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Theorem numclwwlkovgel 30852
Description: Properties of an element of the value of operation  G. (Contributed by Alexander van der Vekens, 24-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
numclwwlkovgel  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( P  e.  ( X G N )  <-> 
( P  e.  ( ( V ClWWalksN  E ) `  N )  /\  ( P `  0 )  =  X  /\  ( P `  ( N  -  2 ) )  =  ( P ` 
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, P
Allowed substitution hints:    P( v, n)    E( v)    F( v, n)    G( w, v, n)

Proof of Theorem numclwwlkovgel
StepHypRef Expression
1 numclwwlk.c . . . . 5  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
2 numclwwlk.f . . . . 5  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
3 numclwwlk.g . . . . 5  |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>=
`  2 )  |->  { w  e.  ( C `
 n )  |  ( ( w ` 
0 )  =  v  /\  ( w `  ( n  -  2
) )  =  ( w `  0 ) ) } )
41, 2, 3numclwwlkovg 30851 . . . 4  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( X G N )  =  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) } )
54eleq2d 2524 . . 3  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( P  e.  ( X G N )  <-> 
P  e.  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) } ) )
6 fveq1 5801 . . . . . 6  |-  ( w  =  P  ->  (
w `  0 )  =  ( P ` 
0 ) )
76eqeq1d 2456 . . . . 5  |-  ( w  =  P  ->  (
( w `  0
)  =  X  <->  ( P `  0 )  =  X ) )
8 fveq1 5801 . . . . . 6  |-  ( w  =  P  ->  (
w `  ( N  -  2 ) )  =  ( P `  ( N  -  2
) ) )
98, 6eqeq12d 2476 . . . . 5  |-  ( w  =  P  ->  (
( w `  ( N  -  2 ) )  =  ( w `
 0 )  <->  ( P `  ( N  -  2 ) )  =  ( P `  0 ) ) )
107, 9anbi12d 710 . . . 4  |-  ( w  =  P  ->  (
( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) )  <->  ( ( P `
 0 )  =  X  /\  ( P `
 ( N  - 
2 ) )  =  ( P `  0
) ) ) )
1110elrab 3224 . . 3  |-  ( P  e.  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) }  <-> 
( P  e.  ( C `  N )  /\  ( ( P `
 0 )  =  X  /\  ( P `
 ( N  - 
2 ) )  =  ( P `  0
) ) ) )
125, 11syl6bb 261 . 2  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( P  e.  ( X G N )  <-> 
( P  e.  ( C `  N )  /\  ( ( P `
 0 )  =  X  /\  ( P `
 ( N  - 
2 ) )  =  ( P `  0
) ) ) ) )
13 eluzge2nn0 30363 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  NN0 )
1413adantl 466 . . . . . 6  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  N  e.  NN0 )
151numclwwlkfvc 30841 . . . . . 6  |-  ( N  e.  NN0  ->  ( C `
 N )  =  ( ( V ClWWalksN  E ) `
 N ) )
1614, 15syl 16 . . . . 5  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( C `  N
)  =  ( ( V ClWWalksN  E ) `  N
) )
1716eleq2d 2524 . . . 4  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( P  e.  ( C `  N )  <-> 
P  e.  ( ( V ClWWalksN  E ) `  N
) ) )
1817anbi1d 704 . . 3  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( P  e.  ( C `  N
)  /\  ( ( P `  0 )  =  X  /\  ( P `  ( N  -  2 ) )  =  ( P ` 
0 ) ) )  <-> 
( P  e.  ( ( V ClWWalksN  E ) `  N )  /\  (
( P `  0
)  =  X  /\  ( P `  ( N  -  2 ) )  =  ( P ` 
0 ) ) ) ) )
19 3anass 969 . . 3  |-  ( ( P  e.  ( ( V ClWWalksN  E ) `  N
)  /\  ( P `  0 )  =  X  /\  ( P `
 ( N  - 
2 ) )  =  ( P `  0
) )  <->  ( P  e.  ( ( V ClWWalksN  E ) `
 N )  /\  ( ( P ` 
0 )  =  X  /\  ( P `  ( N  -  2
) )  =  ( P `  0 ) ) ) )
2018, 19syl6bbr 263 . 2  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( P  e.  ( C `  N
)  /\  ( ( P `  0 )  =  X  /\  ( P `  ( N  -  2 ) )  =  ( P ` 
0 ) ) )  <-> 
( P  e.  ( ( V ClWWalksN  E ) `  N )  /\  ( P `  0 )  =  X  /\  ( P `  ( N  -  2 ) )  =  ( P ` 
0 ) ) ) )
2112, 20bitrd 253 1  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( P  e.  ( X G N )  <-> 
( P  e.  ( ( V ClWWalksN  E ) `  N )  /\  ( P `  0 )  =  X  /\  ( P `  ( N  -  2 ) )  =  ( P ` 
0 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   {crab 2803    |-> cmpt 4461   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   0cc0 9397    - cmin 9710   2c2 10486   NN0cn0 10694   ZZ>=cuz 10976   ClWWalksN cclwwlkn 30585
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-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
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-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-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-n0 10695  df-z 10762  df-uz 10977
This theorem is referenced by:  numclwwlkovgelim  30853
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