2 ) )  =  ( ( V ClWWalksN  E ) `  ( N  +  2 ) ) )
61603ad2ant3 1019 . . . . . . . . 9  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  ( C `  ( N  +  2 ) )  =  ( ( V ClWWalksN  E ) `  ( N  +  2 ) ) )
6261ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E
) `  N )  /\  ( ( W ` 
0 )  =  X  /\  ( lastS  `  W
)  =/=  X ) ) )  /\  v  e.  V )  /\  ( { ( lastS  `  W ) ,  v }  e.  ran  E  /\  { v ,  ( W ` 
0 ) }  e.  ran  E ) )  -> 
( C `  ( N  +  2 ) )  =  ( ( V ClWWalksN  E ) `  ( N  +  2 ) ) )
6355, 62eleqtrrd 2558 . . . . . . 7  |-  ( ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E
) `  N )  /\  ( ( W ` 
0 )  =  X  /\  ( lastS  `  W
)  =/=  X ) ) )  /\  v  e.  V )  /\  ( { ( lastS  `  W ) ,  v }  e.  ran  E  /\  { v ,  ( W ` 
0 ) }  e.  ran  E ) )  -> 
( W concat  <" v "> )  e.  ( C `  ( N  +  2 ) ) )
64 wwlknprop 24500 . . . . . . . . . . 11  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  W  e. Word  V ) ) )
65 simprr 756 . . . . . . . . . . 11  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  W  e. Word  V ) )  ->  W  e. Word  V )
6664, 65syl 16 . . . . . . . . . 10  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  W  e. Word  V )
6766ad2antrl 727 . . . . . . . . 9  |-  ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  W  e. Word  V
)
6867ad2antrr 725 . . . . . . . 8  |-  ( ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E
) `  N )  /\  ( ( W ` 
0 )  =  X  /\  ( lastS  `  W
)  =/=  X ) ) )  /\  v  e.  V )  /\  ( W concat  <" v "> )  e.  ( C `  ( N  +  2 ) ) )  ->  W  e. Word  V )
6949adantr 465 . . . . . . . 8  |-  ( ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E
) `  N )  /\  ( ( W ` 
0 )  =  X  /\  ( lastS  `  W
)  =/=  X ) ) )  /\  v  e.  V )  /\  ( W concat  <" v "> )  e.  ( C `  ( N  +  2 ) ) )  ->  v  e.  V )
70 2z 10908 . . . . . . . . . . 11  |-  2  e.  ZZ
71 nn0pzuz 11150 . . . . . . . . . . 11  |-  ( ( N  e.  NN0  /\  2  e.  ZZ )  ->  ( N  +  2 )  e.  ( ZZ>= ` 
2 ) )
721, 70, 71sylancl 662 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( N  +  2 )  e.  ( ZZ>= `  2
) )
73723ad2ant3 1019 . . . . . . . . 9  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  ( N  +  2 )  e.  ( ZZ>= `  2
) )
7473ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E
) `  N )  /\  ( ( W ` 
0 )  =  X  /\  ( lastS  `  W
)  =/=  X ) ) )  /\  v  e.  V )  /\  ( W concat  <" v "> )  e.  ( C `  ( N  +  2 ) ) )  ->  ( N  +  2 )  e.  ( ZZ>= `  2 )
)
7559eleq2d 2537 . . . . . . . . . . . 12  |-  ( ( N  +  2 )  e.  NN0  ->  ( ( W concat  <" v "> )  e.  ( C `  ( N  +  2 ) )  <-> 
( W concat  <" v "> )  e.  ( ( V ClWWalksN  E ) `  ( N  +  2 ) ) ) )
7658, 75syl 16 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
( W concat  <" v "> )  e.  ( C `  ( N  +  2 ) )  <-> 
( W concat  <" v "> )  e.  ( ( V ClWWalksN  E ) `  ( N  +  2 ) ) ) )
77763ad2ant3 1019 . . . . . . . . . 10  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  (
