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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 14831 | . . . 4 ⊢ (∅ cyclShift 0) = ∅ | |
| 2 | oveq1 7438 | . . . 4 ⊢ (∅ = 𝑊 → (∅ cyclShift 0) = (𝑊 cyclShift 0)) | |
| 3 | id 22 | . . . 4 ⊢ (∅ = 𝑊 → ∅ = 𝑊) | |
| 4 | 1, 2, 3 | 3eqtr3a 2801 | . . 3 ⊢ (∅ = 𝑊 → (𝑊 cyclShift 0) = 𝑊) |
| 5 | 4 | a1d 25 | . 2 ⊢ (∅ = 𝑊 → (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊)) |
| 6 | 0z 12624 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
| 7 | cshword 14829 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 0 ∈ ℤ) → (𝑊 cyclShift 0) = ((𝑊 substr 〈(0 mod (♯‘𝑊)), (♯‘𝑊)〉) ++ (𝑊 prefix (0 mod (♯‘𝑊))))) | |
| 8 | 6, 7 | mpan2 691 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = ((𝑊 substr 〈(0 mod (♯‘𝑊)), (♯‘𝑊)〉) ++ (𝑊 prefix (0 mod (♯‘𝑊))))) |
| 9 | 8 | adantr 480 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 cyclShift 0) = ((𝑊 substr 〈(0 mod (♯‘𝑊)), (♯‘𝑊)〉) ++ (𝑊 prefix (0 mod (♯‘𝑊))))) |
| 10 | necom 2994 | . . . . . 6 ⊢ (∅ ≠ 𝑊 ↔ 𝑊 ≠ ∅) | |
| 11 | lennncl 14572 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (♯‘𝑊) ∈ ℕ) | |
| 12 | nnrp 13046 | . . . . . . 7 ⊢ ((♯‘𝑊) ∈ ℕ → (♯‘𝑊) ∈ ℝ+) | |
| 13 | 0mod 13942 | . . . . . . . . . 10 ⊢ ((♯‘𝑊) ∈ ℝ+ → (0 mod (♯‘𝑊)) = 0) | |
| 14 | 13 | opeq1d 4879 | . . . . . . . . 9 ⊢ ((♯‘𝑊) ∈ ℝ+ → 〈(0 mod (♯‘𝑊)), (♯‘𝑊)〉 = 〈0, (♯‘𝑊)〉) |
| 15 | 14 | oveq2d 7447 | . . . . . . . 8 ⊢ ((♯‘𝑊) ∈ ℝ+ → (𝑊 substr 〈(0 mod (♯‘𝑊)), (♯‘𝑊)〉) = (𝑊 substr 〈0, (♯‘𝑊)〉)) |
| 16 | 13 | oveq2d 7447 | . . . . . . . 8 ⊢ ((♯‘𝑊) ∈ ℝ+ → (𝑊 prefix (0 mod (♯‘𝑊))) = (𝑊 prefix 0)) |
| 17 | 15, 16 | oveq12d 7449 | . . . . . . 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 594 | . . . . 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 2777 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 cyclShift 0) = ((𝑊 substr 〈0, (♯‘𝑊)〉) ++ (𝑊 prefix 0))) |
| 21 | lencl 14571 | . . . . . . . 8 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ0) | |
| 22 | pfxval 14711 | . . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℕ0) → (𝑊 prefix (♯‘𝑊)) = (𝑊 substr 〈0, (♯‘𝑊)〉)) | |
| 23 | 21, 22 | mpdan 687 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 prefix (♯‘𝑊)) = (𝑊 substr 〈0, (♯‘𝑊)〉)) |
| 24 | pfxid 14722 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 prefix (♯‘𝑊)) = 𝑊) | |
| 25 | 23, 24 | eqtr3d 2779 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 substr 〈0, (♯‘𝑊)〉) = 𝑊) |
| 26 | 25 | adantr 480 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 substr 〈0, (♯‘𝑊)〉) = 𝑊) |
| 27 | pfx00 14712 | . . . . . 6 ⊢ (𝑊 prefix 0) = ∅ | |
| 28 | 27 | a1i 11 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 prefix 0) = ∅) |
| 29 | 26, 28 | oveq12d 7449 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → ((𝑊 substr 〈0, (♯‘𝑊)〉) ++ (𝑊 prefix 0)) = (𝑊 ++ ∅)) |
| 30 | ccatrid 14625 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 ++ ∅) = 𝑊) | |
| 31 | 30 | adantr 480 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 ++ ∅) = 𝑊) |
| 32 | 20, 29, 31 | 3eqtrd 2781 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 cyclShift 0) = 𝑊) |
| 33 | 32 | expcom 413 | . 2 ⊢ (∅ ≠ 𝑊 → (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊)) |
| 34 | 5, 33 | pm2.61ine 3025 | 1 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 〈cop 4632 ‘cfv 6561 (class class class)co 7431 0cc0 11155 ℕcn 12266 ℕ0cn0 12526 ℤcz 12613 ℝ+crp 13034 mod cmo 13909 ♯chash 14369 Word cword 14552 ++ cconcat 14608 substr csubstr 14678 prefix cpfx 14708 cyclShift ccsh 14826 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-hash 14370 df-word 14553 df-concat 14609 df-substr 14679 df-pfx 14709 df-csh 14827 |
| This theorem is referenced by: cshwn 14835 2cshwcshw 14864 scshwfzeqfzo 14865 cshwrepswhash1 17140 crctcshlem4 29840 clwwisshclwws 30034 erclwwlkref 30039 erclwwlknref 30088 1cshid 32944 |
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