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Mirrors > Home > MPE Home > Th. List > cshwidxm | Structured version Visualization version GIF version |
Description: The symbol at index (n-N) of a word of length n (not 0) cyclically shifted by N positions (not 0) is the symbol at index 0 of the original word. (Contributed by AV, 18-May-2018.) (Revised by AV, 21-May-2018.) (Revised by AV, 30-Oct-2018.) |
Ref | Expression |
---|---|
cshwidxm | ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(♯‘𝑊))) → ((𝑊 cyclShift 𝑁)‘((♯‘𝑊) − 𝑁)) = (𝑊‘0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 481 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(♯‘𝑊))) → 𝑊 ∈ Word 𝑉) | |
2 | elfzelz 13505 | . . . 4 ⊢ (𝑁 ∈ (1...(♯‘𝑊)) → 𝑁 ∈ ℤ) | |
3 | 2 | adantl 480 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(♯‘𝑊))) → 𝑁 ∈ ℤ) |
4 | ubmelfzo 13701 | . . . 4 ⊢ (𝑁 ∈ (1...(♯‘𝑊)) → ((♯‘𝑊) − 𝑁) ∈ (0..^(♯‘𝑊))) | |
5 | 4 | adantl 480 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(♯‘𝑊))) → ((♯‘𝑊) − 𝑁) ∈ (0..^(♯‘𝑊))) |
6 | cshwidxmod 14757 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ ∧ ((♯‘𝑊) − 𝑁) ∈ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 𝑁)‘((♯‘𝑊) − 𝑁)) = (𝑊‘((((♯‘𝑊) − 𝑁) + 𝑁) mod (♯‘𝑊)))) | |
7 | 1, 3, 5, 6 | syl3anc 1369 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(♯‘𝑊))) → ((𝑊 cyclShift 𝑁)‘((♯‘𝑊) − 𝑁)) = (𝑊‘((((♯‘𝑊) − 𝑁) + 𝑁) mod (♯‘𝑊)))) |
8 | elfz1b 13574 | . . . . . . . 8 ⊢ (𝑁 ∈ (1...(♯‘𝑊)) ↔ (𝑁 ∈ ℕ ∧ (♯‘𝑊) ∈ ℕ ∧ 𝑁 ≤ (♯‘𝑊))) | |
9 | nncn 12224 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
10 | nncn 12224 | . . . . . . . . . 10 ⊢ ((♯‘𝑊) ∈ ℕ → (♯‘𝑊) ∈ ℂ) | |
11 | 9, 10 | anim12ci 612 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (♯‘𝑊) ∈ ℕ) → ((♯‘𝑊) ∈ ℂ ∧ 𝑁 ∈ ℂ)) |
12 | 11 | 3adant3 1130 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (♯‘𝑊) ∈ ℕ ∧ 𝑁 ≤ (♯‘𝑊)) → ((♯‘𝑊) ∈ ℂ ∧ 𝑁 ∈ ℂ)) |
13 | 8, 12 | sylbi 216 | . . . . . . 7 ⊢ (𝑁 ∈ (1...(♯‘𝑊)) → ((♯‘𝑊) ∈ ℂ ∧ 𝑁 ∈ ℂ)) |
14 | npcan 11473 | . . . . . . 7 ⊢ (((♯‘𝑊) ∈ ℂ ∧ 𝑁 ∈ ℂ) → (((♯‘𝑊) − 𝑁) + 𝑁) = (♯‘𝑊)) | |
15 | 13, 14 | syl 17 | . . . . . 6 ⊢ (𝑁 ∈ (1...(♯‘𝑊)) → (((♯‘𝑊) − 𝑁) + 𝑁) = (♯‘𝑊)) |
16 | 15 | oveq1d 7426 | . . . . 5 ⊢ (𝑁 ∈ (1...(♯‘𝑊)) → ((((♯‘𝑊) − 𝑁) + 𝑁) mod (♯‘𝑊)) = ((♯‘𝑊) mod (♯‘𝑊))) |
17 | 16 | adantl 480 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(♯‘𝑊))) → ((((♯‘𝑊) − 𝑁) + 𝑁) mod (♯‘𝑊)) = ((♯‘𝑊) mod (♯‘𝑊))) |
18 | nnrp 12989 | . . . . . . . 8 ⊢ ((♯‘𝑊) ∈ ℕ → (♯‘𝑊) ∈ ℝ+) | |
19 | modid0 13866 | . . . . . . . 8 ⊢ ((♯‘𝑊) ∈ ℝ+ → ((♯‘𝑊) mod (♯‘𝑊)) = 0) | |
20 | 18, 19 | syl 17 | . . . . . . 7 ⊢ ((♯‘𝑊) ∈ ℕ → ((♯‘𝑊) mod (♯‘𝑊)) = 0) |
21 | 20 | 3ad2ant2 1132 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (♯‘𝑊) ∈ ℕ ∧ 𝑁 ≤ (♯‘𝑊)) → ((♯‘𝑊) mod (♯‘𝑊)) = 0) |
22 | 8, 21 | sylbi 216 | . . . . 5 ⊢ (𝑁 ∈ (1...(♯‘𝑊)) → ((♯‘𝑊) mod (♯‘𝑊)) = 0) |
23 | 22 | adantl 480 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(♯‘𝑊))) → ((♯‘𝑊) mod (♯‘𝑊)) = 0) |
24 | 17, 23 | eqtrd 2770 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(♯‘𝑊))) → ((((♯‘𝑊) − 𝑁) + 𝑁) mod (♯‘𝑊)) = 0) |
25 | 24 | fveq2d 6894 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(♯‘𝑊))) → (𝑊‘((((♯‘𝑊) − 𝑁) + 𝑁) mod (♯‘𝑊))) = (𝑊‘0)) |
26 | 7, 25 | eqtrd 2770 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(♯‘𝑊))) → ((𝑊 cyclShift 𝑁)‘((♯‘𝑊) − 𝑁)) = (𝑊‘0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 class class class wbr 5147 ‘cfv 6542 (class class class)co 7411 ℂcc 11110 0cc0 11112 1c1 11113 + caddc 11115 ≤ cle 11253 − cmin 11448 ℕcn 12216 ℤcz 12562 ℝ+crp 12978 ...cfz 13488 ..^cfzo 13631 mod cmo 13838 ♯chash 14294 Word cword 14468 cyclShift ccsh 14742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-fz 13489 df-fzo 13632 df-fl 13761 df-mod 13839 df-hash 14295 df-word 14469 df-concat 14525 df-substr 14595 df-pfx 14625 df-csh 14743 |
This theorem is referenced by: (None) |
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