Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  numclwwlk2lem1 Structured version   Unicode version

Theorem numclwwlk2lem1 30648
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 10578 . . . . . . 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 30645 . . . . 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 2505 . . 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 5685 . . . . . 6  |-  ( w  =  W  ->  (
w `  0 )  =  ( W ` 
0 ) )
1211eqeq1d 2446 . . . . 5  |-  ( w  =  W  ->  (
( w `  0
)  =  X  <->  ( W `  0 )  =  X ) )
13 fveq2 5686 . . . . . 6  |-  ( w  =  W  ->  ( lastS  `  w )  =  ( lastS  `  W ) )
1413neeq1d 2616 . . . . 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 3112 . . 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 30274 . . . . . . . . . . . 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 10326 . . . . . . . . . . . . . . . 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 12263 . . . . . . . . . . 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 2498 . . . . . . . . . . 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 2612 . . . . . . . . . 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 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 30541 . . . . 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 30418 . . . . . . . . . 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 10588 . . . . . . . . . . . . 13  |-  2  e.  NN0
5756a1i 11 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  2  e.  NN0 )
581, 57nn0addcld 10632 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  ( N  +  2 )  e.  NN0 )
594numclwwlkfvc 30623 . . . . . . . . . . 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 2515 . . . . . . 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 30273 . . . . . . . . . . 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 10670 . . . . . . . . . . 11  |-  2  e.  ZZ
71 nn0pzuz 30149 . . . . . . . . . . 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 2505 . . . . . . . . . . . 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 30417 . . . . . . . 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 1221 . . . . . . 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 30622 . . . . . . . . 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 2612 . . . . . . . . . . . 12  |-  ( X  =  ( ( W concat  <" v "> ) `  0 )  ->  ( ( ( W concat  <" v "> ) `  N )  =/=  X  <->  ( ( W concat  <" v "> ) `  N )  =/=  ( ( W concat  <" v "> ) `  0
) ) )
9594eqcoms 2441 . . . . . . . . . . 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 10322 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  e.  CC )
100 2cnd 10386 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  2  e.  CC )
10199, 100pncand 9712 . . . . . . . . . . . . 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 5690 . . . . . . . . . 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 2616 . . . . . . . . 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 10607 . . . . . . . . . . . . 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 30647 . . . . . . . . 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 2505 . . . . . . 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 5685 . . . . . . . . . 10  |-  ( w  =  ( W concat  <" v "> )  ->  (
w `  0 )  =  ( ( W concat  <" v "> ) `  0 )
)
119118eqeq1d 2446 . . . . . . . . 9  |-  ( w  =  ( W concat  <" v "> )  ->  (
( w `  0
)  =  X  <->  ( ( W concat  <" v "> ) `  0
)  =  X ) )
120 fveq1 5685 . . . . . . . . . 10  |-  ( w  =  ( W concat  <" v "> )  ->  (
w `  ( ( N  +  2 )  -  2 ) )  =  ( ( W concat  <" v "> ) `  ( ( N  +  2 )  -  2 ) ) )
121120, 118neeq12d 2618 . . . . . . . . 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 3112 . . . . . . 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 2899 . . . 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 1369    e. wcel 1756    =/= wne 2601   A.wral 2710   E!wreu 2712   {crab 2714   _Vcvv 2967   {cpr 3874   class class class wbr 4287    e. cmpt 4345   ran crn 4836   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   0cc0 9274   1c1 9275    + caddc 9277    - cmin 9587   NNcn 10314   2c2 10363   NN0cn0 10571   ZZcz 10638   ZZ>=cuz 10853  ..^cfzo 11540   #chash 12095  Word cword 12213   lastS clsw 12214   concat cconcat 12215   <"cs1 12216   WWalksN cwwlkn 30265   ClWWalksN cclwwlkn 30367   FriendGrph cfrgra 30533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-fzo 11541  df-hash 12096  df-word 12221  df-lsw 12222  df-concat 12223  df-s1 12224  df-wwlk 30266  df-wwlkn 30267  df-clwwlk 30369  df-clwwlkn 30370  df-frgra 30534
This theorem is referenced by:  numclwlk2lem2f1o  30651
  Copyright terms: Public domain W3C validator