Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cshwcl | Structured version Visualization version GIF version |
Description: A cyclically shifted word is a word over the same set as for the original word. (Contributed by AV, 16-May-2018.) (Revised by AV, 21-May-2018.) (Revised by AV, 27-Oct-2018.) |
Ref | Expression |
---|---|
cshwcl | ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 𝑁) ∈ Word 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cshword 14448 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = ((𝑊 substr 〈(𝑁 mod (♯‘𝑊)), (♯‘𝑊)〉) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊))))) | |
2 | swrdcl 14302 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 substr 〈(𝑁 mod (♯‘𝑊)), (♯‘𝑊)〉) ∈ Word 𝑉) | |
3 | pfxcl 14334 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 prefix (𝑁 mod (♯‘𝑊))) ∈ Word 𝑉) | |
4 | ccatcl 14221 | . . . . . 6 ⊢ (((𝑊 substr 〈(𝑁 mod (♯‘𝑊)), (♯‘𝑊)〉) ∈ Word 𝑉 ∧ (𝑊 prefix (𝑁 mod (♯‘𝑊))) ∈ Word 𝑉) → ((𝑊 substr 〈(𝑁 mod (♯‘𝑊)), (♯‘𝑊)〉) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊)))) ∈ Word 𝑉) | |
5 | 2, 3, 4 | syl2anc 583 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → ((𝑊 substr 〈(𝑁 mod (♯‘𝑊)), (♯‘𝑊)〉) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊)))) ∈ Word 𝑉) |
6 | 5 | adantr 480 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → ((𝑊 substr 〈(𝑁 mod (♯‘𝑊)), (♯‘𝑊)〉) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊)))) ∈ Word 𝑉) |
7 | 1, 6 | eqeltrd 2837 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) ∈ Word 𝑉) |
8 | 7 | expcom 413 | . 2 ⊢ (𝑁 ∈ ℤ → (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 𝑁) ∈ Word 𝑉)) |
9 | cshnz 14449 | . . . 4 ⊢ (¬ 𝑁 ∈ ℤ → (𝑊 cyclShift 𝑁) = ∅) | |
10 | wrd0 14186 | . . . 4 ⊢ ∅ ∈ Word 𝑉 | |
11 | 9, 10 | eqeltrdi 2845 | . . 3 ⊢ (¬ 𝑁 ∈ ℤ → (𝑊 cyclShift 𝑁) ∈ Word 𝑉) |
12 | 11 | a1d 25 | . 2 ⊢ (¬ 𝑁 ∈ ℤ → (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 𝑁) ∈ Word 𝑉)) |
13 | 8, 12 | pm2.61i 182 | 1 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 𝑁) ∈ Word 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2107 ∅c0 4258 〈cop 4569 ‘cfv 6423 (class class class)co 7260 ℤcz 12265 mod cmo 13533 ♯chash 13988 Word cword 14161 ++ cconcat 14217 substr csubstr 14297 prefix cpfx 14327 cyclShift ccsh 14445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5210 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7571 ax-cnex 10874 ax-resscn 10875 ax-1cn 10876 ax-icn 10877 ax-addcl 10878 ax-addrcl 10879 ax-mulcl 10880 ax-mulrcl 10881 ax-mulcom 10882 ax-addass 10883 ax-mulass 10884 ax-distr 10885 ax-i2m1 10886 ax-1ne0 10887 ax-1rid 10888 ax-rnegex 10889 ax-rrecex 10890 ax-cnre 10891 ax-pre-lttri 10892 ax-pre-lttrn 10893 ax-pre-ltadd 10894 ax-pre-mulgt0 10895 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3429 df-sbc 3717 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6259 df-on 6260 df-lim 6261 df-suc 6262 df-iota 6381 df-fun 6425 df-fn 6426 df-f 6427 df-f1 6428 df-fo 6429 df-f1o 6430 df-fv 6431 df-riota 7217 df-ov 7263 df-oprab 7264 df-mpo 7265 df-om 7693 df-1st 7809 df-2nd 7810 df-frecs 8073 df-wrecs 8104 df-recs 8178 df-rdg 8217 df-1o 8272 df-er 8461 df-en 8697 df-dom 8698 df-sdom 8699 df-fin 8700 df-card 9644 df-pnf 10958 df-mnf 10959 df-xr 10960 df-ltxr 10961 df-le 10962 df-sub 11153 df-neg 11154 df-nn 11920 df-n0 12180 df-z 12266 df-uz 12528 df-fz 13185 df-fzo 13328 df-hash 13989 df-word 14162 df-concat 14218 df-substr 14298 df-pfx 14328 df-csh 14446 |
This theorem is referenced by: cshwf 14457 2cshw 14470 3cshw 14475 cshwsiun 16745 crctcshwlkn0 28127 clwwisshclwws 28320 tocycfv 31318 |
Copyright terms: Public domain | W3C validator |