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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 14739 | . . . 4 ⊢ (∅ cyclShift 0) = ∅ | |
2 | oveq1 7412 | . . . 4 ⊢ (∅ = 𝑊 → (∅ cyclShift 0) = (𝑊 cyclShift 0)) | |
3 | id 22 | . . . 4 ⊢ (∅ = 𝑊 → ∅ = 𝑊) | |
4 | 1, 2, 3 | 3eqtr3a 2796 | . . 3 ⊢ (∅ = 𝑊 → (𝑊 cyclShift 0) = 𝑊) |
5 | 4 | a1d 25 | . 2 ⊢ (∅ = 𝑊 → (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊)) |
6 | 0z 12565 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
7 | cshword 14737 | . . . . . . 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 481 | . . . . 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 14480 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (♯‘𝑊) ∈ ℕ) | |
12 | nnrp 12981 | . . . . . . 7 ⊢ ((♯‘𝑊) ∈ ℕ → (♯‘𝑊) ∈ ℝ+) | |
13 | 0mod 13863 | . . . . . . . . . 10 ⊢ ((♯‘𝑊) ∈ ℝ+ → (0 mod (♯‘𝑊)) = 0) | |
14 | 13 | opeq1d 4878 | . . . . . . . . 9 ⊢ ((♯‘𝑊) ∈ ℝ+ → 〈(0 mod (♯‘𝑊)), (♯‘𝑊)〉 = 〈0, (♯‘𝑊)〉) |
15 | 14 | oveq2d 7421 | . . . . . . . 8 ⊢ ((♯‘𝑊) ∈ ℝ+ → (𝑊 substr 〈(0 mod (♯‘𝑊)), (♯‘𝑊)〉) = (𝑊 substr 〈0, (♯‘𝑊)〉)) |
16 | 13 | oveq2d 7421 | . . . . . . . 8 ⊢ ((♯‘𝑊) ∈ ℝ+ → (𝑊 prefix (0 mod (♯‘𝑊))) = (𝑊 prefix 0)) |
17 | 15, 16 | oveq12d 7423 | . . . . . . 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 2772 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 cyclShift 0) = ((𝑊 substr 〈0, (♯‘𝑊)〉) ++ (𝑊 prefix 0))) |
21 | lencl 14479 | . . . . . . . 8 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ0) | |
22 | pfxval 14619 | . . . . . . . 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 14630 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 prefix (♯‘𝑊)) = 𝑊) | |
25 | 23, 24 | eqtr3d 2774 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 substr 〈0, (♯‘𝑊)〉) = 𝑊) |
26 | 25 | adantr 481 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 substr 〈0, (♯‘𝑊)〉) = 𝑊) |
27 | pfx00 14620 | . . . . . 6 ⊢ (𝑊 prefix 0) = ∅ | |
28 | 27 | a1i 11 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 prefix 0) = ∅) |
29 | 26, 28 | oveq12d 7423 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → ((𝑊 substr 〈0, (♯‘𝑊)〉) ++ (𝑊 prefix 0)) = (𝑊 ++ ∅)) |
30 | ccatrid 14533 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 ++ ∅) = 𝑊) | |
31 | 30 | adantr 481 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 ++ ∅) = 𝑊) |
32 | 20, 29, 31 | 3eqtrd 2776 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 cyclShift 0) = 𝑊) |
33 | 32 | expcom 414 | . 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 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∅c0 4321 〈cop 4633 ‘cfv 6540 (class class class)co 7405 0cc0 11106 ℕcn 12208 ℕ0cn0 12468 ℤcz 12554 ℝ+crp 12970 mod cmo 13830 ♯chash 14286 Word cword 14460 ++ cconcat 14516 substr csubstr 14586 prefix cpfx 14616 cyclShift ccsh 14734 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-hash 14287 df-word 14461 df-concat 14517 df-substr 14587 df-pfx 14617 df-csh 14735 |
This theorem is referenced by: cshwn 14743 2cshwcshw 14772 scshwfzeqfzo 14773 cshwrepswhash1 17032 crctcshlem4 29063 clwwisshclwws 29257 erclwwlkref 29262 erclwwlknref 29311 1cshid 32110 |
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