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Theorem numclwwlkovfel2 25229
Description: Properties of an element of the value of operation  F. (Contributed by Alexander van der Vekens, 20-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 } )
Assertion
Ref Expression
numclwwlkovfel2  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( A  e.  ( X F N )  <->  ( ( A  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) { ( A `
 i ) ,  ( A `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  A
) ,  ( A `
 0 ) }  e.  ran  E )  /\  ( # `  A
)  =  N  /\  ( A `  0 )  =  X ) ) )
Distinct variable groups:    n, E    n, N    n, V    i, E    i, N    i, V    w, i    w, C    w, N    C, n, v, w   
v, N    n, X, v, w    v, V    A, i, w
Allowed substitution hints:    A( v, n)    C( i)    E( w, v)    F( w, v, i, n)    V( w)    X( i)

Proof of Theorem numclwwlkovfel2
StepHypRef Expression
1 pm3.22 447 . . . . 5  |-  ( ( N  e.  NN0  /\  X  e.  V )  ->  ( X  e.  V  /\  N  e.  NN0 ) )
213adant1 1012 . . . 4  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( X  e.  V  /\  N  e.  NN0 ) )
3 numclwwlk.c . . . . 5  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
4 numclwwlk.f . . . . 5  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
53, 4numclwwlkovf 25227 . . . 4  |-  ( ( X  e.  V  /\  N  e.  NN0 )  -> 
( X F N )  =  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  X } )
62, 5syl 16 . . 3  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( X F N )  =  { w  e.  ( C `  N )  |  ( w ` 
0 )  =  X } )
76eleq2d 2466 . 2  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( A  e.  ( X F N )  <->  A  e.  { w  e.  ( C `
 N )  |  ( w `  0
)  =  X }
) )
83numclwwlkfvc 25223 . . . . . . 7  |-  ( N  e.  NN0  ->  ( C `
 N )  =  ( ( V ClWWalksN  E ) `
 N ) )
983ad2ant2 1016 . . . . . 6  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( C `  N )  =  ( ( V ClWWalksN  E ) `  N
) )
109eleq2d 2466 . . . . 5  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( A  e.  ( C `  N )  <->  A  e.  ( ( V ClWWalksN  E ) `
 N ) ) )
11 usgrav 24484 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
1211anim1i 566 . . . . . . . 8  |-  ( ( V USGrph  E  /\  N  e. 
NN0 )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 ) )
13 df-3an 973 . . . . . . . 8  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  <->  ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 ) )
1412, 13sylibr 212 . . . . . . 7  |-  ( ( V USGrph  E  /\  N  e. 
NN0 )  ->  ( V  e.  _V  /\  E  e.  _V  /\  N  e. 
NN0 ) )
15143adant3 1014 . . . . . 6  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( V  e.  _V  /\  E  e.  _V  /\  N  e. 
NN0 ) )
16 isclwwlkn 24915 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  ( A  e.  ( ( V ClWWalksN  E ) `  N
)  <->  ( A  e.  ( V ClWWalks  E )  /\  ( # `  A
)  =  N ) ) )
1715, 16syl 16 . . . . 5  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( A  e.  ( ( V ClWWalksN  E ) `  N
)  <->  ( A  e.  ( V ClWWalks  E )  /\  ( # `  A
)  =  N ) ) )
18 isclwwlk 24914 . . . . . . . 8  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( A  e.  ( V ClWWalks  E )  <->  ( A  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  A )  -  1 ) ) { ( A `  i ) ,  ( A `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  A
) ,  ( A `
 0 ) }  e.  ran  E ) ) )
1911, 18syl 16 . . . . . . 7  |-  ( V USGrph  E  ->  ( A  e.  ( V ClWWalks  E )  <->  ( A  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) { ( A `
 i ) ,  ( A `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  A
) ,  ( A `
 0 ) }  e.  ran  E ) ) )
20193ad2ant1 1015 . . . . . 6  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( A  e.  ( V ClWWalks  E )  <->  ( A  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  A )  -  1 ) ) { ( A `  i ) ,  ( A `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  A
) ,  ( A `
 0 ) }  e.  ran  E ) ) )
2120anbi1d 702 . . . . 5  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  (
( A  e.  ( V ClWWalks  E )  /\  ( # `
 A )  =  N )  <->  ( ( A  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) { ( A `
