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Theorem cshwssizelem3 28315
Description: If cyclically shifting a word of length being a prime number not consisting of identical symbols by at least one position (and not by as many positions as the length of the word), the result will not be the word itself. (Contributed by Alexander van der Vekens, 19-May-2018.) (Revised by Alexander van der Vekens, 8-Jun-2018.)
Hypothesis
Ref Expression
cshwssize.0  |-  ( ph  ->  ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )
)
Assertion
Ref Expression
cshwssizelem3  |-  ( (
ph  /\  E. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =/=  ( W ` 
0 )  /\  L  e.  ( 1..^ ( # `  W ) ) )  ->  ( W CyclShift  L )  =/=  W )
Distinct variable groups:    i, L    i, V    i, W    ph, i

Proof of Theorem cshwssizelem3
StepHypRef Expression
1 df-ne 2603 . . . . . . 7  |-  ( ( W `  i )  =/=  ( W ` 
0 )  <->  -.  ( W `  i )  =  ( W ` 
0 ) )
21rexbii 2732 . . . . . 6  |-  ( E. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =/=  ( W `  0
)  <->  E. i  e.  ( 0..^ ( # `  W
) )  -.  ( W `  i )  =  ( W ` 
0 ) )
3 rexnal 2718 . . . . . 6  |-  ( E. i  e.  ( 0..^ ( # `  W
) )  -.  ( W `  i )  =  ( W ` 
0 )  <->  -.  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) )
42, 3bitri 242 . . . . 5  |-  ( E. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =/=  ( W `  0
)  <->  -.  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) )
5 simpll 732 . . . . . . . . . . 11  |-  ( ( ( ph  /\  L  e.  ( 1..^ ( # `  W ) ) )  /\  ( W CyclShift  L )  =  W )  ->  ph )
6 fzo0ss1 28145 . . . . . . . . . . . . . 14  |-  ( 1..^ ( # `  W
) )  C_  (
0..^ ( # `  W
) )
7 fzossfz 11162 . . . . . . . . . . . . . 14  |-  ( 0..^ ( # `  W
) )  C_  (
0 ... ( # `  W
) )
86, 7sstri 3359 . . . . . . . . . . . . 13  |-  ( 1..^ ( # `  W
) )  C_  (
0 ... ( # `  W
) )
98sseli 3346 . . . . . . . . . . . 12  |-  ( L  e.  ( 1..^ (
# `  W )
)  ->  L  e.  ( 0 ... ( # `
 W ) ) )
109ad2antlr 709 . . . . . . . . . . 11  |-  ( ( ( ph  /\  L  e.  ( 1..^ ( # `  W ) ) )  /\  ( W CyclShift  L )  =  W )  ->  L  e.  ( 0 ... ( # `  W
) ) )
11 simpr 449 . . . . . . . . . . 11  |-  ( ( ( ph  /\  L  e.  ( 1..^ ( # `  W ) ) )  /\  ( W CyclShift  L )  =  W )  -> 
( W CyclShift  L )  =  W )
12 cshwssize.0 . . . . . . . . . . . 12  |-  ( ph  ->  ( W  e. Word  V  /\  ( # `  W
)  e.  Prime )
)
1312cshwssizelem2 28314 . . . . . . . . . . 11  |-  ( (
ph  /\  L  e.  ( 0 ... ( # `
 W ) )  /\  ( W CyclShift  L )  =  W )  -> 
( L  =  0  \/  L  =  (
# `  W )  \/  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
145, 10, 11, 13syl3anc 1185 . . . . . . . . . 10  |-  ( ( ( ph  /\  L  e.  ( 1..^ ( # `  W ) ) )  /\  ( W CyclShift  L )  =  W )  -> 
( L  =  0  \/  L  =  (
# `  W )  \/  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
15 elfzo1 11178 . . . . . . . . . . . . . 14  |-  ( L  e.  ( 1..^ (
# `  W )
)  <->  ( L  e.  NN  /\  ( # `  W )  e.  NN  /\  L  <  ( # `  W ) ) )
16 nnne0 10037 . . . . . . . . . . . . . . . 16  |-  ( L  e.  NN  ->  L  =/=  0 )
17 df-ne 2603 . . . . . . . . . . . . . . . . 17  |-  ( L  =/=  0  <->  -.  L  =  0 )
18 pm2.21 103 . . . . . . . . . . . . . . . . 17  |-  ( -.  L  =  0  -> 
