MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cshwidx0 Structured version   Unicode version

Theorem cshwidx0 12830
Description: The symbol at index 0 of a cyclically shifted nonempty word is the symbol at index N of the original word. (Contributed by AV, 15-May-2018.) (Revised by AV, 21-May-2018.) (Revised by AV, 30-Oct-2018.)
Assertion
Ref Expression
cshwidx0  |-  ( ( W  e. Word  V  /\  N  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( W cyclShift  N ) `
 0 )  =  ( W `  N
) )

Proof of Theorem cshwidx0
StepHypRef Expression
1 hasheq0 12479 . . . . . 6  |-  ( W  e. Word  V  ->  (
( # `  W )  =  0  <->  W  =  (/) ) )
2 elfzo0 11893 . . . . . . . 8  |-  ( N  e.  ( 0..^ (
# `  W )
)  <->  ( N  e. 
NN0  /\  ( # `  W
)  e.  NN  /\  N  <  ( # `  W
) ) )
3 elnnne0 10849 . . . . . . . . . 10  |-  ( (
# `  W )  e.  NN  <->  ( ( # `  W )  e.  NN0  /\  ( # `  W
)  =/=  0 ) )
4 eqneqall 2610 . . . . . . . . . . . 12  |-  ( (
# `  W )  =  0  ->  (
( # `  W )  =/=  0  ->  ( W  e. Word  V  ->  (
( W cyclShift  N ) ` 
0 )  =  ( W `  N ) ) ) )
54com12 29 . . . . . . . . . . 11  |-  ( (
# `  W )  =/=  0  ->  ( (
# `  W )  =  0  ->  ( W  e. Word  V  ->  (
( W cyclShift  N ) ` 
0 )  =  ( W `  N ) ) ) )
65adantl 464 . . . . . . . . . 10  |-  ( ( ( # `  W
)  e.  NN0  /\  ( # `  W )  =/=  0 )  -> 
( ( # `  W
)  =  0  -> 
( W  e. Word  V  ->  ( ( W cyclShift  N ) `
 0 )  =  ( W `  N
) ) ) )
73, 6sylbi 195 . . . . . . . . 9  |-  ( (
# `  W )  e.  NN  ->  ( ( # `
 W )  =  0  ->  ( W  e. Word  V  ->  ( ( W cyclShift  N ) `  0
)  =  ( W `
 N ) ) ) )
873ad2ant2 1019 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( # `  W )  e.  NN  /\  N  <  ( # `  W
) )  ->  (
( # `  W )  =  0  ->  ( W  e. Word  V  ->  (
( W cyclShift  N ) ` 
0 )  =  ( W `  N ) ) ) )
92, 8sylbi 195 . . . . . . 7  |-  ( N  e.  ( 0..^ (
# `  W )
)  ->  ( ( # `
 W )  =  0  ->  ( W  e. Word  V  ->  ( ( W cyclShift  N ) `  0
)  =  ( W `
 N ) ) ) )
109com13 80 . . . . . 6  |-  ( W  e. Word  V  ->  (
( # `  W )  =  0  ->  ( N  e.  ( 0..^ ( # `  W
) )  ->  (
( W cyclShift  N ) ` 
0 )  =  ( W `  N ) ) ) )
111, 10sylbird 235 . . . . 5  |-  ( W  e. Word  V  ->  ( W  =  (/)  ->  ( N  e.  ( 0..^ ( # `  W
) )  ->  (
( W cyclShift  N ) ` 
0 )  =  ( W `  N ) ) ) )
1211com23 78 . . . 4  |-  ( W  e. Word  V  ->  ( N  e.  ( 0..^ ( # `  W
) )  ->  ( W  =  (/)  ->  (
( W cyclShift  N ) ` 
0 )  =  ( W `  N ) ) ) )
1312imp 427 . . 3  |-  ( ( W  e. Word  V  /\  N  e.  ( 0..^ ( # `  W
) ) )  -> 
( W  =  (/)  ->  ( ( W cyclShift  N ) `
 0 )  =  ( W `  N
) ) )
1413com12 29 . 2  |-  ( W  =  (/)  ->  ( ( W  e. Word  V  /\  N  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( W cyclShift  N ) `