( W concat  <" v "> )  e.  ( C `  ( N  +  2 ) )  <-> 
( W concat  <" v "> )  e.  ( ( V ClWWalksN  E ) `  ( N  +  2 ) ) ) )
7877ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  /\  v  e.  V
)  ->  ( ( W concat  <" v "> )  e.  ( C `  ( N  +  2 ) )  <-> 
( W concat  <" v "> )  e.  ( ( V ClWWalksN  E ) `  ( N  +  2 ) ) ) )
7978biimpa 484 . . . . . . . 8  |-  ( ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E
) `  N )  /\  ( ( W ` 
0 )  =  X  /\  ( lastS  `  W
)  =/=  X ) ) )  /\  v  e.  V )  /\  ( W concat  <" v "> )  e.  ( C `  ( N  +  2 ) ) )  ->  ( W concat  <" v "> )  e.  ( ( V ClWWalksN  E ) `  ( N  +  2 ) ) )
80 clwwlkext2edg 24616 . . . . . . . 8  |-  ( ( ( W  e. Word  V  /\  v  e.  V  /\  ( N  +  2 )  e.  ( ZZ>= ` 
2 ) )  /\  ( W concat  <" v "> )  e.  ( ( V ClWWalksN  E ) `  ( N  +  2 ) ) )  -> 
( { ( lastS  `  W
) ,  v }  e.  ran  E  /\  { v ,  ( W `
 0 ) }  e.  ran  E ) )
8168, 69, 74, 79, 80syl31anc 1231 . . . . . . 7  |-  ( ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E
) `  N )  /\  ( ( W ` 
0 )  =  X  /\  ( lastS  `  W
)  =/=  X ) ) )  /\  v  e.  V )  /\  ( W concat  <" v "> )  e.  ( C `  ( N  +  2 ) ) )  ->  ( {
( lastS  `  W ) ,  v }  e.  ran  E  /\  { v ,  ( W `  0
) }  e.  ran  E ) )
8263, 81impbida 830 . . . . . 6  |-  ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  /\  v  e.  V
)  ->  ( ( { ( lastS  `  W ) ,  v }  e.  ran  E  /\  { v ,  ( W ` 
0 ) }  e.  ran  E )  <->  ( W concat  <" v "> )  e.  ( C `  ( N  +  2 ) ) ) )
83 df-3an 975 . . . . . . . . . . . . . 14  |-  ( ( X  e.  V  /\  N  e.  NN  /\  v  e.  V )  <->  ( ( X  e.  V  /\  N  e.  NN )  /\  v  e.  V
) )
8483simplbi2 625 . . . . . . . . . . . . 13  |-  ( ( X  e.  V  /\  N  e.  NN )  ->  ( v  e.  V  ->  ( X  e.  V  /\  N  e.  NN  /\  v  e.  V ) ) )
85843adant1 1014 . . . . . . . . . . . 12  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  (
v  e.  V  -> 
( X  e.  V  /\  N  e.  NN  /\  v  e.  V ) ) )
8685adantr 465 . . . . . . . . . . 11  |-  ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  ( v  e.  V  ->  ( X  e.  V  /\  N  e.  NN  /\  v  e.  V ) ) )
8786imp 429 . . . . . . . . . 10  |-  ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  /\  v  e.  V
)  ->  ( X  e.  V  /\  N  e.  NN  /\  v  e.  V ) )
88 3anass 977 . . . . . . . . . . . 12  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( W `  0 )  =  X  /\  ( lastS  `  W
)  =/=  X )  <-> 
( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )
8988biimpri 206 . . . . . . . . . . 11  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( ( W `  0 )  =  X  /\  ( lastS  `  W )  =/=  X
) )  ->  ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( W `  0 )  =  X  /\  ( lastS  `  W
)  =/=  X ) )
9089ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  /\  v  e.  V
)  ->  ( W  e.  ( ( V WWalksN  E
) `  N )  /\  ( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) )
9187, 90jca 532 . . . . . . . . 9  |-  ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  /\  v  e.  V
