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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 13390 | . . . 4 ⊢ (∅ cyclShift 0) = ∅ | |
2 | oveq1 6556 | . . . 4 ⊢ (∅ = 𝑊 → (∅ cyclShift 0) = (𝑊 cyclShift 0)) | |
3 | id 22 | . . . 4 ⊢ (∅ = 𝑊 → ∅ = 𝑊) | |
4 | 1, 2, 3 | 3eqtr3a 2668 | . . 3 ⊢ (∅ = 𝑊 → (𝑊 cyclShift 0) = 𝑊) |
5 | 4 | a1d 25 | . 2 ⊢ (∅ = 𝑊 → (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊)) |
6 | 0z 11265 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
7 | cshword 13388 | . . . . . . 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 703 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = ((𝑊 substr 〈(0 mod (#‘𝑊)), (#‘𝑊)〉) ++ (𝑊 substr 〈0, (0 mod (#‘𝑊))〉))) |
9 | 8 | adantr 480 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 cyclShift 0) = ((𝑊 substr 〈(0 mod (#‘𝑊)), (#‘𝑊)〉) ++ (𝑊 substr 〈0, (0 mod (#‘𝑊))〉))) |
10 | necom 2835 | . . . . . 6 ⊢ (∅ ≠ 𝑊 ↔ 𝑊 ≠ ∅) | |
11 | lennncl 13180 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (#‘𝑊) ∈ ℕ) | |
12 | nnrp 11718 | . . . . . . 7 ⊢ ((#‘𝑊) ∈ ℕ → (#‘𝑊) ∈ ℝ+) | |
13 | 0mod 12563 | . . . . . . . . . 10 ⊢ ((#‘𝑊) ∈ ℝ+ → (0 mod (#‘𝑊)) = 0) | |
14 | 13 | opeq1d 4346 | . . . . . . . . 9 ⊢ ((#‘𝑊) ∈ ℝ+ → 〈(0 mod (#‘𝑊)), (#‘𝑊)〉 = 〈0, (#‘𝑊)〉) |
15 | 14 | oveq2d 6565 | . . . . . . . 8 ⊢ ((#‘𝑊) ∈ ℝ+ → (𝑊 substr 〈(0 mod (#‘𝑊)), (#‘𝑊)〉) = (𝑊 substr 〈0, (#‘𝑊)〉)) |
16 | 13 | opeq2d 4347 | . . . . . . . . 9 ⊢ ((#‘𝑊) ∈ ℝ+ → 〈0, (0 mod (#‘𝑊))〉 = 〈0, 0〉) |
17 | 16 | oveq2d 6565 | . . . . . . . 8 ⊢ ((#‘𝑊) ∈ ℝ+ → (𝑊 substr 〈0, (0 mod (#‘𝑊))〉) = (𝑊 substr 〈0, 0〉)) |
18 | 15, 17 | oveq12d 6567 | . . . . . . 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 491 | . . . . 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 2644 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 cyclShift 0) = ((𝑊 substr 〈0, (#‘𝑊)〉) ++ (𝑊 substr 〈0, 0〉))) |
22 | swrdid 13280 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 substr 〈0, (#‘𝑊)〉) = 𝑊) | |
23 | 22 | adantr 480 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 substr 〈0, (#‘𝑊)〉) = 𝑊) |
24 | swrd00 13270 | . . . . . 6 ⊢ (𝑊 substr 〈0, 0〉) = ∅ | |
25 | 24 | a1i 11 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 substr 〈0, 0〉) = ∅) |
26 | 23, 25 | oveq12d 6567 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → ((𝑊 substr 〈0, (#‘𝑊)〉) ++ (𝑊 substr 〈0, 0〉)) = (𝑊 ++ ∅)) |
27 | ccatrid 13223 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 ++ ∅) = 𝑊) | |
28 | 27 | adantr 480 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 ++ ∅) = 𝑊) |
29 | 21, 26, 28 | 3eqtrd 2648 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∅ ≠ 𝑊) → (𝑊 cyclShift 0) = 𝑊) |
30 | 29 | expcom 450 | . 2 ⊢ (∅ ≠ 𝑊 → (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊)) |
31 | 5, 30 | pm2.61ine 2865 | 1 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 〈cop 4131 ‘cfv 5804 (class class class)co 6549 0cc0 9815 ℕcn 10897 ℤcz 11254 ℝ+crp 11708 mod cmo 12530 #chash 12979 Word cword 13146 ++ cconcat 13148 substr csubstr 13150 cyclShift ccsh 13385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-hash 12980 df-word 13154 df-concat 13156 df-substr 13158 df-csh 13386 |
This theorem is referenced by: cshwn 13394 2cshwcshw 13422 scshwfzeqfzo 13423 cshwrepswhash1 15647 clwwisshclww 26335 erclwwlkref 26341 erclwwlknref 26353 crctcshlem4 41023 clwwisshclwws 41235 erclwwlksref 41241 erclwwlksnref 41253 |
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