![]() |
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 14792 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = ((𝑊 substr 〈(𝑁 mod (♯‘𝑊)), (♯‘𝑊)〉) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊))))) | |
2 | swrdcl 14646 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 substr 〈(𝑁 mod (♯‘𝑊)), (♯‘𝑊)〉) ∈ Word 𝑉) | |
3 | pfxcl 14678 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 prefix (𝑁 mod (♯‘𝑊))) ∈ Word 𝑉) | |
4 | ccatcl 14575 | . . . . . 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 582 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → ((𝑊 substr 〈(𝑁 mod (♯‘𝑊)), (♯‘𝑊)〉) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊)))) ∈ Word 𝑉) |
6 | 5 | adantr 479 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → ((𝑊 substr 〈(𝑁 mod (♯‘𝑊)), (♯‘𝑊)〉) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊)))) ∈ Word 𝑉) |
7 | 1, 6 | eqeltrd 2826 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) ∈ Word 𝑉) |
8 | 7 | expcom 412 | . 2 ⊢ (𝑁 ∈ ℤ → (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 𝑁) ∈ Word 𝑉)) |
9 | cshnz 14793 | . . . 4 ⊢ (¬ 𝑁 ∈ ℤ → (𝑊 cyclShift 𝑁) = ∅) | |
10 | wrd0 14540 | . . . 4 ⊢ ∅ ∈ Word 𝑉 | |
11 | 9, 10 | eqeltrdi 2834 | . . 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 394 ∈ wcel 2099 ∅c0 4323 〈cop 4630 ‘cfv 6544 (class class class)co 7414 ℤcz 12602 mod cmo 13881 ♯chash 14340 Word cword 14515 ++ cconcat 14571 substr csubstr 14641 prefix cpfx 14671 cyclShift ccsh 14789 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-int 4948 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 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 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9973 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-nn 12257 df-n0 12517 df-z 12603 df-uz 12867 df-fz 13531 df-fzo 13674 df-hash 14341 df-word 14516 df-concat 14572 df-substr 14642 df-pfx 14672 df-csh 14790 |
This theorem is referenced by: cshwf 14801 2cshw 14814 3cshw 14819 cshwsiun 17095 crctcshwlkn0 29750 clwwisshclwws 29943 tocycfv 32989 |
Copyright terms: Public domain | W3C validator |