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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 14358 | . . . 4 ⊢ (∅ cyclShift 0) = ∅ | |
2 | oveq1 7220 | . . . 4 ⊢ (∅ = 𝑊 → (∅ cyclShift 0) = (𝑊 cyclShift 0)) | |
3 | id 22 | . . . 4 ⊢ (∅ = 𝑊 → ∅ = 𝑊) | |
4 | 1, 2, 3 | 3eqtr3a 2802 | . . 3 ⊢ (∅ = 𝑊 → (𝑊 cyclShift 0) = 𝑊) |
5 | 4 | a1d 25 | . 2 ⊢ (∅ = 𝑊 → (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊)) |
6 | 0z 12187 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
7 | cshword 14356 | . . . . . . 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 484 | . . . . 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 14089 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (♯‘𝑊) ∈ ℕ) | |
12 | nnrp 12597 | . . . . . . 7 ⊢ ((♯‘𝑊) ∈ ℕ → (♯‘𝑊) ∈ ℝ+) | |
13 | 0mod 13475 | . . . . . . . . . 10 ⊢ ((♯‘𝑊) ∈ ℝ+ → (0 mod (♯‘𝑊)) = 0) | |
14 | 13 | opeq1d 4790 | . . . . . . . . 9 ⊢ ((♯‘𝑊) ∈ ℝ+ → 〈(0 mod (♯‘𝑊)), (♯‘𝑊)〉 = 〈0, (♯‘𝑊)〉) |
15 | 14 | oveq2d 7229 | . . . . . . . 8 ⊢ ((♯‘𝑊) ∈ ℝ+ → (𝑊 substr 〈(0 mod (♯‘𝑊)), (♯‘𝑊)〉) = (𝑊 substr 〈0, (♯‘𝑊)〉)) |
16 | 13 | oveq2d 7229 | . . . . . . . 8 ⊢ ((♯‘𝑊) ∈ ℝ+ → (𝑊 prefix (0 mod (♯‘𝑊))) = (𝑊 prefix 0)) |
17 | 15, 16 | oveq12d 7231 | . . . . . . 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 597 | . . . . 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 14088 | . . . . . . . 8 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ0) | |
22 | pfxval 14238 | . . . . . . . 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 14249 | . . . . . . 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 484 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 substr 〈0, (♯‘𝑊)〉) = 𝑊) |
27 | pfx00 14239 | . . . . . 6 ⊢ (𝑊 prefix 0) = ∅ | |
28 | 27 | a1i 11 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 prefix 0) = ∅) |
29 | 26, 28 | oveq12d 7231 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → ((𝑊 substr 〈0, (♯‘𝑊)〉) ++ (𝑊 prefix 0)) = (𝑊 ++ ∅)) |
30 | ccatrid 14144 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 ++ ∅) = 𝑊) | |
31 | 30 | adantr 484 | . . . 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 417 | . 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 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ∅c0 4237 〈cop 4547 ‘cfv 6380 (class class class)co 7213 0cc0 10729 ℕcn 11830 ℕ0cn0 12090 ℤcz 12176 ℝ+crp 12586 mod cmo 13442 ♯chash 13896 Word cword 14069 ++ cconcat 14125 substr csubstr 14205 prefix cpfx 14235 cyclShift ccsh 14353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-inf 9059 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-fz 13096 df-fzo 13239 df-fl 13367 df-mod 13443 df-hash 13897 df-word 14070 df-concat 14126 df-substr 14206 df-pfx 14236 df-csh 14354 |
This theorem is referenced by: cshwn 14362 2cshwcshw 14390 scshwfzeqfzo 14391 cshwrepswhash1 16656 crctcshlem4 27904 clwwisshclwws 28098 erclwwlkref 28103 erclwwlknref 28152 1cshid 30951 |
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