)  ->  ( ( X  e.  V  /\  N  e.  NN  /\  v  e.  V )  /\  ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( W `  0 )  =  X  /\  ( lastS  `  W
)  =/=  X ) ) )
92 clwwlkextfrlem1 24891 . . . . . . . . 9  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  v  e.  V )  /\  ( W  e.  ( ( V WWalksN  E
) `  N )  /\  ( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) )  -> 
( ( ( W concat  <" v "> ) `  0 )  =  X  /\  (
( W concat  <" v "> ) `  N
)  =/=  X ) )
93 simpl 457 . . . . . . . . . 10  |-  ( ( ( ( W concat  <" v "> ) `  0
)  =  X  /\  ( ( W concat  <" v "> ) `  N
)  =/=  X )  ->  ( ( W concat  <" v "> ) `  0 )  =  X )
94 neeq2 2750 . . . . . . . . . . . 12  |-  ( X  =  ( ( W concat  <" v "> ) `  0 )  ->  ( ( ( W concat  <" v "> ) `  N )  =/=  X  <->  ( ( W concat  <" v "> ) `  N )  =/=  ( ( W concat  <" v "> ) `  0
) ) )
9594eqcoms 2479 . . . . . . . . . . 11  |-  ( ( ( W concat  <" v "> ) `  0
)  =  X  -> 
( ( ( W concat  <" v "> ) `  N )  =/=  X  <->  ( ( W concat  <" v "> ) `  N )  =/=  ( ( W concat  <" v "> ) `  0
) ) )
9695biimpa 484 . . . . . . . . . 10  |-  ( ( ( ( W concat  <" v "> ) `  0
)  =  X  /\  ( ( W concat  <" v "> ) `  N
)  =/=  X )  ->  ( ( W concat  <" v "> ) `  N )  =/=  ( ( W concat  <" v "> ) `  0
) )
9793, 96jca 532 . . . . . . . . 9  |-  ( ( ( ( W concat  <" v "> ) `  0
)  =  X  /\  ( ( W concat  <" v "> ) `  N
)  =/=  X )  ->  ( ( ( W concat  <" v "> ) `  0
)  =  X  /\  ( ( W concat  <" v "> ) `  N
)  =/=  ( ( W concat  <" v "> ) `  0
) ) )
9891, 92, 973syl 20 . . . . . . . 8  |-  ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  /\  v  e.  V
)  ->  ( (
( W concat  <" v "> ) `  0
)  =  X  /\  ( ( W concat  <" v "> ) `  N
)  =/=  ( ( W concat  <" v "> ) `  0
) ) )
99 nncn 10556 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  e.  CC )
100 2cnd 10620 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  2  e.  CC )
10199, 100pncand 9943 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  (
( N  +  2 )  -  2 )  =  N )
1021013ad2ant3 1019 . . . . . . . . . . . 12  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  (
( N  +  2 )  -  2 )  =  N )
103102ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  /\  v  e.  V
)  ->  ( ( N  +  2 )  -  2 )  =  N )
104103fveq2d 5876 . . . . . . . . . 10  |-  ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  /\  v  e.  V
)  ->  ( ( W concat  <" v "> ) `  (
( N  +  2 )  -  2 ) )  =  ( ( W concat  <" v "> ) `  N
) )
105104neeq1d 2744 . . . . . . . . 9  |-  ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  /\  v  e.  V
)  ->  ( (
( W concat  <" v "> ) `  (
( N  +  2 )  -  2 ) )  =/=  ( ( W concat  <" v "> ) `  0
)  <->  ( ( W concat  <" v "> ) `  N )  =/=  ( ( W concat  <" v "> ) `  0
) ) )
106105anbi2d 703 . . . . . . . 8  |-  ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  /\  v  e.  V
)  ->  ( (
( ( W concat  <" v "> ) `  0
)  =  X  /\  ( ( W concat  <" v "> ) `  (
