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Mirrors > Home > MPE Home > Th. List > cshwn | Structured version Visualization version GIF version |
Description: A word cyclically shifted by its length is the word itself. (Contributed by AV, 16-May-2018.) (Revised by AV, 20-May-2018.) (Revised by AV, 26-Oct-2018.) |
Ref | Expression |
---|---|
cshwn | ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift (♯‘𝑊)) = 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0csh0 14146 | . . . 4 ⊢ (∅ cyclShift (♯‘𝑊)) = ∅ | |
2 | oveq1 7142 | . . . 4 ⊢ (∅ = 𝑊 → (∅ cyclShift (♯‘𝑊)) = (𝑊 cyclShift (♯‘𝑊))) | |
3 | id 22 | . . . 4 ⊢ (∅ = 𝑊 → ∅ = 𝑊) | |
4 | 1, 2, 3 | 3eqtr3a 2857 | . . 3 ⊢ (∅ = 𝑊 → (𝑊 cyclShift (♯‘𝑊)) = 𝑊) |
5 | 4 | a1d 25 | . 2 ⊢ (∅ = 𝑊 → (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift (♯‘𝑊)) = 𝑊)) |
6 | lencl 13876 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ0) | |
7 | 6 | nn0zd 12073 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℤ) |
8 | cshwmodn 14148 | . . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℤ) → (𝑊 cyclShift (♯‘𝑊)) = (𝑊 cyclShift ((♯‘𝑊) mod (♯‘𝑊)))) | |
9 | 7, 8 | mpdan 686 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift (♯‘𝑊)) = (𝑊 cyclShift ((♯‘𝑊) mod (♯‘𝑊)))) |
10 | 9 | adantl 485 | . . . 4 ⊢ ((∅ ≠ 𝑊 ∧ 𝑊 ∈ Word 𝑉) → (𝑊 cyclShift (♯‘𝑊)) = (𝑊 cyclShift ((♯‘𝑊) mod (♯‘𝑊)))) |
11 | necom 3040 | . . . . . . . . 9 ⊢ (∅ ≠ 𝑊 ↔ 𝑊 ≠ ∅) | |
12 | lennncl 13877 | . . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (♯‘𝑊) ∈ ℕ) | |
13 | 11, 12 | sylan2b 596 | . . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (♯‘𝑊) ∈ ℕ) |
14 | 13 | nnrpd 12417 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (♯‘𝑊) ∈ ℝ+) |
15 | 14 | ancoms 462 | . . . . . 6 ⊢ ((∅ ≠ 𝑊 ∧ 𝑊 ∈ Word 𝑉) → (♯‘𝑊) ∈ ℝ+) |
16 | modid0 13260 | . . . . . 6 ⊢ ((♯‘𝑊) ∈ ℝ+ → ((♯‘𝑊) mod (♯‘𝑊)) = 0) | |
17 | 15, 16 | syl 17 | . . . . 5 ⊢ ((∅ ≠ 𝑊 ∧ 𝑊 ∈ Word 𝑉) → ((♯‘𝑊) mod (♯‘𝑊)) = 0) |
18 | 17 | oveq2d 7151 | . . . 4 ⊢ ((∅ ≠ 𝑊 ∧ 𝑊 ∈ Word 𝑉) → (𝑊 cyclShift ((♯‘𝑊) mod (♯‘𝑊))) = (𝑊 cyclShift 0)) |
19 | cshw0 14147 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊) | |
20 | 19 | adantl 485 | . . . 4 ⊢ ((∅ ≠ 𝑊 ∧ 𝑊 ∈ Word 𝑉) → (𝑊 cyclShift 0) = 𝑊) |
21 | 10, 18, 20 | 3eqtrd 2837 | . . 3 ⊢ ((∅ ≠ 𝑊 ∧ 𝑊 ∈ Word 𝑉) → (𝑊 cyclShift (♯‘𝑊)) = 𝑊) |
22 | 21 | ex 416 | . 2 ⊢ (∅ ≠ 𝑊 → (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift (♯‘𝑊)) = 𝑊)) |
23 | 5, 22 | pm2.61ine 3070 | 1 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift (♯‘𝑊)) = 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 ‘cfv 6324 (class class class)co 7135 0cc0 10526 ℕcn 11625 ℤcz 11969 ℝ+crp 12377 mod cmo 13232 ♯chash 13686 Word cword 13857 cyclShift ccsh 14141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-hash 13687 df-word 13858 df-concat 13914 df-substr 13994 df-pfx 14024 df-csh 14142 |
This theorem is referenced by: 2cshwid 14167 cshweqdif2 14172 scshwfzeqfzo 14179 cshwcshid 14180 clwwisshclwwsn 27801 eucrct2eupth 28030 |
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