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Theorem clwlkfclwwlk2wrd 25269
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 3058 . 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 25193 . . . 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 12616 . . . . . 6  |-  ( A  e. Word  dom  E  ->  (
# `  A )  e.  NN0 )
7 nn0z 10930 . . . . . . . . . 10  |-  ( (
# `  A )  e.  NN0  ->  ( # `  A
)  e.  ZZ )
8 fzval3 11923 . . . . . . . . . 10  |-  ( (
# `  A )  e.  ZZ  ->  ( 0 ... ( # `  A
) )  =  ( 0..^ ( ( # `  A )  +  1 ) ) )
97, 8syl 17 . . . . . . . . 9  |-  ( (
# `  A )  e.  NN0  ->  ( 0 ... ( # `  A
) )  =  ( 0..^ ( ( # `  A )  +  1 ) ) )
109feq2d 5703 . . . . . . . 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 12604 . . . . . . 7  |-  ( B : ( 0..^ ( ( # `  A
)  +  1 ) ) --> V  ->  B  e. Word  V )
1311, 12syl 17 . . . . . 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 17 . . 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 197 1  |-  ( c  e.  C  ->  B  e. Word  V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1407    e. wcel 1844   A.wral 2756   {crab 2760   {cpr 3976   <.cop 3980    |-> cmpt 4455   dom cdm 4825   -->wf 5567   ` cfv 5571  (class class class)co 6280   1stc1st 6784   2ndc2nd 6785   0cc0 9524   1c1 9525    + caddc 9527   NN0cn0 10838   ZZcz 10907   ...cfz 11728  ..^cfzo 11856   #chash 12454  Word cword 12585   substr csubstr 12589   ClWalks cclwlk 25176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-map 7461  df-pm 7462  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-card 8354  df-cda 8582  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-n0 10839  df-z 10908  df-uz 11130  df-fz 11729  df-fzo 11857  df-hash 12455  df-word 12593  df-wlk 24937  df-clwlk 25179
This theorem is referenced by:  clwlkfclwwlk2sswd  25272  clwlkfclwwlk  25273
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