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Theorem clwlkf1clwwlklem3 24521
Description: Lemma 3 for clwlkf1clwwlklem 24522. (Contributed by Alexander van der Vekens, 5-Jul-2018.)
Hypotheses
Ref Expression
clwlkfclwwlk.1  |-  A  =  ( 1st `  c
)
clwlkfclwwlk.2  |-  B  =  ( 2nd `  c
)
clwlkfclwwlk.c  |-  C  =  { c  e.  ( V ClWalks  E )  |  (
# `  A )  =  N }
clwlkfclwwlk.f  |-  F  =  ( c  e.  C  |->  ( B substr  <. 0 ,  ( # `  A
) >. ) )
Assertion
Ref Expression
clwlkf1clwwlklem3  |-  ( W  e.  C  ->  ( 2nd `  W )  e. Word  V )
Distinct variable groups:    E, c    N, c    V, c    W, c    C, c    F, c
Allowed substitution hints:    A( c)    B( c)

Proof of Theorem clwlkf1clwwlklem3
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 clwlkfclwwlk.1 . . . . . 6  |-  A  =  ( 1st `  c
)
2 fveq2 5864 . . . . . 6  |-  ( c  =  W  ->  ( 1st `  c )  =  ( 1st `  W
) )
31, 2syl5eq 2520 . . . . 5  |-  ( c  =  W  ->  A  =  ( 1st `  W
) )
43fveq2d 5868 . . . 4  |-  ( c  =  W  ->  ( # `
 A )  =  ( # `  ( 1st `  W ) ) )
54eqeq1d 2469 . . 3  |-  ( c  =  W  ->  (
( # `  A )  =  N  <->  ( # `  ( 1st `  W ) )  =  N ) )
6 clwlkfclwwlk.c . . 3  |-  C  =  { c  e.  ( V ClWalks  E )  |  (
# `  A )  =  N }
75, 6elrab2 3263 . 2  |-  ( W  e.  C  <->  ( W  e.  ( V ClWalks  E )  /\  ( # `  ( 1st `  W ) )  =  N ) )
8 eqid 2467 . . . . 5  |-  ( 1st `  W )  =  ( 1st `  W )
9 eqid 2467 . . . . 5  |-  ( 2nd `  W )  =  ( 2nd `  W )
108, 9clwlkcompim 24437 . . . 4  |-  ( W  e.  ( V ClWalks  E
)  ->  ( (
( 1st `  W
)  e. Word  dom  E  /\  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V )  /\  ( A. i  e.  ( 0..^ ( # `  ( 1st `  W ) ) ) ( E `  ( ( 1st `  W
) `  i )
)  =  { ( ( 2nd `  W
) `  i ) ,  ( ( 2nd `  W ) `  (
i  +  1 ) ) }  /\  (
( 2nd `  W
) `  0 )  =  ( ( 2nd `  W ) `  ( # `
 ( 1st `  W
) ) ) ) ) )
11 lencl 12522 . . . . . 6  |-  ( ( 1st `  W )  e. Word  dom  E  ->  (
# `  ( 1st `  W ) )  e. 
NN0 )
12 nn0z 10883 . . . . . . . . . 10  |-  ( (
# `  ( 1st `  W ) )  e. 
NN0  ->  ( # `  ( 1st `  W ) )  e.  ZZ )
13 fzval3 11849 . . . . . . . . . 10  |-  ( (
# `  ( 1st `  W ) )  e.  ZZ  ->  ( 0 ... ( # `  ( 1st `  W ) ) )  =  ( 0..^ ( ( # `  ( 1st `  W ) )  +  1 ) ) )
1412, 13syl 16 . . . . . . . . 9  |-  ( (
# `  ( 1st `  W ) )  e. 
NN0  ->  ( 0 ... ( # `  ( 1st `  W ) ) )  =  ( 0..^ ( ( # `  ( 1st `  W ) )  +  1 ) ) )
1514feq2d 5716 . . . . . . . 8  |-  ( (
# `  ( 1st `  W ) )  e. 
NN0  ->  ( ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W
) ) ) --> V  <-> 
( 2nd `  W
) : ( 0..^ ( ( # `  ( 1st `  W ) )  +  1 ) ) --> V ) )
1615biimpa 484 . . . . . . 7  |-  ( ( ( # `  ( 1st `  W ) )  e.  NN0  /\  ( 2nd `  W ) : ( 0 ... ( # `
 ( 1st `  W
) ) ) --> V )  ->  ( 2nd `  W ) : ( 0..^ ( ( # `  ( 1st `  W
) )  +  1 ) ) --> V )
17 iswrdi 12512 . . . . . . 7  |-  ( ( 2nd `  W ) : ( 0..^ ( ( # `  ( 1st `  W ) )  +  1 ) ) --> V  ->  ( 2nd `  W )  e. Word  V
)
1816, 17syl 16 . . . . . 6  |-  ( ( ( # `  ( 1st `  W ) )  e.  NN0  /\  ( 2nd `  W ) : ( 0 ... ( # `
 ( 1st `  W
) ) ) --> V )  ->  ( 2nd `  W )  e. Word  V
)
1911, 18sylan 471 . . . . 5  |-  ( ( ( 1st `  W
)  e. Word  dom  E  /\  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V )  -> 
( 2nd `  W
)  e. Word  V )
2019adantr 465 . . . 4  |-  ( ( ( ( 1st `  W
)  e. Word  dom  E  /\  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V )  /\  ( A. i  e.  ( 0..^ ( # `  ( 1st `  W ) ) ) ( E `  ( ( 1st `  W
) `  i )
)  =  { ( ( 2nd `  W
) `  i ) ,  ( ( 2nd `  W ) `  (
i  +  1 ) ) }  /\  (
( 2nd `  W
) `  0 )  =  ( ( 2nd `  W ) `  ( # `
 ( 1st `  W
) ) ) ) )  ->  ( 2nd `  W )  e. Word  V
)
2110, 20syl 16 . . 3  |-  ( W  e.  ( V ClWalks  E
)  ->  ( 2nd `  W )  e. Word  V
)
2221adantr 465 . 2  |-  ( ( W  e.  ( V ClWalks  E )  /\  ( # `
 ( 1st `  W
) )  =  N )  ->  ( 2nd `  W )  e. Word  V
)
237, 22sylbi 195 1  |-  ( W  e.  C  ->  ( 2nd `  W )  e. Word  V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818   {cpr 4029   <.cop 4033    |-> cmpt 4505   dom cdm 4999   -->wf 5582   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780   0cc0 9488   1c1 9489    + caddc 9491   NN0cn0 10791   ZZcz 10860   ...cfz 11668  ..^cfzo 11788   #chash 12367  Word cword 12494   substr csubstr 12498   ClWalks cclwlk 24420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-hash 12368  df-word 12502  df-wlk 24181  df-clwlk 24423
This theorem is referenced by:  clwlkf1clwwlklem  24522
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