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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 13585 | . . . 4 ⊢ (∅ cyclShift 0) = ∅ | |
2 | oveq1 6697 | . . . 4 ⊢ (∅ = 𝑊 → (∅ cyclShift 0) = (𝑊 cyclShift 0)) | |
3 | id 22 | . . . 4 ⊢ (∅ = 𝑊 → ∅ = 𝑊) | |
4 | 1, 2, 3 | 3eqtr3a 2709 | . . 3 ⊢ (∅ = 𝑊 → (𝑊 cyclShift 0) = 𝑊) |
5 | 4 | a1d 25 | . 2 ⊢ (∅ = 𝑊 → (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊)) |
6 | 0z 11426 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
7 | cshword 13583 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 0 ∈ ℤ) → (𝑊 cyclShift 0) = ((𝑊 substr 〈(0 mod (#‘𝑊)), (#‘𝑊)〉) ++ (𝑊 substr 〈0, (0 mod (#‘𝑊))〉))) | |
8 | 6, 7 | mpan2 707 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = ((𝑊 substr 〈(0 mod (#‘𝑊)), (#‘𝑊)〉) ++ (𝑊 substr 〈0, (0 mod (#‘𝑊))〉))) |
9 | 8 | adantr 480 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 cyclShift 0) = ((𝑊 substr 〈(0 mod (#‘𝑊)), (#‘𝑊)〉) ++ (𝑊 substr 〈0, (0 mod (#‘𝑊))〉))) |
10 | necom 2876 | . . . . . 6 ⊢ (∅ ≠ 𝑊 ↔ 𝑊 ≠ ∅) | |
11 | lennncl 13357 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (#‘𝑊) ∈ ℕ) | |
12 | nnrp 11880 | . . . . . . 7 ⊢ ((#‘𝑊) ∈ ℕ → (#‘𝑊) ∈ ℝ+) | |
13 | 0mod 12741 | . . . . . . . . . 10 ⊢ ((#‘𝑊) ∈ ℝ+ → (0 mod (#‘𝑊)) = 0) | |
14 | 13 | opeq1d 4439 | . . . . . . . . 9 ⊢ ((#‘𝑊) ∈ ℝ+ → 〈(0 mod (#‘𝑊)), (#‘𝑊)〉 = 〈0, (#‘𝑊)〉) |
15 | 14 | oveq2d 6706 | . . . . . . . 8 ⊢ ((#‘𝑊) ∈ ℝ+ → (𝑊 substr 〈(0 mod (#‘𝑊)), (#‘𝑊)〉) = (𝑊 substr 〈0, (#‘𝑊)〉)) |
16 | 13 | opeq2d 4440 | . . . . . . . . 9 ⊢ ((#‘𝑊) ∈ ℝ+ → 〈0, (0 mod (#‘𝑊))〉 = 〈0, 0〉) |
17 | 16 | oveq2d 6706 | . . . . . . . 8 ⊢ ((#‘𝑊) ∈ ℝ+ → (𝑊 substr 〈0, (0 mod (#‘𝑊))〉) = (𝑊 substr 〈0, 0〉)) |
18 | 15, 17 | oveq12d 6708 | . . . . . . 7 ⊢ ((#‘𝑊) ∈ ℝ+ → ((𝑊 substr 〈(0 mod (#‘𝑊)), (#‘𝑊)〉) ++ (𝑊 substr 〈0, (0 mod (#‘𝑊))〉)) = ((𝑊 substr 〈0, (#‘𝑊)〉) ++ (𝑊 substr 〈0, 0〉))) |
19 | 11, 12, 18 | 3syl 18 | . . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → ((𝑊 substr 〈(0 mod (#‘𝑊)), (#‘𝑊)〉) ++ (𝑊 substr 〈0, (0 mod (#‘𝑊))〉)) = ((𝑊 substr 〈0, (#‘𝑊)〉) ++ (𝑊 substr 〈0, 0〉))) |
20 | 10, 19 | sylan2b 491 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → ((𝑊 substr 〈(0 mod (#‘𝑊)), (#‘𝑊)〉) ++ (𝑊 substr 〈0, (0 mod (#‘𝑊))〉)) = ((𝑊 substr 〈0, (#‘𝑊)〉) ++ (𝑊 substr 〈0, 0〉))) |
21 | 9, 20 | eqtrd 2685 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 cyclShift 0) = ((𝑊 substr 〈0, (#‘𝑊)〉) ++ (𝑊 substr 〈0, 0〉))) |
22 | swrdid 13474 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 substr 〈0, (#‘𝑊)〉) = 𝑊) | |
23 | 22 | adantr 480 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 substr 〈0, (#‘𝑊)〉) = 𝑊) |
24 | swrd00 13463 | . . . . . 6 ⊢ (𝑊 substr 〈0, 0〉) = ∅ | |
25 | 24 | a1i 11 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 substr 〈0, 0〉) = ∅) |
26 | 23, 25 | oveq12d 6708 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → ((𝑊 substr 〈0, (#‘𝑊)〉) ++ (𝑊 substr 〈0, 0〉)) = (𝑊 ++ ∅)) |
27 | ccatrid 13405 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 ++ ∅) = 𝑊) | |
28 | 27 | adantr 480 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 ++ ∅) = 𝑊) |
29 | 21, 26, 28 | 3eqtrd 2689 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 cyclShift 0) = 𝑊) |
30 | 29 | expcom 450 | . 2 ⊢ (∅ ≠ 𝑊 → (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊)) |
31 | 5, 30 | pm2.61ine 2906 | 1 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∅c0 3948 〈cop 4216 ‘cfv 5926 (class class class)co 6690 0cc0 9974 ℕcn 11058 ℤcz 11415 ℝ+crp 11870 mod cmo 12708 #chash 13157 Word cword 13323 ++ cconcat 13325 substr csubstr 13327 cyclShift ccsh 13580 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-inf 8390 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-n0 11331 df-z 11416 df-uz 11726 df-rp 11871 df-fz 12365 df-fzo 12505 df-fl 12633 df-mod 12709 df-hash 13158 df-word 13331 df-concat 13333 df-substr 13335 df-csh 13581 |
This theorem is referenced by: cshwn 13589 2cshwcshw 13617 scshwfzeqfzo 13618 cshwrepswhash1 15856 crctcshlem4 26768 clwwisshclwws 26972 erclwwlkref 26977 erclwwlknref 27033 |
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