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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 472 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(#‘𝑊))) → 𝑊 ∈ Word 𝑉) | |
2 | elfzelz 12213 | . . . 4 ⊢ (𝑁 ∈ (1...(#‘𝑊)) → 𝑁 ∈ ℤ) | |
3 | 2 | adantl 481 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(#‘𝑊))) → 𝑁 ∈ ℤ) |
4 | ubmelfzo 12400 | . . . 4 ⊢ (𝑁 ∈ (1...(#‘𝑊)) → ((#‘𝑊) − 𝑁) ∈ (0..^(#‘𝑊))) | |
5 | 4 | adantl 481 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(#‘𝑊))) → ((#‘𝑊) − 𝑁) ∈ (0..^(#‘𝑊))) |
6 | cshwidxmod 13400 | . . 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 1318 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(#‘𝑊))) → ((𝑊 cyclShift 𝑁)‘((#‘𝑊) − 𝑁)) = (𝑊‘((((#‘𝑊) − 𝑁) + 𝑁) mod (#‘𝑊)))) |
8 | elfz1b 12279 | . . . . . . . 8 ⊢ (𝑁 ∈ (1...(#‘𝑊)) ↔ (𝑁 ∈ ℕ ∧ (#‘𝑊) ∈ ℕ ∧ 𝑁 ≤ (#‘𝑊))) | |
9 | nncn 10905 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
10 | nncn 10905 | . . . . . . . . . 10 ⊢ ((#‘𝑊) ∈ ℕ → (#‘𝑊) ∈ ℂ) | |
11 | 9, 10 | anim12ci 589 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (#‘𝑊) ∈ ℕ) → ((#‘𝑊) ∈ ℂ ∧ 𝑁 ∈ ℂ)) |
12 | 11 | 3adant3 1074 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (#‘𝑊) ∈ ℕ ∧ 𝑁 ≤ (#‘𝑊)) → ((#‘𝑊) ∈ ℂ ∧ 𝑁 ∈ ℂ)) |
13 | 8, 12 | sylbi 206 | . . . . . . 7 ⊢ (𝑁 ∈ (1...(#‘𝑊)) → ((#‘𝑊) ∈ ℂ ∧ 𝑁 ∈ ℂ)) |
14 | npcan 10169 | . . . . . . 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 6564 | . . . . 5 ⊢ (𝑁 ∈ (1...(#‘𝑊)) → ((((#‘𝑊) − 𝑁) + 𝑁) mod (#‘𝑊)) = ((#‘𝑊) mod (#‘𝑊))) |
17 | 16 | adantl 481 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(#‘𝑊))) → ((((#‘𝑊) − 𝑁) + 𝑁) mod (#‘𝑊)) = ((#‘𝑊) mod (#‘𝑊))) |
18 | nnrp 11718 | . . . . . . . 8 ⊢ ((#‘𝑊) ∈ ℕ → (#‘𝑊) ∈ ℝ+) | |
19 | modid0 12558 | . . . . . . . 8 ⊢ ((#‘𝑊) ∈ ℝ+ → ((#‘𝑊) mod (#‘𝑊)) = 0) | |
20 | 18, 19 | syl 17 | . . . . . . 7 ⊢ ((#‘𝑊) ∈ ℕ → ((#‘𝑊) mod (#‘𝑊)) = 0) |
21 | 20 | 3ad2ant2 1076 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (#‘𝑊) ∈ ℕ ∧ 𝑁 ≤ (#‘𝑊)) → ((#‘𝑊) mod (#‘𝑊)) = 0) |
22 | 8, 21 | sylbi 206 | . . . . 5 ⊢ (𝑁 ∈ (1...(#‘𝑊)) → ((#‘𝑊) mod (#‘𝑊)) = 0) |
23 | 22 | adantl 481 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(#‘𝑊))) → ((#‘𝑊) mod (#‘𝑊)) = 0) |
24 | 17, 23 | eqtrd 2644 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(#‘𝑊))) → ((((#‘𝑊) − 𝑁) + 𝑁) mod (#‘𝑊)) = 0) |
25 | 24 | fveq2d 6107 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(#‘𝑊))) → (𝑊‘((((#‘𝑊) − 𝑁) + 𝑁) mod (#‘𝑊))) = (𝑊‘0)) |
26 | 7, 25 | eqtrd 2644 | 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 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 ≤ cle 9954 − cmin 10145 ℕcn 10897 ℤcz 11254 ℝ+crp 11708 ...cfz 12197 ..^cfzo 12334 mod cmo 12530 #chash 12979 Word cword 13146 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-2 10956 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: (None) |
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