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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 14749 | . . . 4 ⊢ (∅ cyclShift 0) = ∅ | |
2 | oveq1 7420 | . . . 4 ⊢ (∅ = 𝑊 → (∅ cyclShift 0) = (𝑊 cyclShift 0)) | |
3 | id 22 | . . . 4 ⊢ (∅ = 𝑊 → ∅ = 𝑊) | |
4 | 1, 2, 3 | 3eqtr3a 2794 | . . 3 ⊢ (∅ = 𝑊 → (𝑊 cyclShift 0) = 𝑊) |
5 | 4 | a1d 25 | . 2 ⊢ (∅ = 𝑊 → (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊)) |
6 | 0z 12575 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
7 | cshword 14747 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 0 ∈ ℤ) → (𝑊 cyclShift 0) = ((𝑊 substr ⟨(0 mod (♯‘𝑊)), (♯‘𝑊)⟩) ++ (𝑊 prefix (0 mod (♯‘𝑊))))) | |
8 | 6, 7 | mpan2 687 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = ((𝑊 substr ⟨(0 mod (♯‘𝑊)), (♯‘𝑊)⟩) ++ (𝑊 prefix (0 mod (♯‘𝑊))))) |
9 | 8 | adantr 479 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 cyclShift 0) = ((𝑊 substr ⟨(0 mod (♯‘𝑊)), (♯‘𝑊)⟩) ++ (𝑊 prefix (0 mod (♯‘𝑊))))) |
10 | necom 2992 | . . . . . 6 ⊢ (∅ ≠ 𝑊 ↔ 𝑊 ≠ ∅) | |
11 | lennncl 14490 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (♯‘𝑊) ∈ ℕ) | |
12 | nnrp 12991 | . . . . . . 7 ⊢ ((♯‘𝑊) ∈ ℕ → (♯‘𝑊) ∈ ℝ+) | |
13 | 0mod 13873 | . . . . . . . . . 10 ⊢ ((♯‘𝑊) ∈ ℝ+ → (0 mod (♯‘𝑊)) = 0) | |
14 | 13 | opeq1d 4880 | . . . . . . . . 9 ⊢ ((♯‘𝑊) ∈ ℝ+ → ⟨(0 mod (♯‘𝑊)), (♯‘𝑊)⟩ = ⟨0, (♯‘𝑊)⟩) |
15 | 14 | oveq2d 7429 | . . . . . . . 8 ⊢ ((♯‘𝑊) ∈ ℝ+ → (𝑊 substr ⟨(0 mod (♯‘𝑊)), (♯‘𝑊)⟩) = (𝑊 substr ⟨0, (♯‘𝑊)⟩)) |
16 | 13 | oveq2d 7429 | . . . . . . . 8 ⊢ ((♯‘𝑊) ∈ ℝ+ → (𝑊 prefix (0 mod (♯‘𝑊))) = (𝑊 prefix 0)) |
17 | 15, 16 | oveq12d 7431 | . . . . . . 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 592 | . . . . 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 2770 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 cyclShift 0) = ((𝑊 substr ⟨0, (♯‘𝑊)⟩) ++ (𝑊 prefix 0))) |
21 | lencl 14489 | . . . . . . . 8 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ0) | |
22 | pfxval 14629 | . . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℕ0) → (𝑊 prefix (♯‘𝑊)) = (𝑊 substr ⟨0, (♯‘𝑊)⟩)) | |
23 | 21, 22 | mpdan 683 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 prefix (♯‘𝑊)) = (𝑊 substr ⟨0, (♯‘𝑊)⟩)) |
24 | pfxid 14640 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 prefix (♯‘𝑊)) = 𝑊) | |
25 | 23, 24 | eqtr3d 2772 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 substr ⟨0, (♯‘𝑊)⟩) = 𝑊) |
26 | 25 | adantr 479 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 substr ⟨0, (♯‘𝑊)⟩) = 𝑊) |
27 | pfx00 14630 | . . . . . 6 ⊢ (𝑊 prefix 0) = ∅ | |
28 | 27 | a1i 11 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 prefix 0) = ∅) |
29 | 26, 28 | oveq12d 7431 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → ((𝑊 substr ⟨0, (♯‘𝑊)⟩) ++ (𝑊 prefix 0)) = (𝑊 ++ ∅)) |
30 | ccatrid 14543 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 ++ ∅) = 𝑊) | |
31 | 30 | adantr 479 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 ++ ∅) = 𝑊) |
32 | 20, 29, 31 | 3eqtrd 2774 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 cyclShift 0) = 𝑊) |
33 | 32 | expcom 412 | . 2 ⊢ (∅ ≠ 𝑊 → (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊)) |
34 | 5, 33 | pm2.61ine 3023 | 1 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∅c0 4323 ⟨cop 4635 ‘cfv 6544 (class class class)co 7413 0cc0 11114 ℕcn 12218 ℕ0cn0 12478 ℤcz 12564 ℝ+crp 12980 mod cmo 13840 ♯chash 14296 Word cword 14470 ++ cconcat 14526 substr csubstr 14596 prefix cpfx 14626 cyclShift ccsh 14744 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-inf 9442 df-card 9938 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-div 11878 df-nn 12219 df-n0 12479 df-z 12565 df-uz 12829 df-rp 12981 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-hash 14297 df-word 14471 df-concat 14527 df-substr 14597 df-pfx 14627 df-csh 14745 |
This theorem is referenced by: cshwn 14753 2cshwcshw 14782 scshwfzeqfzo 14783 cshwrepswhash1 17042 crctcshlem4 29339 clwwisshclwws 29533 erclwwlkref 29538 erclwwlknref 29587 1cshid 32388 |
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