( L  =  0  ->  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
1917, 18sylbi 189 . . . . . . . . . . . . . . . 16  |-  ( L  =/=  0  ->  ( L  =  0  ->  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
2016, 19syl 16 . . . . . . . . . . . . . . 15  |-  ( L  e.  NN  ->  ( L  =  0  ->  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
21203ad2ant1 979 . . . . . . . . . . . . . 14  |-  ( ( L  e.  NN  /\  ( # `  W )  e.  NN  /\  L  <  ( # `  W
) )  ->  ( L  =  0  ->  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
2215, 21sylbi 189 . . . . . . . . . . . . 13  |-  ( L  e.  ( 1..^ (
# `  W )
)  ->  ( L  =  0  ->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) ) )
2322ad2antlr 709 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  L  e.  ( 1..^ ( # `  W ) ) )  /\  ( W CyclShift  L )  =  W )  -> 
( L  =  0  ->  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
2423com12 30 . . . . . . . . . . 11  |-  ( L  =  0  ->  (
( ( ph  /\  L  e.  ( 1..^ ( # `  W
) ) )  /\  ( W CyclShift  L )  =  W )  ->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) ) )
25 nnre 10012 . . . . . . . . . . . . . . . . 17  |-  ( L  e.  NN  ->  L  e.  RR )
26 ltne 9175 . . . . . . . . . . . . . . . . 17  |-  ( ( L  e.  RR  /\  L  <  ( # `  W
) )  ->  ( # `
 W )  =/= 
L )
2725, 26sylan 459 . . . . . . . . . . . . . . . 16  |-  ( ( L  e.  NN  /\  L  <  ( # `  W
) )  ->  ( # `
 W )  =/= 
L )
28 df-ne 2603 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  W )  =/=  L  <->  -.  ( # `  W
)  =  L )
29 eqcom 2440 . . . . . . . . . . . . . . . . . 18  |-  ( L  =  ( # `  W
)  <->  ( # `  W
)  =  L )
30 pm2.21 103 . . . . . . . . . . . . . . . . . 18  |-  ( -.  ( # `  W
)  =  L  -> 
( ( # `  W
)  =  L  ->  A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
3129, 30syl5bi 210 . . . . . . . . . . . . . . . . 17  |-  ( -.  ( # `  W
)  =  L  -> 
( L  =  (
# `  W )  ->  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
3228, 31sylbi 189 . . . . . . . . . . . . . . . 16  |-  ( (
# `  W )  =/=  L  ->  ( L  =  ( # `  W
)  ->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) ) )
3327, 32syl 16 . . . . . . . . . . . . . . 15  |-  ( ( L  e.  NN  /\  L  <  ( # `  W
) )  ->  ( L  =  ( # `  W
)  ->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) ) )
34333adant2 977 . . . . . . . . . . . . . 14  |-  ( ( L  e.  NN  /\  ( # `  W )  e.  NN  /\  L  <  ( # `  W
) )  ->  ( L  =  ( # `  W
)  ->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) ) )
3515, 34sylbi 189 . . . . . . . . . . . . 13  |-  ( L  e.  ( 1..^ (
# `  W )
)  ->  ( L  =  ( # `  W
)  ->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) ) )
3635ad2antlr 709 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  L  e.  ( 1..^ ( # `  W ) ) )  /\  ( W CyclShift  L )  =  W )  -> 
( L  =  (
# `  W )  ->  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
3736com12 30 . . . . . . . . . . 11  |-  ( L  =  ( # `  W
)  ->  ( (
( ph  /\  L  e.  ( 1..^ ( # `  W ) ) )  /\  ( W CyclShift  L )  =  W )  ->  A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
38 ax-1 6 . . . . . . . . . . 11  |-  ( A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
)  ->  ( (
