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Theorem numclwwlk2lem1 25683
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 ++  <" 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 10876 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  NN0 )
21anim2i 571 . . . . . 6  |-  ( ( X  e.  V  /\  N  e.  NN )  ->  ( X  e.  V  /\  N  e.  NN0 ) )
323adant1 1023 . . . . 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 25680 . . . . 5  |-  ( ( X  e.  V  /\  N  e.  NN0 )  -> 
( X Q N )  =  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( ( w `
 0 )  =  X  /\  ( lastS  `  w
)  =/=  X ) } )
93, 8syl 17 . . . 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 2499 . . 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 5880 . . . . . 6  |-  ( w  =  W  ->  (
w `  0 )  =  ( W ` 
0 ) )
1211eqeq1d 2431 . . . . 5  |-  ( w  =  W  ->  (
( w `  0
)  =  X  <->  ( W `  0 )  =  X ) )
13 fveq2 5881 . . . . . 6  |-  ( w  =  W  ->  ( lastS  `  w )  =  ( lastS  `  W ) )
1413neeq1d 2708 . . . . 5  |-  ( w  =  W  ->  (
( lastS  `  w )  =/= 
X  <->  ( lastS  `  W )  =/=  X ) )
1512, 14anbi12d 715 . . . 4  |-  ( w  =  W  ->  (
( ( w ` 
0 )  =  X  /\  ( lastS  `  w
)  =/=  X )  <-> 
( ( W ` 
0 )  =  X  /\  ( lastS  `  W
)  =/=  X ) ) )
1615elrab 3235 . . 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 264 . 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 1008 . . . . 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 25268 . . . . . . . . . . . 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 10621 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  NN )
2120adantl 467 . . . . . . . . . . . . . . 15  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) )  /\  N  e.  NN )  ->  ( N  + 
1 )  e.  NN )
22 simpl 458 . . . . . . . . . . . . . . 15  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) )  /\  N  e.  NN )  ->  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )
2321, 22jca 534 . . . . . . . . . . . . . 14  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) )  /\  N  e.  NN )  ->  ( ( N  +  1 )  e.  NN  /\  ( W  e. Word  V  /\  ( # `
 W )  =  ( N  +  1 ) ) ) )
2423ex 435 . . . . . . . . . . . . 13  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( N  + 
1 ) )  -> 
( N  e.  NN  ->  ( ( N  + 
1 )  e.  NN  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) ) ) )
25243adant3 1025 . . . . . . . . . . . 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 17 . . . . . . . . . . 11  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( N  e.  NN  ->  ( ( N  +  1 )  e.  NN  /\  ( W  e. Word  V  /\  ( # `
 W )  =  ( N  +  1 ) ) ) ) )
27 lswlgt0cl 12703 . . . . . . . . . . 11  |-  ( ( ( N  +  1 )  e.  NN  /\  ( W  e. Word  V  /\  ( # `  W )  =  ( N  + 
1 ) ) )  ->  ( lastS  `  W )  e.  V )
2826, 27syl6 34 . . . . . . . . . 10  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( N  e.  NN  ->  ( lastS  `  W
)  e.  V ) )
2928adantr 466 . . . . . . . . 9  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( ( W `  0 )  =  X  /\  ( lastS  `  W )  =/=  X
) )  ->  ( N  e.  NN  ->  ( lastS  `  W )  e.  V
) )
3029com12 32 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) )  -> 
( lastS  `  W )  e.  V ) )
31303ad2ant3 1028 . . . . . . 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 430 . . . . . 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 2501 . . . . . . . . . . 11  |-  ( ( W `  0 )  =  X  ->  (
( W `  0
)  e.  V  <->  X  e.  V ) )
3433biimprd 226 . . . . . . . . . 10  |-  ( ( W `  0 )  =  X  ->  ( X  e.  V  ->  ( W `  0 )  e.  V ) )