( N  +  2 )  -  2 ) )  =/=  ( ( W concat  <" v "> ) `  0
) )  <->  ( (
( W concat  <" v "> ) `  0
)  =  X  /\  ( ( W concat  <" v "> ) `  N
)  =/=  ( ( W concat  <" v "> ) `  0
) ) ) )
10798, 106mpbird 232 . . . . . . 7  |-  ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  /\  v  e.  V
)  ->  ( (
( W concat  <" v "> ) `  0
)  =  X  /\  ( ( W concat  <" v "> ) `  (
( N  +  2 )  -  2 ) )  =/=  ( ( W concat  <" v "> ) `  0
) ) )
108107biantrud 507 . . . . . 6  |-  ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  /\  v  e.  V
)  ->  ( ( W concat  <" v "> )  e.  ( C `  ( N  +  2 ) )  <-> 
( ( W concat  <" v "> )  e.  ( C `  ( N  +  2 ) )  /\  ( ( ( W concat  <" v "> ) `  0
)  =  X  /\  ( ( W concat  <" v "> ) `  (
( N  +  2 )  -  2 ) )  =/=  ( ( W concat  <" v "> ) `  0
) ) ) ) )
109 nn0addcl 10843 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  2  e.  NN0 )  -> 
( N  +  2 )  e.  NN0 )
1101, 56, 109sylancl 662 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  ( N  +  2 )  e.  NN0 )
111110anim2i 569 . . . . . . . . . . 11  |-  ( ( X  e.  V  /\  N  e.  NN )  ->  ( X  e.  V  /\  ( N  +  2 )  e.  NN0 )
)
1121113adant1 1014 . . . . . . . . . 10  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  ( X  e.  V  /\  ( N  +  2
)  e.  NN0 )
)
113112ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  /\  v  e.  V
)  ->  ( X  e.  V  /\  ( N  +  2 )  e.  NN0 ) )
114 numclwwlk.h . . . . . . . . . 10  |-  H  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( ( w `
 0 )  =  v  /\  ( w `
 ( n  - 
2 ) )  =/=  ( w `  0
) ) } )
1154, 5, 6, 7, 114numclwwlkovh 24916 . . . . . . . . 9  |-  ( ( X  e.  V  /\  ( N  +  2
)  e.  NN0 )  ->  ( X H ( N  +  2 ) )  =  { w  e.  ( C `  ( N  +  2 ) )  |  ( ( w `  0 )  =  X  /\  (
w `  ( ( N  +  2 )  -  2 ) )  =/=  ( w ` 
0 ) ) } )
116113, 115syl 16 . . . . . . . 8  |-  ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  /\  v  e.  V
)  ->  ( X H ( N  + 
2 ) )  =  { w  e.  ( C `  ( N  +  2 ) )  |  ( ( w `
 0 )  =  X  /\  ( w `
 ( ( N  +  2 )  - 
2 ) )  =/=  ( w `  0
) ) } )
117116eleq2d 2537 . . . . . . 7  |-  ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  /\  v  e.  V
)  ->  ( ( W concat  <" v "> )  e.  ( X H ( N  +  2 ) )  <-> 
( W concat  <" v "> )  e.  {
w  e.  ( C `
 ( N  + 
2 ) )  |  ( ( w ` 
0 )  =  X  /\  ( w `  ( ( N  + 
2 )  -  2 ) )  =/=  (
w `  0 )
) } ) )
118 fveq1 5871 . . . . . . . . . 10  |-  ( w  =  ( W concat  <" v "> )  ->  (
w `  0 )  =  ( ( W concat  <" v "> ) `  0 )
)
119118eqeq1d 2469 . . . . . . . . 9  |-  ( w  =  ( W concat  <" v "> )  ->  (
( w `  0
)  =  X  <->  ( ( W concat  <" v "> ) `  0
)  =  X ) )
120 fveq1 5871 . . . . . . . . . 10  |-  ( w  =  ( W concat  <" v "> )  ->  (
w `  ( ( N  +  2 )  -  2 ) )  =  ( ( W concat  <" v "> ) `  ( ( N  +  2 )  -  2 ) ) )
121120, 118neeq12d 2746 . . . . . . . . 9  |-  ( w  =  ( W concat  <" v "> )  ->  (
( w `  (
( N  +  2 )  -  2 ) )  =/=  ( w `
 0 )  <->  ( ( W concat  <" v "> ) `  (
