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Theorem numclwwlkovgel 25386
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 25385 . . . 4  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( X G N )  =  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) } )
54eleq2d 2472 . . 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 5804 . . . . . 6  |-  ( w  =  P  ->  (
w `  0 )  =  ( P ` 
0 ) )
76eqeq1d 2404 . . . . 5  |-  ( w  =  P  ->  (
( w `  0
)  =  X  <->  ( P `  0 )  =  X ) )
8 fveq1 5804 . . . . . 6  |-  ( w  =  P  ->  (
w `  ( N  -  2 ) )  =  ( P `  ( N  -  2
) ) )
98, 6eqeq12d 2424 . . . . 5  |-  ( w  =  P  ->  (
( w `  ( N  -  2 ) )  =  ( w `
 0 )  <->  ( P `  ( N  -  2 ) )  =  ( P `  0 ) ) )
107, 9anbi12d 709 . . . 4  |-  ( w  =  P  ->  (
( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) )  <->  ( ( P `
 0 )  =  X  /\  ( P `
 ( N  - 
2 ) )  =  ( P `  0
) ) ) )
1110elrab 3206 . . 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 11084 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  NN0 )
1413adantl 464 . . . . . 6  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  N  e.  NN0 )
151numclwwlkfvc 25375 . . . . . 6  |-  ( N  e.  NN0  ->  ( C `
 N )  =  ( ( V ClWWalksN  E ) `
 N ) )
1614, 15syl 17 . . . . 5  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( C `  N
)  =  ( ( V ClWWalksN  E ) `  N
) )
1716eleq2d 2472 . . . 4  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( P  e.  ( C `  N )  <-> 
P  e.  ( ( V ClWWalksN  E ) `  N
) ) )
1817anbi1d 703 . . 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 978 . . 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   {crab 2757    |-> cmpt 4452   ` cfv 5525  (class class class)co 6234    |-> cmpt2 6236   0cc0 9442    - cmin 9761   2c2 10546   NN0cn0 10756   ZZ>=cuz 11045   ClWWalksN cclwwlkn 25047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-recs 6999  df-rdg 7033  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-nn 10497  df-2 10555  df-n0 10757  df-z 10826  df-uz 11046
This theorem is referenced by:  numclwwlkovgelim  25387
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