3534ad2antrl 732 . . . . . . . . 9  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( ( W `  0 )  =  X  /\  ( lastS  `  W )  =/=  X
) )  ->  ( X  e.  V  ->  ( W `  0 )  e.  V ) )
3635com12 32 . . . . . . . 8  |-  ( X  e.  V  ->  (
( W  e.  ( ( V WWalksN  E ) `  N )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) )  -> 
( W `  0
)  e.  V ) )
37363ad2ant2 1027 . . . . . . 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 430 . . . . . 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 2714 . . . . . . . . . 10  |-  ( X  =  ( W ` 
0 )  ->  (
( lastS  `  W )  =/= 
X  <->  ( lastS  `  W )  =/=  ( W ` 
0 ) ) )
4039eqcoms 2441 . . . . . . . . 9  |-  ( ( W `  0 )  =  X  ->  (
( lastS  `  W )  =/= 
X  <->  ( lastS  `  W )  =/=  ( W ` 
0 ) ) )
4140biimpa 486 . . . . . . . 8  |-  ( ( ( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X )  ->  ( lastS  `  W )  =/=  ( W `  0 )
)
4241adantl 467 . . . . . . 7  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( ( W `  0 )  =  X  /\  ( lastS  `  W )  =/=  X
) )  ->  ( lastS  `  W )  =/=  ( W `  0 )
)
4342adantl 467 . . . . . 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 1185 . . . . 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 25577 . . . . 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 62 . . . 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 458 . . . . . . . . . . 11  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( ( W `  0 )  =  X  /\  ( lastS  `  W )  =/=  X
) )  ->  W  e.  ( ( V WWalksN  E
) `  N )
)
4847ad2antlr 731 . . . . . . . . . 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 462 . . . . . . . . . 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 1028 . . . . . . . . . . 11  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  N  e.  NN0 )
5150ad2antrr 730 . . . . . . . . . 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 1185 . . . . . . . . 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 25384 . . . . . . . . . 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 ++  <" v "> )  e.  ( ( V ClWWalksN  E ) `  ( N  +  2 ) ) ) )
5453imp 430 . . . . . . . . 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 ++  <" v "> )  e.  ( ( V ClWWalksN  E ) `  ( N  +  2 ) ) )
5552, 54sylan 473 . . . . . . . 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 ++  <" v "> )  e.  ( ( V ClWWalksN  E ) `  ( N  +  2 ) ) )
56 2nn0 10886 . . . . . . . . . . . . 13  |-  2  e.  NN0
5756a1i 11 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  2  e.  NN0 )
581, 57nn0addcld 10929 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  ( N  +  2 )  e.  NN0 )
594numclwwlkfvc 25658 . . . . . . . . . . 11  |-  ( ( N  +  2 )  e.  NN0  ->  ( C `
 ( N  + 
2 ) )  =  ( ( V ClWWalksN  E ) `
 ( N  + 
2 ) ) )
6058, 59syl 17 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( C `  ( N  +  2 ) )  =  ( ( V ClWWalksN  E ) `  ( N  +  2 ) ) )
61603ad2ant3 1028 . . . . . . . . 9  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  ( C `  ( N  +  2 ) )  =  ( ( V ClWWalksN  E ) `  ( N  +  2 ) ) )
6261ad3antrrr 734 . . . . . . . 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 2520 . . . . . . 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 ++  <" v "> )  e.  ( C `  ( N  +  2 ) ) )
64 wwlknprop 25267 . . . . . . . . . . 11  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  W  e. Word  V ) ) )
6564simprrd 765 . . . . . . . . . 10  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  W  e. Word  V )
6665ad2antrl 732 . . . . . . . . 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
)
6766ad2antrr 730 . . . . . . . 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 ++  <" v "> )  e.  ( C `  ( N  +  2 ) ) )  ->  W  e. Word  V )
