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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 13539 | . . . 4 ⊢ (∅ cyclShift 0) = ∅ | |
2 | oveq1 6657 | . . . 4 ⊢ (∅ = 𝑊 → (∅ cyclShift 0) = (𝑊 cyclShift 0)) | |
3 | id 22 | . . . 4 ⊢ (∅ = 𝑊 → ∅ = 𝑊) | |
4 | 1, 2, 3 | 3eqtr3a 2680 | . . 3 ⊢ (∅ = 𝑊 → (𝑊 cyclShift 0) = 𝑊) |
5 | 4 | a1d 25 | . 2 ⊢ (∅ = 𝑊 → (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊)) |
6 | 0z 11388 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
7 | cshword 13537 | . . . . . . 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 481 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 cyclShift 0) = ((𝑊 substr 〈(0 mod (#‘𝑊)), (#‘𝑊)〉) ++ (𝑊 substr 〈0, (0 mod (#‘𝑊))〉))) |
10 | necom 2847 | . . . . . 6 ⊢ (∅ ≠ 𝑊 ↔ 𝑊 ≠ ∅) | |
11 | lennncl 13325 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (#‘𝑊) ∈ ℕ) | |
12 | nnrp 11842 | . . . . . . 7 ⊢ ((#‘𝑊) ∈ ℕ → (#‘𝑊) ∈ ℝ+) | |
13 | 0mod 12701 | . . . . . . . . . 10 ⊢ ((#‘𝑊) ∈ ℝ+ → (0 mod (#‘𝑊)) = 0) | |
14 | 13 | opeq1d 4408 | . . . . . . . . 9 ⊢ ((#‘𝑊) ∈ ℝ+ → 〈(0 mod (#‘𝑊)), (#‘𝑊)〉 = 〈0, (#‘𝑊)〉) |
15 | 14 | oveq2d 6666 | . . . . . . . 8 ⊢ ((#‘𝑊) ∈ ℝ+ → (𝑊 substr 〈(0 mod (#‘𝑊)), (#‘𝑊)〉) = (𝑊 substr 〈0, (#‘𝑊)〉)) |
16 | 13 | opeq2d 4409 | . . . . . . . . 9 ⊢ ((#‘𝑊) ∈ ℝ+ → 〈0, (0 mod (#‘𝑊))〉 = 〈0, 0〉) |
17 | 16 | oveq2d 6666 | . . . . . . . 8 ⊢ ((#‘𝑊) ∈ ℝ+ → (𝑊 substr 〈0, (0 mod (#‘𝑊))〉) = (𝑊 substr 〈0, 0〉)) |
18 | 15, 17 | oveq12d 6668 | . . . . . . 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 492 | . . . . 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 2656 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 cyclShift 0) = ((𝑊 substr 〈0, (#‘𝑊)〉) ++ (𝑊 substr 〈0, 0〉))) |
22 | swrdid 13428 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 substr 〈0, (#‘𝑊)〉) = 𝑊) | |
23 | 22 | adantr 481 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 substr 〈0, (#‘𝑊)〉) = 𝑊) |
24 | swrd00 13418 | . . . . . 6 ⊢ (𝑊 substr 〈0, 0〉) = ∅ | |
25 | 24 | a1i 11 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 substr 〈0, 0〉) = ∅) |
26 | 23, 25 | oveq12d 6668 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → ((𝑊 substr 〈0, (#‘𝑊)〉) ++ (𝑊 substr 〈0, 0〉)) = (𝑊 ++ ∅)) |
27 | ccatrid 13370 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 ++ ∅) = 𝑊) | |
28 | 27 | adantr 481 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 ++ ∅) = 𝑊) |
29 | 21, 26, 28 | 3eqtrd 2660 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 cyclShift 0) = 𝑊) |
30 | 29 | expcom 451 | . 2 ⊢ (∅ ≠ 𝑊 → (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊)) |
31 | 5, 30 | pm2.61ine 2877 | 1 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∅c0 3915 〈cop 4183 ‘cfv 5888 (class class class)co 6650 0cc0 9936 ℕcn 11020 ℤcz 11377 ℝ+crp 11832 mod cmo 12668 #chash 13117 Word cword 13291 ++ cconcat 13293 substr csubstr 13295 cyclShift ccsh 13534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-hash 13118 df-word 13299 df-concat 13301 df-substr 13303 df-csh 13535 |
This theorem is referenced by: cshwn 13543 2cshwcshw 13571 scshwfzeqfzo 13572 cshwrepswhash1 15809 crctcshlem4 26712 clwwisshclwws 26928 erclwwlksref 26934 erclwwlksnref 26946 |
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