 i ) ,  ( A `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  A
) ,  ( A `
 0 ) }  e.  ran  E )  /\  ( # `  A
)  =  N ) ) )
2210, 17, 213bitrd 279 . . . 4  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( A  e.  ( C `  N )  <->  ( ( A  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) { ( A `
 i ) ,  ( A `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  A
) ,  ( A `
 0 ) }  e.  ran  E )  /\  ( # `  A
)  =  N ) ) )
2322anbi1d 702 . . 3  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  (
( A  e.  ( C `  N )  /\  ( A ` 
0 )  =  X )  <->  ( ( ( A  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) { ( A `
 i ) ,  ( A `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  A
) ,  ( A `
 0 ) }  e.  ran  E )  /\  ( # `  A
)  =  N )  /\  ( A ` 
0 )  =  X ) ) )
24 fveq1 5790 . . . . 5  |-  ( w  =  A  ->  (
w `  0 )  =  ( A ` 
0 ) )
2524eqeq1d 2398 . . . 4  |-  ( w  =  A  ->  (
( w `  0
)  =  X  <->  ( A `  0 )  =  X ) )
2625elrab 3199 . . 3  |-  ( A  e.  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  X }  <->  ( A  e.  ( C `  N
)  /\  ( A `  0 )  =  X ) )
27 df-3an 973 . . 3  |-  ( ( ( A  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  A )  -  1 ) ) { ( A `  i ) ,  ( A `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  A ) ,  ( A `  0 ) }  e.  ran  E
)  /\  ( # `  A
)  =  N  /\  ( A `  0 )  =  X )  <->  ( (
( A  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  A )  -  1 ) ) { ( A `  i ) ,  ( A `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  A ) ,  ( A `  0 ) }  e.  ran  E
)  /\  ( # `  A
)  =  N )  /\  ( A ` 
0 )  =  X ) )
2823, 26, 273bitr4g 288 . 2  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( A  e.  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  X }  <->  ( ( A  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) { ( A `
 i ) ,  ( A `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  A
) ,  ( A `
 0 ) }  e.  ran  E )  /\  ( # `  A
)  =  N  /\  ( A `  0 )  =  X ) ) )
297, 28bitrd 253 1  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( A  e.  ( X F N )  <->  ( ( A  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) { ( A `
 i ) ,  ( A `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  A
) ,  ( A `
 0 ) }  e.  ran  E )  /\  ( # `  A
)  =  N  /\  ( A `  0 )  =  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1836   A.wral 2746   {crab 2750   _Vcvv 3051   {cpr 3963   class class class wbr 4384    |-> cmpt 4442   ran crn 4931   ` cfv 5513  (class class class)co 6218    |-> cmpt2 6220   0cc0 9425   1c1 9426    + caddc 9428    - cmin 9740   NN0cn0 10734  ..^cfzo 11739   #chash 12330  Word cword 12461   lastS clsw 12462   USGrph cusg 24476   ClWWalks cclwwlk 24894   ClWWalksN cclwwlkn 24895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-cnex 9481  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-addrcl 9486  ax-mulcl 9487  ax-mulrcl 9488  ax-mulcom 9489  ax-addass 9490  ax-mulass 9491  ax-distr 9492  ax-i2m1 9493  ax-1ne0 9494  ax-1rid 9495  ax-rnegex 9496  ax-rrecex 9497  ax-cnre 9498  ax-pre-lttri 9499  ax-pre-lttrn 9500  ax-pre-ltadd 9501  ax-pre-mulgt0 9502
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4181  df-int 4217  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-tr 4478  df-eprel 4722  df-id 4726  df-po 4731  df-so 4732  df-fr 4769  df-we 4771  df-ord 4812  df-on 4813  df-lim 4814  df-suc 4815  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-om 6622  df-1st 6721  df-2nd 6722  df-recs 6982  df-rdg 7016  df-1o 7070  df-oadd 7074  df-er 7251  df-map 7362  df-pm 7363  df-en 7458  df-dom 7459  df-sdom 7460  df-fin 7461  df-card 8255  df-cda 8483  df-pnf 9563  df-mnf 9564  df-xr 9565  df-ltxr 9566  df-le 9567  df-sub 9742  df-neg 9743  df-nn 10475  df-2 10533  df-n0 10735  df-z 10804  df-uz 11024  df-fz 11616  df-fzo 11740  df-hash 12331  df-word 12469  df-usgra 24479  df-clwwlk 24897  df-clwwlkn 24898
This theorem is referenced by:  numclwwlkovf2ex  25232  numclwlk1lem2foa  25237  numclwlk1lem2fo  25241
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