( N  +  2 )  -  2 ) )  =/=  ( ( W concat  <" v "> ) `  0
) ) )
122119, 121anbi12d 710 . . . . . . . 8  |-  ( w  =  ( W concat  <" v "> )  ->  (
( ( w ` 
0 )  =  X  /\  ( w `  ( ( N  + 
2 )  -  2 ) )  =/=  (
w `  0 )
)  <->  ( ( ( W concat  <" v "> ) `  0
)  =  X  /\  ( ( W concat  <" v "> ) `  (
( N  +  2 )  -  2 ) )  =/=  ( ( W concat  <" v "> ) `  0
) ) ) )
123122elrab 3266 . . . . . . 7  |-  ( ( W concat  <" v "> )  e.  {
w  e.  ( C `
 ( N  + 
2 ) )  |  ( ( w ` 
0 )  =  X  /\  ( w `  ( ( N  + 
2 )  -  2 ) )  =/=  (
w `  0 )
) }  <->  ( ( W concat  <" v "> )  e.  ( C `  ( N  +  2 ) )  /\  ( ( ( W concat  <" v "> ) `  0
)  =  X  /\  ( ( W concat  <" v "> ) `  (
( N  +  2 )  -  2 ) )  =/=  ( ( W concat  <" v "> ) `  0
) ) ) )
124117, 123syl6rbb 262 . . . . . 6  |-  ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  /\  v  e.  V
)  ->  ( (
( W concat  <" v "> )  e.  ( C `  ( N  +  2 ) )  /\  ( ( ( W concat  <" v "> ) `  0
)  =  X  /\  ( ( W concat  <" v "> ) `  (
( N  +  2 )  -  2 ) )  =/=  ( ( W concat  <" v "> ) `  0
) ) )  <->  ( W concat  <" v "> )  e.  ( X H ( N  + 
2 ) ) ) )
12582, 108, 1243bitrd 279 . . . . 5  |-  ( ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  /\  v  e.  V
)  ->  ( ( { ( lastS  `  W ) ,  v }  e.  ran  E  /\  { v ,  ( W ` 
0 ) }  e.  ran  E )  <->  ( W concat  <" v "> )  e.  ( X H ( N  + 
2 ) ) ) )
126125reubidva 3050 . . . 4  |-  ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  ( E! v  e.  V  ( { ( lastS  `  W ) ,  v }  e.  ran  E  /\  { v ,  ( W ` 
0 ) }  e.  ran  E )  <->  E! v  e.  V  ( W concat  <" v "> )  e.  ( X H ( N  + 
2 ) ) ) )
12746, 126mpbid 210 . . 3  |-  ( ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  E! v  e.  V  ( W concat  <" v "> )  e.  ( X H ( N  +  2 ) ) )
128127ex 434 . 2  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  (
( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) )  ->  E! v  e.  V  ( W concat  <" v "> )  e.  ( X H ( N  +  2 ) ) ) )
12917, 128sylbid 215 1  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  ( W  e.  ( X Q N )  ->  E! v  e.  V  ( W concat  <" v "> )  e.  ( X H ( N  +  2 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E!wreu 2819   {crab 2821   _Vcvv 3118   {cpr 4035   class class class wbr 4453    |-> cmpt 4511   ran crn 5006   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   0cc0 9504   1c1 9505    + caddc 9507    - cmin 9817   NNcn 10548   2c2 10597   NN0cn0 10807   ZZcz 10876   ZZ>=cuz 11094  ..^cfzo 11804   #chash 12385  Word cword 12515   lastS clsw 12516   concat cconcat 12517   <"cs1 12518   WWalksN cwwlkn 24492   ClWWalksN cclwwlkn 24563   FriendGrph cfrgra 24802
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-lsw 12524  df-concat 12525  df-s1 12526  df-wwlk 24493  df-wwlkn 24494  df-clwwlk 24565  df-clwwlkn 24566  df-frgra 24803
This theorem is referenced by:  numclwlk2lem2f1o  24920
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