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Theorem numclwwlkovh 25303
Description: Value of operation H, 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 definition 7 in [Huneke] p. 2. (Contributed by Alexander van der Vekens, 26-Aug-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
numclwwlkovh  |-  ( ( X  e.  V  /\  N  e.  NN0 )  -> 
( X H 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    w, Q    w, G    v, E
Allowed substitution hints:    Q( v, n)    F( v, n)    G( v, n)    H( w, v, n)

Proof of Theorem numclwwlkovh
StepHypRef Expression
1 fveq2 5848 . . . 4  |-  ( n  =  N  ->  ( C `  n )  =  ( C `  N ) )
21adantl 464 . . 3  |-  ( ( v  =  X  /\  n  =  N )  ->  ( C `  n
)  =  ( C `
 N ) )
3 eqeq2 2469 . . . . 5  |-  ( v  =  X  ->  (
( w `  0
)  =  v  <->  ( w `  0 )  =  X ) )
43adantr 463 . . . 4  |-  ( ( v  =  X  /\  n  =  N )  ->  ( ( w ` 
0 )  =  v  <-> 
( w `  0
)  =  X ) )
5 oveq1 6277 . . . . . . 7  |-  ( n  =  N  ->  (
n  -  2 )  =  ( N  - 
2 ) )
65fveq2d 5852 . . . . . 6  |-  ( n  =  N  ->  (
w `  ( n  -  2 ) )  =  ( w `  ( N  -  2
) ) )
76adantl 464 . . . . 5  |-  ( ( v  =  X  /\  n  =  N )  ->  ( w `  (
n  -  2 ) )  =  ( w `
 ( N  - 
2 ) ) )
87neeq1d 2731 . . . 4  |-  ( ( v  =  X  /\  n  =  N )  ->  ( ( w `  ( n  -  2
) )  =/=  (
w `  0 )  <->  ( w `  ( N  -  2 ) )  =/=  ( w ` 
0 ) ) )
94, 8anbi12d 708 . . 3  |-  ( ( v  =  X  /\  n  =  N )  ->  ( ( ( w `
 0 )  =  v  /\  ( w `
 ( n  - 
2 ) )  =/=  ( w `  0
) )  <->  ( (
w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =/=  ( w ` 
0 ) ) ) )
102, 9rabeqbidv 3101 . 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
) ) } )
11 numclwwlk.h . 2  |-  H  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( ( w `
 0 )  =  v  /\  ( w `
 ( n  - 
2 ) )  =/=  ( w `  0
) ) } )
12 fvex 5858 . . 3  |-  ( C `
 N )  e. 
_V
1312rabex 4588 . 2  |-  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =/=  ( w ` 
0 ) ) }  e.  _V
1410, 11, 13ovmpt2a 6406 1  |-  ( ( X  e.  V  /\  N  e.  NN0 )  -> 
( X H N )  =  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =/=  ( w ` 
0 ) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   {crab 2808    |-> cmpt 4497   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   0cc0 9481    - cmin 9796   2c2 10581   NN0cn0 10791   ZZ>=cuz 11082   lastS clsw 12519   WWalksN cwwlkn 24880   ClWWalksN cclwwlkn 24951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275
This theorem is referenced by:  numclwwlk2lem1  25304  numclwlk2lem2f  25305  numclwlk2lem2f1o  25307  numclwwlk3lem  25310
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