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Theorem numclwwlkovg 24911
Description: Value of operation  G, mapping a vertex v and a nonnegative integer n to the "closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) with v(n-2) = v" according to Huneke. (Contributed by Alexander van der Vekens, 14-Sep-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 ) ) } )
Assertion
Ref Expression
numclwwlkovg  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( X G N )  =  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) } )
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
Allowed substitution hints:    E( v)    F( v, n)    G( w, v, n)

Proof of Theorem numclwwlkovg
StepHypRef Expression
1 fveq2 5872 . . . 4  |-  ( n  =  N  ->  ( C `  n )  =  ( C `  N ) )
21adantl 466 . . 3  |-  ( ( v  =  X  /\  n  =  N )  ->  ( C `  n
)  =  ( C `
 N ) )
3 eqeq2 2482 . . . 4  |-  ( v  =  X  ->  (
( w `  0
)  =  v  <->  ( w `  0 )  =  X ) )
4 oveq1 6302 . . . . . 6  |-  ( n  =  N  ->  (
n  -  2 )  =  ( N  - 
2 ) )
54fveq2d 5876 . . . . 5  |-  ( n  =  N  ->  (
w `  ( n  -  2 ) )  =  ( w `  ( N  -  2
) ) )
65eqeq1d 2469 . . . 4  |-  ( n  =  N  ->  (
( w `  (
n  -  2 ) )  =  ( w `
 0 )  <->  ( w `  ( N  -  2 ) )  =  ( w `  0 ) ) )
73, 6bi2anan9 871 . . 3  |-  ( ( v  =  X  /\  n  =  N )  ->  ( ( ( w `
 0 )  =  v  /\  ( w `
 ( n  - 
2 ) )  =  ( w `  0
) )  <->  ( (
w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) ) )
82, 7rabeqbidv 3113 . 2  |-  ( ( v  =  X  /\  n  =  N )  ->  { w  e.  ( C `  n )  |  ( ( w `
 0 )  =  v  /\  ( w `
 ( n  - 
2 ) )  =  ( w `  0
) ) }  =  { w  e.  ( C `  N )  |  ( ( w `
 0 )  =  X  /\  ( w `
 ( N  - 
2 ) )  =  ( w `  0
) ) } )
9 numclwwlk.g . 2  |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>=
`  2 )  |->  { w  e.  ( C `
 n )  |  ( ( w ` 
0 )  =  v  /\  ( w `  ( n  -  2
) )  =  ( w `  0 ) ) } )
10 fvex 5882 . . 3  |-  ( C `
 N )  e. 
_V
1110rabex 4604 . 2  |-  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) }  e.  _V
128, 9, 11ovmpt2a 6428 1  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( X G N )  =  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2821    |-> cmpt 4511   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   0cc0 9504    - cmin 9817   2c2 10597   NN0cn0 10807   ZZ>=cuz 11094   ClWWalksN cclwwlkn 24572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300
This theorem is referenced by:  numclwwlkovgel  24912  extwwlkfab  24914  numclwwlk3lem  24932
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