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Theorem numclwwlk2lem1 30863
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 10700 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  NN0 )
21anim2i 569 . . . . . 6  |-  ( ( X  e.  V  /\  N  e.  NN )  ->  ( X  e.  V  /\  N  e.  NN0 ) )
323adant1 1006 . . . . 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 30860 . . . . 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 2524 . . 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 5801 . . . . . 6  |-  ( w  =  W  ->  (
w `  0 )  =  ( W ` 
0 ) )
1211eqeq1d 2456 . . . . 5  |-  ( w  =  W  ->  (
( w `  0
)  =  X  <->  ( W `  0 )  =  X ) )
13 fveq2 5802 . . . . . 6  |-  ( w  =  W  ->  ( lastS  `  w )  =  ( lastS  `  W ) )
1413neeq1d 2729 . . . . 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 3224 . . 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 991 . . . . 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 30489 . . . . . . . . . . . 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 10448 . . . . . . . . . . . . . . . 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 1008 . . . . . . . . . . . 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 12392 . . . . . . . . . . 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 1011 . . . . . . 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 2526 . . . . . . . . . . 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 1010 . . . . . . 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 2735 . . . . . . . . . 10  |-  ( X  =  ( W ` 
0 )  ->  (
( lastS  `  W )  =/= 
X  <->  ( lastS  `  W )  =/=  ( W ` 
0 ) ) )
4039eqcoms 2466 . . . . . . . . 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 1168 . . . . 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 30756 . . . . 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 1011 . . . . . . . . . . 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 1168 . . . . . . . . 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 30633 . . . . . . . . . 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 10710 . . . . . . . . . . . . 13  |-  2  e.  NN0
5756a1i 11 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  2  e.  NN0 )
581, 57nn0addcld 10754 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  ( N  +  2 )  e.  NN0 )
594numclwwlkfvc 30838 . . . . . . . . . . 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 1011 . . . . . . . . 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 2545 . . . . . . 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 30488 . . . . . . . . . . 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 10792 . . . . . . . . . . 11  |-  2  e.  ZZ
71 nn0pzuz 30364 . . . . . . . . . . 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 1011 . . . . . . . . 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 2524 . . . . . . . . . . . 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 1011 . . . . . . . . . 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 30632 . . . . . . . 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 1222 . . . . . . 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 828 . . . . . 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 967 . . . . . . . . . . . . . 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 1006 . . . . . . . . . . . 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 969 . . . . . . . . . . . 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 30837 . . . . . . . . 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 2735 . . . . . . . . . . . 12  |-  ( X  =  ( ( W concat  <" v "> ) `  0 )  ->  ( ( ( W concat  <" v "> ) `  N )  =/=  X  <->  ( ( W concat  <" v "> ) `  N )  =/=  ( ( W concat  <" v "> ) `  0
) ) )
9594eqcoms 2466 . . . . . . . . . . 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 10444 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  e.  CC )
100 2cnd 10508 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  2  e.  CC )
10199, 100pncand 9834 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  (
( N  +  2 )  -  2 )  =  N )
1021013ad2ant3 1011 . . . . . . . . . . . 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 5806 . . . . . . . . . 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 2729 . . . . . . . . 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 10729 . . . . . . . . . . . . 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 1006 . . . . . . . . . 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 30862 . . . . . . . . 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 2524 . . . . . . 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 5801 . . . . . . . . . 10  |-  ( w  =  ( W concat  <" v "> )  ->  (
w `  0 )  =  ( ( W concat  <" v "> ) `  0 )
)
119118eqeq1d 2456 . . . . . . . . 9  |-  ( w  =  ( W concat  <" v "> )  ->  (
( w `  0
)  =  X  <->  ( ( W concat  <" v "> ) `  0
)  =  X ) )
120 fveq1 5801 . . . . . . . . . 10  |-  ( w  =  ( W concat  <" v "> )  ->  (
w `  ( ( N  +  2 )  -  2 ) )  =  ( ( W concat  <" v "> ) `  ( ( N  +  2 )  -  2 ) ) )
121120, 118neeq12d 2731 . . . . . . . . 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 3224 . . . . . . 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 3010 . . . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   E!wreu 2801   {crab 2803   _Vcvv 3078   {cpr 3990   class class class wbr 4403    |-> cmpt 4461   ran crn 4952   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   0cc0 9396   1c1 9397    + caddc 9399    - cmin 9709   NNcn 10436   2c2 10485   NN0cn0 10693   ZZcz 10760   ZZ>=cuz 10975  ..^cfzo 11668   #chash 12223  Word cword 12342   lastS clsw 12343   concat cconcat 12344   <"cs1 12345   WWalksN cwwlkn 30480   ClWWalksN cclwwlkn 30582   FriendGrph cfrgra 30748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-fzo 11669  df-hash 12224  df-word 12350  df-lsw 12351  df-concat 12352  df-s1 12353  df-wwlk 30481  df-wwlkn 30482  df-clwwlk 30584  df-clwwlkn 30585  df-frgra 30749
This theorem is referenced by:  numclwlk2lem2f1o  30866
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