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Theorem clwlkf1clwwlklem3 25146
Description: Lemma 3 for clwlkf1clwwlklem 25147. (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 5805 . . . . . 6  |-  ( c  =  W  ->  ( 1st `  c )  =  ( 1st `  W
) )
31, 2syl5eq 2455 . . . . 5  |-  ( c  =  W  ->  A  =  ( 1st `  W
) )
43fveq2d 5809 . . . 4  |-  ( c  =  W  ->  ( # `
 A )  =  ( # `  ( 1st `  W ) ) )
54eqeq1d 2404 . . 3  |-  ( c  =  W  ->  (
( # `  A )  =  N  <->  ( # `  ( 1st `  W ) )  =  N ) )
6 clwlkfclwwlk.c . . 3  |-  C  =  { c  e.  ( V ClWalks  E )  |  (
# `  A )  =  N }
75, 6elrab2 3208 . 2  |-  ( W  e.  C  <->  ( W  e.  ( V ClWalks  E )  /\  ( # `  ( 1st `  W ) )  =  N ) )
8 eqid 2402 . . . . 5  |-  ( 1st `  W )  =  ( 1st `  W )
9 eqid 2402 . . . . 5  |-  ( 2nd `  W )  =  ( 2nd `  W )
108, 9clwlkcompim 25062 . . . 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 12521 . . . . . 6  |-  ( ( 1st `  W )  e. Word  dom  E  ->  (
# `  ( 1st `  W ) )  e. 
NN0 )
12 nn0z 10848 . . . . . . . . . 10  |-  ( (
# `  ( 1st `  W ) )  e. 
NN0  ->  ( # `  ( 1st `  W ) )  e.  ZZ )
13 fzval3 11834 . . . . . . . . . 10  |-  ( (
# `  ( 1st `  W ) )  e.  ZZ  ->  ( 0 ... ( # `  ( 1st `  W ) ) )  =  ( 0..^ ( ( # `  ( 1st `  W ) )  +  1 ) ) )
1412, 13syl 17 . . . . . . . . 9  |-  ( (
# `  ( 1st `  W ) )  e. 
NN0  ->  ( 0 ... ( # `  ( 1st `  W ) ) )  =  ( 0..^ ( ( # `  ( 1st `  W ) )  +  1 ) ) )
1514feq2d 5657 . . . . . . . 8  |-  ( (
# `  ( 1st `  W ) )  e. 
NN0  ->  ( ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W
) ) ) --> V  <-> 
( 2nd `  W
) : ( 0..^ ( ( # `  ( 1st `  W ) )  +  1 ) ) --> V ) )
1615biimpa 482 . . . . . . 7  |-  ( ( ( # `  ( 1st `  W ) )  e.  NN0  /\  ( 2nd `  W ) : ( 0 ... ( # `
 ( 1st `  W
) ) ) --> V )  ->  ( 2nd `  W ) : ( 0..^ ( ( # `  ( 1st `  W
) )  +  1 ) ) --> V )
17 iswrdi 12509 . . . . . . 7  |-  ( ( 2nd `  W ) : ( 0..^ ( ( # `  ( 1st `  W ) )  +  1 ) ) --> V  ->  ( 2nd `  W )  e. Word  V
)
1816, 17syl 17 . . . . . 6  |-  ( ( ( # `  ( 1st `  W ) )  e.  NN0  /\  ( 2nd `  W ) : ( 0 ... ( # `
 ( 1st `  W
) ) ) --> V )  ->  ( 2nd `  W )  e. Word  V
)
1911, 18sylan 469 . . . . 5  |-  ( ( ( 1st `  W
)  e. Word  dom  E  /\  ( 2nd `  W ) : ( 0 ... ( # `  ( 1st `  W ) ) ) --> V )  -> 
( 2nd `  W
)  e. Word  V )
2019adantr 463 . . . 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 17 . . 3  |-  ( W  e.  ( V ClWalks  E
)  ->  ( 2nd `  W )  e. Word  V
)
2221adantr 463 . 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 367    = wceq 1405    e. wcel 1842   A.wral 2753   {crab 2757   {cpr 3973   <.cop 3977    |-> cmpt 4452   dom cdm 4942   -->wf 5521   ` cfv 5525  (class class class)co 6234   1stc1st 6736   2ndc2nd 6737   0cc0 9442   1c1 9443    + caddc 9445   NN0cn0 10756   ZZcz 10825   ...cfz 11643  ..^cfzo 11767   #chash 12359  Word cword 12490   substr csubstr 12494   ClWalks cclwlk 25045
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-rep 4506  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-rmo 2761  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-int 4227  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-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-oadd 7091  df-er 7268  df-map 7379  df-pm 7380  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-card 8272  df-cda 8500  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  df-fz 11644  df-fzo 11768  df-hash 12360  df-word 12498  df-wlk 24806  df-clwlk 25048
This theorem is referenced by:  clwlkf1clwwlklem  25147
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