( ph  /\  L  e.  ( 1..^ ( # `  W ) ) )  /\  ( W CyclShift  L )  =  W )  ->  A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) ) )
3924, 37, 383jaoi 1248 . . . . . . . . . 10  |-  ( ( L  =  0  \/  L  =  ( # `  W )  \/  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) )  ->  (
( ( ph  /\  L  e.  ( 1..^ ( # `  W
) ) )  /\  ( W CyclShift  L )  =  W )  ->  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 ) ) )
4014, 39mpcom 35 . . . . . . . . 9  |-  ( ( ( ph  /\  L  e.  ( 1..^ ( # `  W ) ) )  /\  ( W CyclShift  L )  =  W )  ->  A. i  e.  (
0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
) )
4140pm2.24d 138 . . . . . . . 8  |-  ( ( ( ph  /\  L  e.  ( 1..^ ( # `  W ) ) )  /\  ( W CyclShift  L )  =  W )  -> 
( -.  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 )  ->  ( W CyclShift  L )  =/=  W
) )
4241exp31 589 . . . . . . 7  |-  ( ph  ->  ( L  e.  ( 1..^ ( # `  W
) )  ->  (
( W CyclShift  L )  =  W  ->  ( -.  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
)  ->  ( W CyclShift  L )  =/=  W ) ) ) )
4342com34 80 . . . . . 6  |-  ( ph  ->  ( L  e.  ( 1..^ ( # `  W
) )  ->  ( -.  A. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =  ( W `  0
)  ->  ( ( W CyclShift  L )  =  W  ->  ( W CyclShift  L )  =/=  W ) ) ) )
4443com23 75 . . . . 5  |-  ( ph  ->  ( -.  A. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =  ( W ` 
0 )  ->  ( L  e.  ( 1..^ ( # `  W
) )  ->  (
( W CyclShift  L )  =  W  ->  ( W CyclShift  L )  =/=  W ) ) ) )
454, 44syl5bi 210 . . . 4  |-  ( ph  ->  ( E. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =/=  ( W ` 
0 )  ->  ( L  e.  ( 1..^ ( # `  W
) )  ->  (
( W CyclShift  L )  =  W  ->  ( W CyclShift  L )  =/=  W ) ) ) )
46453imp 1148 . . 3  |-  ( (
ph  /\  E. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =/=  ( W ` 
0 )  /\  L  e.  ( 1..^ ( # `  W ) ) )  ->  ( ( W CyclShift  L )  =  W  ->  ( W CyclShift  L )  =/=  W ) )
4746com12 30 . 2  |-  ( ( W CyclShift  L )  =  W  ->  ( ( ph  /\ 
E. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =/=  ( W `  0
)  /\  L  e.  ( 1..^ ( # `  W
) ) )  -> 
( W CyclShift  L )  =/= 
W ) )
48 ax-1 6 . 2  |-  ( ( W CyclShift  L )  =/=  W  ->  ( ( ph  /\  E. i  e.  ( 0..^ ( # `  W
) ) ( W `
 i )  =/=  ( W `  0
)  /\  L  e.  ( 1..^ ( # `  W
) ) )  -> 
( W CyclShift  L )  =/= 
W ) )
4947, 48pm2.61ine 2682 1  |-  ( (
ph  /\  E. i  e.  ( 0..^ ( # `  W ) ) ( W `  i )  =/=  ( W ` 
0 )  /\  L  e.  ( 1..^ ( # `  W ) ) )  ->  ( W CyclShift  L )  =/=  W )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    \/ w3o 936    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   RRcr 8994   0cc0 8995   1c1 8996    < clt 9125   NNcn 10005   ...cfz 11048  ..^cfzo 11140   #chash 11623  Word cword 11722   Primecprime 13084   CyclShift ccsh 28263
This theorem is referenced by:  cshwssizelem4a  28316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-fal 1330  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fzo 11141  df-fl 11207  df-mod 11256  df-seq 11329  df-exp 11388  df-hash 11624  df-word 11728  df-concat 11729  df-substr 11731  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-dvds 12858  df-gcd 13012  df-prm 13085  df-phi 13160  df-csh 28265
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