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Mirrors > Home > MPE Home > Th. List > cshwshashlem3 | Structured version Visualization version GIF version |
Description: If cyclically shifting a word of length being a prime number and not of identical symbols by different numbers of positions, the resulting words are different. (Contributed by Alexander van der Vekens, 19-May-2018.) (Revised by Alexander van der Vekens, 8-Jun-2018.) |
Ref | Expression |
---|---|
cshwshash.0 | ⊢ (𝜑 → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ)) |
Ref | Expression |
---|---|
cshwshashlem3 | ⊢ ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ≠ 𝐿) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzoelz 12470 | . . . . . 6 ⊢ (𝐾 ∈ (0..^(#‘𝑊)) → 𝐾 ∈ ℤ) | |
2 | 1 | zred 11482 | . . . . 5 ⊢ (𝐾 ∈ (0..^(#‘𝑊)) → 𝐾 ∈ ℝ) |
3 | elfzoelz 12470 | . . . . . 6 ⊢ (𝐿 ∈ (0..^(#‘𝑊)) → 𝐿 ∈ ℤ) | |
4 | 3 | zred 11482 | . . . . 5 ⊢ (𝐿 ∈ (0..^(#‘𝑊)) → 𝐿 ∈ ℝ) |
5 | lttri2 10120 | . . . . 5 ⊢ ((𝐾 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝐾 ≠ 𝐿 ↔ (𝐾 < 𝐿 ∨ 𝐿 < 𝐾))) | |
6 | 2, 4, 5 | syl2anr 495 | . . . 4 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊))) → (𝐾 ≠ 𝐿 ↔ (𝐾 < 𝐿 ∨ 𝐿 < 𝐾))) |
7 | cshwshash.0 | . . . . . . . 8 ⊢ (𝜑 → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ)) | |
8 | 7 | cshwshashlem2 15803 | . . . . . . 7 ⊢ ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾))) |
9 | 8 | com12 32 | . . . . . 6 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾))) |
10 | 9 | 3expia 1267 | . . . . 5 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊))) → (𝐾 < 𝐿 → ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾)))) |
11 | 7 | cshwshashlem2 15803 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → ((𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐿 < 𝐾) → (𝑊 cyclShift 𝐾) ≠ (𝑊 cyclShift 𝐿))) |
12 | 11 | imp 445 | . . . . . . . . 9 ⊢ (((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) ∧ (𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐿 < 𝐾)) → (𝑊 cyclShift 𝐾) ≠ (𝑊 cyclShift 𝐿)) |
13 | 12 | necomd 2849 | . . . . . . . 8 ⊢ (((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) ∧ (𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐿 < 𝐾)) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾)) |
14 | 13 | expcom 451 | . . . . . . 7 ⊢ ((𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐿 < 𝐾) → ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾))) |
15 | 14 | 3expia 1267 | . . . . . 6 ⊢ ((𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐿 ∈ (0..^(#‘𝑊))) → (𝐿 < 𝐾 → ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾)))) |
16 | 15 | ancoms 469 | . . . . 5 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊))) → (𝐿 < 𝐾 → ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾)))) |
17 | 10, 16 | jaod 395 | . . . 4 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊))) → ((𝐾 < 𝐿 ∨ 𝐿 < 𝐾) → ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾)))) |
18 | 6, 17 | sylbid 230 | . . 3 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊))) → (𝐾 ≠ 𝐿 → ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾)))) |
19 | 18 | 3impia 1261 | . 2 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ≠ 𝐿) → ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾))) |
20 | 19 | com12 32 | 1 ⊢ ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ≠ 𝐿) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 ∈ wcel 1990 ≠ wne 2794 ∃wrex 2913 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 < clt 10074 ..^cfzo 12465 #chash 13117 Word cword 13291 cyclShift ccsh 13534 ℙcprime 15385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-word 13299 df-concat 13301 df-substr 13303 df-reps 13306 df-csh 13535 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-gcd 15217 df-prm 15386 df-phi 15471 |
This theorem is referenced by: cshwsdisj 15805 |
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