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Theorem numclwwlkovfel2 25809
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 pm3.22 450 . . . . 5  |-  ( ( N  e.  NN0  /\  X  e.  V )  ->  ( X  e.  V  /\  N  e.  NN0 ) )
213adant1 1023 . . . 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 25807 . . . 4  |-  ( ( X  e.  V  /\  N  e.  NN0 )  -> 
( X F N )  =  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  X } )
62, 5syl 17 . . 3  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( X F N )  =  { w  e.  ( C `  N )  |  ( w ` 
0 )  =  X } )
76eleq2d 2492 . 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 25803 . . . . . . 7  |-  ( N  e.  NN0  ->  ( C `
 N )  =  ( ( V ClWWalksN  E ) `
 N ) )
983ad2ant2 1027 . . . . . 6  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( C `  N )  =  ( ( V ClWWalksN  E ) `  N
) )
109eleq2d 2492 . . . . 5  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( A  e.  ( C `  N )  <->  A  e.  ( ( V ClWWalksN  E ) `
 N ) ) )
11 usgrav 25063 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
1211anim1i 570 . . . . . . . 8  |-  ( ( V USGrph  E  /\  N  e. 
NN0 )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 ) )
13 df-3an 984 . . . . . . . 8  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  <->  ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 ) )
1412, 13sylibr 215 . . . . . . 7  |-  ( ( V USGrph  E  /\  N  e. 
NN0 )  ->  ( V  e.  _V  /\  E  e.  _V  /\  N  e. 
NN0 ) )
15143adant3 1025 . . . . . 6  |-  ( ( V USGrph  E  /\  N  e. 
NN0  /\  X  e.  V )  ->  ( V  e.  _V  /\  E  e.  _V  /\  N  e. 
NN0 ) )
16 isclwwlkn 25495 . . . . . 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 17 . . . . 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 25494 . . . . . . . 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 17 . . . . . . 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 1026 . . . . . 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 709 . . . . 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 282 . . . 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 709 . . 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 5880 . . . . 5  |-  ( w  =  A  ->  (
w `  0 )  =  ( A ` 
0 ) )
2524eqeq1d 2424 . . . 4  |-  ( w  =  A  ->  (
( w `  0
)  =  X  <->  ( A `  0 )  =  X ) )
2625elrab 3228 . . 3  |-  ( A  e.  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  X }  <->  ( A  e.  ( C `  N
)  /\  ( A `  0 )  =  X ) )
27 df-3an 984 . . 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 291 . 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 256 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   A.wral 2771   {crab 2775   _Vcvv 3080   {cpr 4000   class class class wbr 4423    |-> cmpt 4482   ran crn 4854   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   0cc0 9546   1c1 9547    + caddc 9549    - cmin 9867   NN0cn0 10876  ..^cfzo 11922   #chash 12521  Word cword 12660   lastS clsw 12661   USGrph cusg 25055   ClWWalks cclwwlk 25474   ClWWalksN cclwwlkn 25475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-map 7485  df-pm 7486  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-card 8381  df-cda 8605  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923  df-hash 12522  df-word 12668  df-usgra 25058  df-clwwlk 25477  df-clwwlkn 25478
This theorem is referenced by:  numclwwlkovf2ex  25812  numclwlk1lem2foa  25817  numclwlk1lem2fo  25821
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