6849adantr 466 . . . . . . . 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 ++  <" v "> )  e.  ( C `  ( N  +  2 ) ) )  ->  v  e.  V )
69 2z 10969 . . . . . . . . . . 11  |-  2  e.  ZZ
70 nn0pzuz 11216 . . . . . . . . . . 11  |-  ( ( N  e.  NN0  /\  2  e.  ZZ )  ->  ( N  +  2 )  e.  ( ZZ>= ` 
2 ) )
711, 69, 70sylancl 666 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( N  +  2 )  e.  ( ZZ>= `  2
) )
72713ad2ant3 1028 . . . . . . . . 9  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  ( N  +  2 )  e.  ( ZZ>= `  2
) )
7372ad3antrrr 734 . . . . . . . 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 ++  <" v "> )  e.  ( C `  ( N  +  2 ) ) )  ->  ( N  +  2 )  e.  ( ZZ>= `  2 )
)
7459eleq2d 2499 . . . . . . . . . . . 12  |-  ( ( N  +  2 )  e.  NN0  ->  ( ( W ++  <" v "> )  e.  ( C `  ( N  +  2 ) )  <-> 
( W ++  <" v "> )  e.  ( ( V ClWWalksN  E ) `  ( N  +  2 ) ) ) )
7558, 74syl 17 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
( W ++  <" v "> )  e.  ( C `  ( N  +  2 ) )  <-> 
( W ++  <" v "> )  e.  ( ( V ClWWalksN  E ) `  ( N  +  2 ) ) ) )
76753ad2ant3 1028 . . . . . . . . . 10  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  (
( W ++  <" v "> )  e.  ( C `  ( N  +  2 ) )  <-> 
( W ++  <" v "> )  e.  ( ( V ClWWalksN  E ) `  ( N  +  2 ) ) ) )
7776ad2antrr 730 . . . . . . . . 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 ++  <" v "> )  e.  ( C `  ( N  +  2 ) )  <-> 
( W ++  <" v "> )  e.  ( ( V ClWWalksN  E ) `  ( N  +  2 ) ) ) )
7877biimpa 486 . . . . . . . 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 ++  <" v "> )  e.  ( C `  ( N  +  2 ) ) )  ->  ( W ++  <" v "> )  e.  ( ( V ClWWalksN  E ) `  ( N  +  2 ) ) )
79 clwwlkext2edg 25383 . . . . . . . 8  |-  ( ( ( W  e. Word  V  /\  v  e.  V  /\  ( N  +  2 )  e.  ( ZZ>= ` 
2 ) )  /\  ( W ++  <" v "> )  e.  ( ( V ClWWalksN  E ) `  ( N  +  2 ) ) )  -> 
( { ( lastS  `  W
) ,  v }  e.  ran  E  /\  { v ,  ( W `
 0 ) }  e.  ran  E ) )
8067, 68, 73, 78, 79syl31anc 1267 . . . . . . 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 ++  <" v "> )  e.  ( C `  ( N  +  2 ) ) )  ->  ( {
( lastS  `  W ) ,  v }  e.  ran  E  /\  { v ,  ( W `  0
) }  e.  ran  E ) )
8163, 80impbida 840 . . . . . 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 ++  <" v "> )  e.  ( C `  ( N  +  2 ) ) ) )
82 df-3an 984 . . . . . . . . . . . . . 14  |-  ( ( X  e.  V  /\  N  e.  NN  /\  v  e.  V )  <->  ( ( X  e.  V  /\  N  e.  NN )  /\  v  e.  V
) )
8382simplbi2 629 . . . . . . . . . . . . 13  |-  ( ( X  e.  V  /\  N  e.  NN )  ->  ( v  e.  V  ->  ( X  e.  V  /\  N  e.  NN  /\  v  e.  V ) ) )
84833adant1 1023 . . . . . . . . . . . 12  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  (
v  e.  V  -> 
( X  e.  V  /\  N  e.  NN  /\  v  e.  V ) ) )
8584adantr 466 . . . . . . . . . . 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 ) ) )
8685imp 430 . . . . . . . . . 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 ) )
87 3anass 986 . . . . . . . . . . . 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 ) ) )
8887biimpri 209 . . . . . . . . . . 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 ) )
8988ad2antlr 731 . . . . . . . . . 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 ) )
9086, 89jca 534 . . . . . . . . 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 ) ) )
91 clwwlkextfrlem1 25657 . . . . . . . . 9  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  v  e.  V )  /\  ( W  e.  ( ( V WWalksN  E
) `  N )  /\  ( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) )  -> 
( ( ( W ++ 