 0 )  =  ( W `  N
) ) )
15 simpl 455 . . . . . 6  |-  ( ( W  e. Word  V  /\  N  e.  ( 0..^ ( # `  W
) ) )  ->  W  e. Word  V )
1615adantl 464 . . . . 5  |-  ( ( W  =/=  (/)  /\  ( W  e. Word  V  /\  N  e.  ( 0..^ ( # `  W ) ) ) )  ->  W  e. Word  V )
17 simpl 455 . . . . 5  |-  ( ( W  =/=  (/)  /\  ( W  e. Word  V  /\  N  e.  ( 0..^ ( # `  W ) ) ) )  ->  W  =/=  (/) )
18 elfzoelz 11857 . . . . . 6  |-  ( N  e.  ( 0..^ (
# `  W )
)  ->  N  e.  ZZ )
1918ad2antll 727 . . . . 5  |-  ( ( W  =/=  (/)  /\  ( W  e. Word  V  /\  N  e.  ( 0..^ ( # `  W ) ) ) )  ->  N  e.  ZZ )
20 cshwidx0mod 12829 . . . . 5  |-  ( ( W  e. Word  V  /\  W  =/=  (/)  /\  N  e.  ZZ )  ->  (
( W cyclShift  N ) ` 
0 )  =  ( W `  ( N  mod  ( # `  W
) ) ) )
2116, 17, 19, 20syl3anc 1230 . . . 4  |-  ( ( W  =/=  (/)  /\  ( W  e. Word  V  /\  N  e.  ( 0..^ ( # `  W ) ) ) )  ->  ( ( W cyclShift  N ) `  0
)  =  ( W `
 ( N  mod  ( # `  W ) ) ) )
22 zmodidfzoimp 12063 . . . . . 6  |-  ( N  e.  ( 0..^ (
# `  W )
)  ->  ( N  mod  ( # `  W
) )  =  N )
2322ad2antll 727 . . . . 5  |-  ( ( W  =/=  (/)  /\  ( W  e. Word  V  /\  N  e.  ( 0..^ ( # `  W ) ) ) )  ->  ( N  mod  ( # `  W
) )  =  N )
2423fveq2d 5852 . . . 4  |-  ( ( W  =/=  (/)  /\  ( W  e. Word  V  /\  N  e.  ( 0..^ ( # `  W ) ) ) )  ->  ( W `  ( N  mod  ( # `
 W ) ) )  =  ( W `
 N ) )
2521, 24eqtrd 2443 . . 3  |-  ( ( W  =/=  (/)  /\  ( W  e. Word  V  /\  N  e.  ( 0..^ ( # `  W ) ) ) )  ->  ( ( W cyclShift  N ) `  0
)  =  ( W `
 N ) )
2625ex 432 . 2  |-  ( W  =/=  (/)  ->  ( ( W  e. Word  V  /\  N  e.  ( 0..^ ( # `  W ) ) )  ->  ( ( W cyclShift  N ) `  0
)  =  ( W `
 N ) ) )
2714, 26pm2.61ine 2716 1  |-  ( ( W  e. Word  V  /\  N  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( W cyclShift  N ) `
 0 )  =  ( W `  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   (/)c0 3737   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   0cc0 9521    < clt 9657   NNcn 10575   NN0cn0 10835   ZZcz 10904  ..^cfzo 11852    mod cmo 12032   #chash 12450  Word cword 12581   cyclShift ccsh 12813
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 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599
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-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-sup 7934  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-n0 10836  df-z 10905  df-uz 11127  df-rp 11265  df-fz 11725  df-fzo 11853  df-fl 11964  df-mod 12033  df-hash 12451  df-word 12589  df-concat 12591  df-substr 12593  df-csh 12814
This theorem is referenced by:  clwwisshclww  25211
  Copyright terms: Public domain W3C validator