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Theorem clwlkfclwwlk2wrd 30637
Description: The second component of a closed walk is a word over the "vertices". (Contributed by Alexander van der Vekens, 25-Jun-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
clwlkfclwwlk2wrd  |-  ( c  e.  C  ->  B  e. Word  V )

Proof of Theorem clwlkfclwwlk2wrd
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 clwlkfclwwlk.c . . 3  |-  C  =  { c  e.  ( V ClWalks  E )  |  (
# `  A )  =  N }
21rabeq2i 3051 . 2  |-  ( c  e.  C  <->  ( c  e.  ( V ClWalks  E )  /\  ( # `  A
)  =  N ) )
3 clwlkfclwwlk.1 . . . . 5  |-  A  =  ( 1st `  c
)
4 clwlkfclwwlk.2 . . . . 5  |-  B  =  ( 2nd `  c
)
53, 4clwlkcompim 30551 . . . 4  |-  ( c  e.  ( V ClWalks  E
)  ->  ( ( A  e. Word  dom  E  /\  B : ( 0 ... ( # `  A
) ) --> V )  /\  ( A. i  e.  ( 0..^ ( # `  A ) ) ( E `  ( A `
 i ) )  =  { ( B `
 i ) ,  ( B `  (
i  +  1 ) ) }  /\  ( B `  0 )  =  ( B `  ( # `  A ) ) ) ) )
6 lencl 12337 . . . . . 6  |-  ( A  e. Word  dom  E  ->  (
# `  A )  e.  NN0 )
7 nn0z 10756 . . . . . . . . . 10  |-  ( (
# `  A )  e.  NN0  ->  ( # `  A
)  e.  ZZ )
8 fzval3 11692 . . . . . . . . . 10  |-  ( (
# `  A )  e.  ZZ  ->  ( 0 ... ( # `  A
) )  =  ( 0..^ ( ( # `  A )  +  1 ) ) )
97, 8syl 16 . . . . . . . . 9  |-  ( (
# `  A )  e.  NN0  ->  ( 0 ... ( # `  A
) )  =  ( 0..^ ( ( # `  A )  +  1 ) ) )
109feq2d 5631 . . . . . . . 8  |-  ( (
# `  A )  e.  NN0  ->  ( B : ( 0 ... ( # `  A
) ) --> V  <->  B :
( 0..^ ( (
# `  A )  +  1 ) ) --> V ) )
1110biimpa 484 . . . . . . 7  |-  ( ( ( # `  A
)  e.  NN0  /\  B : ( 0 ... ( # `  A
) ) --> V )  ->  B : ( 0..^ ( ( # `  A )  +  1 ) ) --> V )
12 iswrdi 12327 . . . . . . 7  |-  ( B : ( 0..^ ( ( # `  A
)  +  1 ) ) --> V  ->  B  e. Word  V )
1311, 12syl 16 . . . . . 6  |-  ( ( ( # `  A
)  e.  NN0  /\  B : ( 0 ... ( # `  A
) ) --> V )  ->  B  e. Word  V
)
146, 13sylan 471 . . . . 5  |-  ( ( A  e. Word  dom  E  /\  B : ( 0 ... ( # `  A
) ) --> V )  ->  B  e. Word  V
)
1514adantr 465 . . . 4  |-  ( ( ( A  e. Word  dom  E  /\  B : ( 0 ... ( # `  A ) ) --> V )  /\  ( A. i  e.  ( 0..^ ( # `  A
) ) ( E `
 ( A `  i ) )  =  { ( B `  i ) ,  ( B `  ( i  +  1 ) ) }  /\  ( B `
 0 )  =  ( B `  ( # `
 A ) ) ) )  ->  B  e. Word  V )
165, 15syl 16 . . 3  |-  ( c  e.  ( V ClWalks  E
)  ->  B  e. Word  V )
1716adantr 465 . 2  |-  ( ( c  e.  ( V ClWalks  E )  /\  ( # `
 A )  =  N )  ->  B  e. Word  V )
182, 17sylbi 195 1  |-  ( c  e.  C  ->  B  e. Word  V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1757   A.wral 2792   {crab 2796   {cpr 3963   <.cop 3967    |-> cmpt 4434   dom cdm 4924   -->wf 5498   ` cfv 5502  (class class class)co 6176   1stc1st 6661   2ndc2nd 6662   0cc0 9369   1c1 9370    + caddc 9372   NN0cn0 10666   ZZcz 10733   ...cfz 11524  ..^cfzo 11635   #chash 12190  Word cword 12309   substr csubstr 12313   ClWalks cclwlk 30536
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 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446
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 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1st 6663  df-2nd 6664  df-recs 6918  df-rdg 6952  df-1o 7006  df-oadd 7010  df-er 7187  df-map 7302  df-pm 7303  df-en 7397  df-dom 7398  df-sdom 7399  df-fin 7400  df-card 8196  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-nn 10410  df-n0 10667  df-z 10734  df-uz 10949  df-fz 11525  df-fzo 11636  df-hash 12191  df-word 12317  df-wlk 23536  df-clwlk 30539
This theorem is referenced by:  clwlkfclwwlk2sswd  30640  clwlkfclwwlk  30641
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