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Mirrors > Home > MPE Home > Th. List > cshwshash | Structured version Visualization version GIF version |
Description: If a word has a length being a prime number, the size of the set of (different!) words resulting by cyclically shifting the original word equals the length of the original word or 1. (Contributed by AV, 19-May-2018.) (Revised by AV, 10-Nov-2018.) |
Ref | Expression |
---|---|
cshwrepswhash1.m | ⊢ 𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} |
Ref | Expression |
---|---|
cshwshash | ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → ((#‘𝑀) = (#‘𝑊) ∨ (#‘𝑀) = 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | repswsymballbi 13378 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))) | |
2 | 1 | adantr 480 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))) |
3 | prmnn 15226 | . . . . . . . . 9 ⊢ ((#‘𝑊) ∈ ℙ → (#‘𝑊) ∈ ℕ) | |
4 | 3 | nnge1d 10940 | . . . . . . . 8 ⊢ ((#‘𝑊) ∈ ℙ → 1 ≤ (#‘𝑊)) |
5 | wrdsymb1 13197 | . . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑊)) → (𝑊‘0) ∈ 𝑉) | |
6 | 4, 5 | sylan2 490 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → (𝑊‘0) ∈ 𝑉) |
7 | 6 | adantr 480 | . . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ 𝑊 = ((𝑊‘0) repeatS (#‘𝑊))) → (𝑊‘0) ∈ 𝑉) |
8 | 3 | ad2antlr 759 | . . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ 𝑊 = ((𝑊‘0) repeatS (#‘𝑊))) → (#‘𝑊) ∈ ℕ) |
9 | simpr 476 | . . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ 𝑊 = ((𝑊‘0) repeatS (#‘𝑊))) → 𝑊 = ((𝑊‘0) repeatS (#‘𝑊))) | |
10 | cshwrepswhash1.m | . . . . . . 7 ⊢ 𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} | |
11 | 10 | cshwrepswhash1 15647 | . . . . . 6 ⊢ (((𝑊‘0) ∈ 𝑉 ∧ (#‘𝑊) ∈ ℕ ∧ 𝑊 = ((𝑊‘0) repeatS (#‘𝑊))) → (#‘𝑀) = 1) |
12 | 7, 8, 9, 11 | syl3anc 1318 | . . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ 𝑊 = ((𝑊‘0) repeatS (#‘𝑊))) → (#‘𝑀) = 1) |
13 | 12 | ex 449 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) → (#‘𝑀) = 1)) |
14 | 2, 13 | sylbird 249 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → (∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0) → (#‘𝑀) = 1)) |
15 | olc 398 | . . 3 ⊢ ((#‘𝑀) = 1 → ((#‘𝑀) = (#‘𝑊) ∨ (#‘𝑀) = 1)) | |
16 | 14, 15 | syl6com 36 | . 2 ⊢ (∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → ((#‘𝑀) = (#‘𝑊) ∨ (#‘𝑀) = 1))) |
17 | rexnal 2978 | . . . 4 ⊢ (∃𝑖 ∈ (0..^(#‘𝑊)) ¬ (𝑊‘𝑖) = (𝑊‘0) ↔ ¬ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)) | |
18 | df-ne 2782 | . . . . . 6 ⊢ ((𝑊‘𝑖) ≠ (𝑊‘0) ↔ ¬ (𝑊‘𝑖) = (𝑊‘0)) | |
19 | 18 | bicomi 213 | . . . . 5 ⊢ (¬ (𝑊‘𝑖) = (𝑊‘0) ↔ (𝑊‘𝑖) ≠ (𝑊‘0)) |
20 | 19 | rexbii 3023 | . . . 4 ⊢ (∃𝑖 ∈ (0..^(#‘𝑊)) ¬ (𝑊‘𝑖) = (𝑊‘0) ↔ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) |
21 | 17, 20 | bitr3i 265 | . . 3 ⊢ (¬ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0) ↔ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) |
22 | 10 | cshwshashnsame 15648 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → (∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) → (#‘𝑀) = (#‘𝑊))) |
23 | orc 399 | . . . 4 ⊢ ((#‘𝑀) = (#‘𝑊) → ((#‘𝑀) = (#‘𝑊) ∨ (#‘𝑀) = 1)) | |
24 | 22, 23 | syl6com 36 | . . 3 ⊢ (∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → ((#‘𝑀) = (#‘𝑊) ∨ (#‘𝑀) = 1))) |
25 | 21, 24 | sylbi 206 | . 2 ⊢ (¬ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → ((#‘𝑀) = (#‘𝑊) ∨ (#‘𝑀) = 1))) |
26 | 16, 25 | pm2.61i 175 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → ((#‘𝑀) = (#‘𝑊) ∨ (#‘𝑀) = 1)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 ≤ cle 9954 ℕcn 10897 ..^cfzo 12334 #chash 12979 Word cword 13146 repeatS creps 13153 cyclShift ccsh 13385 ℙcprime 15223 |
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-inf2 8421 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-fal 1481 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-disj 4554 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-se 4998 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-isom 5813 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-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 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-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-hash 12980 df-word 13154 df-concat 13156 df-substr 13158 df-reps 13161 df-csh 13386 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-dvds 14822 df-gcd 15055 df-prm 15224 df-phi 15309 |
This theorem is referenced by: hashecclwwlkn1 26361 hashecclwwlksn1 41261 |
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