<" v "> ) `  0 )  =  X  /\  (
( W ++  <" v "> ) `  N
)  =/=  X ) )
92 simpl 458 . . . . . . . . . 10  |-  ( ( ( ( W ++  <" v "> ) `  0 )  =  X  /\  ( ( W ++  <" v "> ) `  N
)  =/=  X )  ->  ( ( W ++ 
<" v "> ) `  0 )  =  X )
93 neeq2 2714 . . . . . . . . . . . 12  |-  ( X  =  ( ( W ++ 
<" v "> ) `  0 )  ->  ( ( ( W ++ 
<" v "> ) `  N )  =/=  X  <->  ( ( W ++ 
<" v "> ) `  N )  =/=  ( ( W ++  <" v "> ) `  0 ) ) )
9493eqcoms 2441 . . . . . . . . . . 11  |-  ( ( ( W ++  <" v "> ) `  0
)  =  X  -> 
( ( ( W ++ 
<" v "> ) `  N )  =/=  X  <->  ( ( W ++ 
<" v "> ) `  N )  =/=  ( ( W ++  <" v "> ) `  0 ) ) )
9594biimpa 486 . . . . . . . . . 10  |-  ( ( ( ( W ++  <" v "> ) `  0 )  =  X  /\  ( ( W ++  <" v "> ) `  N
)  =/=  X )  ->  ( ( W ++ 
<" v "> ) `  N )  =/=  ( ( W ++  <" v "> ) `  0 ) )
9692, 95jca 534 . . . . . . . . 9  |-  ( ( ( ( W ++  <" v "> ) `  0 )  =  X  /\  ( ( W ++  <" v "> ) `  N
)  =/=  X )  ->  ( ( ( W ++  <" v "> ) `  0
)  =  X  /\  ( ( W ++  <" v "> ) `  N )  =/=  (
( W ++  <" v "> ) `  0
) ) )
9790, 91, 963syl 18 . . . . . . . 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 ++  <" v "> ) `  0
)  =  X  /\  ( ( W ++  <" v "> ) `  N )  =/=  (
( W ++  <" v "> ) `  0
) ) )
98 nncn 10617 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  e.  CC )
99 2cnd 10682 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  2  e.  CC )
10098, 99pncand 9986 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  (
( N  +  2 )  -  2 )  =  N )
1011003ad2ant3 1028 . . . . . . . . . . . 12  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  (
( N  +  2 )  -  2 )  =  N )
102101ad2antrr 730 . . . . . . . . . . 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 )
103102fveq2d 5885 . . . . . . . . . 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 ++  <" v "> ) `  (
( N  +  2 )  -  2 ) )  =  ( ( W ++  <" v "> ) `  N
) )
104103neeq1d 2708 . . . . . . . . 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 ++  <" v "> ) `  (
( N  +  2 )  -  2 ) )  =/=  ( ( W ++  <" v "> ) `  0
)  <->  ( ( W ++ 
<" v "> ) `  N )  =/=  ( ( W ++  <" v "> ) `  0 ) ) )
105104anbi2d 708 . . . . . . . 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 ++  <" v "> ) `  0 )  =  X  /\  ( ( W ++  <" v "> ) `  (
( N  +  2 )  -  2 ) )  =/=  ( ( W ++  <" v "> ) `  0
) )  <->  ( (
( W ++  <" v "> ) `  0
)  =  X  /\  ( ( W ++  <" v "> ) `  N )  =/=  (
( W ++  <" v "> ) `  0
) ) ) )
10697, 105mpbird 235 . . . . . . 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 ++  <" v "> ) `  0
)  =  X  /\  ( ( W ++  <" v "> ) `  ( ( N  + 
2 )  -  2 ) )  =/=  (
( W ++  <" v "> ) `  0
) ) )
107106biantrud 509 . . . . . 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 ++  <" v "> )  e.  ( C `  ( N  +  2 ) )  <-> 
( ( W ++  <" v "> )  e.  ( C `  ( N  +  2 ) )  /\  ( ( ( W ++  <" v "> ) `  0
)  =  X  /\  ( ( W ++  <" v "> ) `  ( ( N  + 
2 )  -  2 ) )  =/=  (
( W ++  <" v "> ) `  0
) ) ) ) )
108 nn0addcl 10905 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  2  e.  NN0 )  -> 
( N  +  2 )  e.  NN0 )
1091, 56, 108sylancl 666 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  ( N  +  2 )  e.  NN0 )
110109anim2i 571 . . . . . . . . . . 11  |-  ( ( X  e.  V  /\  N  e.  NN )  ->  ( X  e.  V  /\  ( N  +  2 )  e.  NN0 )
)
1111103adant1 1023 . . . . . . . . . 10  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  ( X  e.  V  /\  ( N  +  2
)  e.  NN0 )
)
112111ad2antrr 730 . . . . . . . . 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 ) )
113 numclwwlk.h . . . . . . . . . 10  |-  H  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( ( w `
 0 )  =  v  /\  ( w `
