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Mirrors > Home > MPE Home > Th. List > cshw0 | Structured version Visualization version GIF version |
Description: A word cyclically shifted by 0 is the word itself. (Contributed by AV, 16-May-2018.) (Revised by AV, 20-May-2018.) (Revised by AV, 26-Oct-2018.) |
Ref | Expression |
---|---|
cshw0 | ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0csh0 14154 | . . . 4 ⊢ (∅ cyclShift 0) = ∅ | |
2 | oveq1 7162 | . . . 4 ⊢ (∅ = 𝑊 → (∅ cyclShift 0) = (𝑊 cyclShift 0)) | |
3 | id 22 | . . . 4 ⊢ (∅ = 𝑊 → ∅ = 𝑊) | |
4 | 1, 2, 3 | 3eqtr3a 2880 | . . 3 ⊢ (∅ = 𝑊 → (𝑊 cyclShift 0) = 𝑊) |
5 | 4 | a1d 25 | . 2 ⊢ (∅ = 𝑊 → (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊)) |
6 | 0z 11991 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
7 | cshword 14152 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 0 ∈ ℤ) → (𝑊 cyclShift 0) = ((𝑊 substr 〈(0 mod (♯‘𝑊)), (♯‘𝑊)〉) ++ (𝑊 prefix (0 mod (♯‘𝑊))))) | |
8 | 6, 7 | mpan2 689 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = ((𝑊 substr 〈(0 mod (♯‘𝑊)), (♯‘𝑊)〉) ++ (𝑊 prefix (0 mod (♯‘𝑊))))) |
9 | 8 | adantr 483 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 cyclShift 0) = ((𝑊 substr 〈(0 mod (♯‘𝑊)), (♯‘𝑊)〉) ++ (𝑊 prefix (0 mod (♯‘𝑊))))) |
10 | necom 3069 | . . . . . 6 ⊢ (∅ ≠ 𝑊 ↔ 𝑊 ≠ ∅) | |
11 | lennncl 13883 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (♯‘𝑊) ∈ ℕ) | |
12 | nnrp 12399 | . . . . . . 7 ⊢ ((♯‘𝑊) ∈ ℕ → (♯‘𝑊) ∈ ℝ+) | |
13 | 0mod 13269 | . . . . . . . . . 10 ⊢ ((♯‘𝑊) ∈ ℝ+ → (0 mod (♯‘𝑊)) = 0) | |
14 | 13 | opeq1d 4808 | . . . . . . . . 9 ⊢ ((♯‘𝑊) ∈ ℝ+ → 〈(0 mod (♯‘𝑊)), (♯‘𝑊)〉 = 〈0, (♯‘𝑊)〉) |
15 | 14 | oveq2d 7171 | . . . . . . . 8 ⊢ ((♯‘𝑊) ∈ ℝ+ → (𝑊 substr 〈(0 mod (♯‘𝑊)), (♯‘𝑊)〉) = (𝑊 substr 〈0, (♯‘𝑊)〉)) |
16 | 13 | oveq2d 7171 | . . . . . . . 8 ⊢ ((♯‘𝑊) ∈ ℝ+ → (𝑊 prefix (0 mod (♯‘𝑊))) = (𝑊 prefix 0)) |
17 | 15, 16 | oveq12d 7173 | . . . . . . 7 ⊢ ((♯‘𝑊) ∈ ℝ+ → ((𝑊 substr 〈(0 mod (♯‘𝑊)), (♯‘𝑊)〉) ++ (𝑊 prefix (0 mod (♯‘𝑊)))) = ((𝑊 substr 〈0, (♯‘𝑊)〉) ++ (𝑊 prefix 0))) |
18 | 11, 12, 17 | 3syl 18 | . . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → ((𝑊 substr 〈(0 mod (♯‘𝑊)), (♯‘𝑊)〉) ++ (𝑊 prefix (0 mod (♯‘𝑊)))) = ((𝑊 substr 〈0, (♯‘𝑊)〉) ++ (𝑊 prefix 0))) |
19 | 10, 18 | sylan2b 595 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → ((𝑊 substr 〈(0 mod (♯‘𝑊)), (♯‘𝑊)〉) ++ (𝑊 prefix (0 mod (♯‘𝑊)))) = ((𝑊 substr 〈0, (♯‘𝑊)〉) ++ (𝑊 prefix 0))) |
20 | 9, 19 | eqtrd 2856 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 cyclShift 0) = ((𝑊 substr 〈0, (♯‘𝑊)〉) ++ (𝑊 prefix 0))) |
21 | lencl 13882 | . . . . . . . 8 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ0) | |
22 | pfxval 14034 | . . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℕ0) → (𝑊 prefix (♯‘𝑊)) = (𝑊 substr 〈0, (♯‘𝑊)〉)) | |
23 | 21, 22 | mpdan 685 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 prefix (♯‘𝑊)) = (𝑊 substr 〈0, (♯‘𝑊)〉)) |
24 | pfxid 14045 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 prefix (♯‘𝑊)) = 𝑊) | |
25 | 23, 24 | eqtr3d 2858 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 substr 〈0, (♯‘𝑊)〉) = 𝑊) |
26 | 25 | adantr 483 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 substr 〈0, (♯‘𝑊)〉) = 𝑊) |
27 | pfx00 14035 | . . . . . 6 ⊢ (𝑊 prefix 0) = ∅ | |
28 | 27 | a1i 11 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 prefix 0) = ∅) |
29 | 26, 28 | oveq12d 7173 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → ((𝑊 substr 〈0, (♯‘𝑊)〉) ++ (𝑊 prefix 0)) = (𝑊 ++ ∅)) |
30 | ccatrid 13940 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 ++ ∅) = 𝑊) | |
31 | 30 | adantr 483 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 ++ ∅) = 𝑊) |
32 | 20, 29, 31 | 3eqtrd 2860 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 cyclShift 0) = 𝑊) |
33 | 32 | expcom 416 | . 2 ⊢ (∅ ≠ 𝑊 → (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊)) |
34 | 5, 33 | pm2.61ine 3100 | 1 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∅c0 4290 〈cop 4572 ‘cfv 6354 (class class class)co 7155 0cc0 10536 ℕcn 11637 ℕ0cn0 11896 ℤcz 11980 ℝ+crp 12388 mod cmo 13236 ♯chash 13689 Word cword 13860 ++ cconcat 13921 substr csubstr 14001 prefix cpfx 14031 cyclShift ccsh 14149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-inf 8906 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-fz 12892 df-fzo 13033 df-fl 13161 df-mod 13237 df-hash 13690 df-word 13861 df-concat 13922 df-substr 14002 df-pfx 14032 df-csh 14150 |
This theorem is referenced by: cshwn 14158 2cshwcshw 14186 scshwfzeqfzo 14187 cshwrepswhash1 16435 crctcshlem4 27597 clwwisshclwws 27792 erclwwlkref 27797 erclwwlknref 27847 1cshid 30633 |
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