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Theorem clwlkf1clwwlklem3 30668
Description: Lemma 3 for clwlkf1clwwlklem 30669. (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 5798 . . . . . 6  |-  ( c  =  W  ->  ( 1st `  c )  =  ( 1st `  W
) )
31, 2syl5eq 2507 . . . . 5  |-  ( c  =  W  ->  A  =  ( 1st `  W
) )
43fveq2d 5802 . . . 4  |-  ( c  =  W  ->  ( # `
 A )  =  ( # `  ( 1st `  W ) ) )
54eqeq1d 2456 . . 3  |-  ( c  =  W  ->  (
( # `  A )  =  N  <->  ( # `  ( 1st `  W ) )  =  N ) )
6 clwlkfclwwlk.c . . 3  |-  C  =  { c  e.  ( V ClWalks  E )  |  (
# `  A )  =  N }
75, 6elrab2 3224 . 2  |-  ( W  e.  C  <->  ( W  e.  ( V ClWalks  E )  /\  ( # `  ( 1st `  W ) )  =  N ) )
8 eqid 2454 . . . . 5  |-  ( 1st `  W )  =  ( 1st `  W )
9 eqid 2454 . . . . 5  |-  ( 2nd `  W )  =  ( 2nd `  W )
108, 9clwlkcompim 30574 . . . 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 12366 . . . . . 6  |-  ( ( 1st `  W )  e. Word  dom  E  ->  (
# `  ( 1st `  W ) )  e. 
NN0 )
12 nn0z 10779 . . . . . . . . . 10  |-  ( (
# `  ( 1st `  W ) )  e. 
NN0  ->  ( # `  ( 1st `  W ) )  e.  ZZ )
13 fzval3 11721 . . . . . . . . . 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 5654 . . . . . . . 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 12356 . . . . . . 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 1370    e. wcel 1758   A.wral 2798   {crab 2802   {cpr 3986   <.cop 3990    |-> cmpt 4457   dom cdm 4947   -->wf 5521   ` cfv 5525  (class class class)co 6199   1stc1st 6684   2ndc2nd 6685   0cc0 9392   1c1 9393    + caddc 9395   NN0cn0 10689   ZZcz 10756   ...cfz 11553  ..^cfzo 11664   #chash 12219  Word cword 12338   substr csubstr 12342   ClWalks cclwlk 30559
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-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
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 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-map 7325  df-pm 7326  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-card 8219  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-n0 10690  df-z 10757  df-uz 10972  df-fz 11554  df-fzo 11665  df-hash 12220  df-word 12346  df-wlk 23566  df-clwlk 30562
This theorem is referenced by:  clwlkf1clwwlklem  30669
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