MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  numclwwlkovg Structured version   Visualization version   Unicode version

Theorem numclwwlkovg 25827
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 definition 6 in [Huneke] p. 2. (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 5870 . . . 4  |-  ( n  =  N  ->  ( C `  n )  =  ( C `  N ) )
21adantl 468 . . 3  |-  ( ( v  =  X  /\  n  =  N )  ->  ( C `  n
)  =  ( C `
 N ) )
3 eqeq2 2464 . . . 4  |-  ( v  =  X  ->  (
( w `  0
)  =  v  <->  ( w `  0 )  =  X ) )
4 oveq1 6302 . . . . . 6  |-  ( n  =  N  ->  (
n  -  2 )  =  ( N  - 
2 ) )
54fveq2d 5874 . . . . 5  |-  ( n  =  N  ->  (
w `  ( n  -  2 ) )  =  ( w `  ( N  -  2
) ) )
65eqeq1d 2455 . . . 4  |-  ( n  =  N  ->  (
( w `  (
n  -  2 ) )  =  ( w `
 0 )  <->  ( w `  ( N  -  2 ) )  =  ( w `  0 ) ) )
73, 6bi2anan9 885 . . 3  |-  ( ( v  =  X  /\  n  =  N )  ->  ( ( ( w `
 0 )  =  v  /\  ( w `
 ( n  - 
2 ) )  =  ( w `  0
) )  <->  ( (
w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) ) )
82, 7rabeqbidv 3042 . 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 5880 . . 3  |-  ( C `
 N )  e. 
_V
1110rabex 4557 . 2  |-  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) }  e.  _V
128, 9, 11ovmpt2a 6432 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 371    = wceq 1446    e. wcel 1889   {crab 2743    |-> cmpt 4464   ` cfv 5585  (class class class)co 6295    |-> cmpt2 6297   0cc0 9544    - cmin 9865   2c2 10666   NN0cn0 10876   ZZ>=cuz 11166   ClWWalksN cclwwlkn 25489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5549  df-fun 5587  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300
This theorem is referenced by:  numclwwlkovgel  25828  extwwlkfab  25830  numclwwlk3lem  25848
  Copyright terms: Public domain W3C validator