 ( n  - 
2 ) )  =/=  ( w `  0
) ) } )
1144, 5, 6, 7, 113numclwwlkovh 25682 . . . . . . . . 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 ) ) } )
115112, 114syl 17 . . . . . . . 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
) ) } )
116115eleq2d 2499 . . . . . . 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 ++  <" v "> )  e.  ( X H ( N  +  2 ) )  <-> 
( W ++  <" v "> )  e.  {
w  e.  ( C `
 ( N  + 
2 ) )  |  ( ( w ` 
0 )  =  X  /\  ( w `  ( ( N  + 
2 )  -  2 ) )  =/=  (
w `  0 )
) } ) )
117 fveq1 5880 . . . . . . . . . 10  |-  ( w  =  ( W ++  <" v "> )  ->  ( w `  0
)  =  ( ( W ++  <" v "> ) `  0
) )
118117eqeq1d 2431 . . . . . . . . 9  |-  ( w  =  ( W ++  <" v "> )  ->  ( ( w ` 
0 )  =  X  <-> 
( ( W ++  <" v "> ) `  0 )  =  X ) )
119 fveq1 5880 . . . . . . . . . 10  |-  ( w  =  ( W ++  <" v "> )  ->  ( w `  (
( N  +  2 )  -  2 ) )  =  ( ( W ++  <" v "> ) `  (
( N  +  2 )  -  2 ) ) )
120119, 117neeq12d 2710 . . . . . . . . 9  |-  ( w  =  ( W ++  <" v "> )  ->  ( ( w `  ( ( N  + 
2 )  -  2 ) )  =/=  (
w `  0 )  <->  ( ( W ++  <" v "> ) `  (
( N  +  2 )  -  2 ) )  =/=  ( ( W ++  <" v "> ) `  0
) ) )
121118, 120anbi12d 715 . . . . . . . 8  |-  ( w  =  ( W ++  <" v "> )  ->  ( ( ( w `
 0 )  =  X  /\  ( w `
 ( ( N  +  2 )  - 
2 ) )  =/=  ( w `  0
) )  <->  ( (
( W ++  <" v "> ) `  0
)  =  X  /\  ( ( W ++  <" v "> ) `  ( ( N  + 
2 )  -  2 ) )  =/=  (
( W ++  <" v "> ) `  0
) ) ) )
122121elrab 3235 . . . . . . 7  |-  ( ( W ++  <" v "> )  e.  {
w  e.  ( C `
 ( N  + 
2 ) )  |  ( ( w ` 
0 )  =  X  /\  ( w `  ( ( N  + 
2 )  -  2 ) )  =/=  (
w `  0 )
) }  <->  ( ( W ++  <" v "> )  e.  ( C `  ( N  +  2 ) )  /\  ( ( ( W ++  <" v "> ) `  0
)  =  X  /\  ( ( W ++  <" v "> ) `  ( ( N  + 
2 )  -  2 ) )  =/=  (
( W ++  <" v "> ) `  0
) ) ) )
123116, 122syl6rbb 265 . . . . . 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 ++  <" v "> )  e.  ( C `  ( N  +  2 ) )  /\  ( ( ( W ++  <" v "> ) `  0
)  =  X  /\  ( ( W ++  <" v "> ) `  ( ( N  + 
2 )  -  2 ) )  =/=  (
( W ++  <" v "> ) `  0
) ) )  <->  ( W ++  <" v "> )  e.  ( X H ( N  + 
2 ) ) ) )
12481, 107, 1233bitrd 282 . . . . 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 ++  <" v "> )  e.  ( X H ( N  + 
2 ) ) ) )
125124reubidva 3019 . . . 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 ++  <" v "> )  e.  ( X H ( N  + 
2 ) ) ) )
12646, 125mpbid 213 . . 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 ++  <" v "> )  e.  ( X H ( N  +  2 ) ) )
127126ex 435 . 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 ++  <" v "> )  e.  ( X H ( N  +  2 ) ) ) )
12817, 127sylbid 218 1  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  ( W  e.  ( X Q N )  ->  E! v  e.  V  ( W ++  <" v "> )  e.  ( X H ( N  +  2 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   E!wreu 2784   {crab 2786   _Vcvv 3087   {cpr 4004   class class class wbr 4426    |-> cmpt 4484   ran crn 4855   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   0cc0 9538   1c1 9539    + caddc 9541    - cmin 9859   NNcn 10609   2c2 10659   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159  ..^cfzo 11913   #chash 12512  Word cword 12643   lastS clsw 12644   ++ cconcat 12645   <"cs1 12646   WWalksN cwwlkn 25259   ClWWalksN cclwwlkn 25330   FriendGrph cfrgra 25569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11783  df-fzo 11914  df-hash 12513  df-word 12651  df-lsw 12652  df-concat 12653  df-s1 12654  df-wwlk 25260  df-wwlkn 25261  df-clwwlk 25332  df-clwwlkn 25333  df-frgra 25570
This theorem is referenced by:  numclwlk2lem2f1o  25686
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