Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cshwshashnsame | Structured version Visualization version GIF version |
Description: If a word (not consisting of identical symbols) 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. (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 |
---|---|
cshwshashnsame | ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → (∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) → (#‘𝑀) = (#‘𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cshwrepswhash1.m | . . . . . 6 ⊢ 𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} | |
2 | 1 | cshwsiun 15644 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → 𝑀 = ∪ 𝑛 ∈ (0..^(#‘𝑊)){(𝑊 cyclShift 𝑛)}) |
3 | 2 | ad2antrr 758 | . . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → 𝑀 = ∪ 𝑛 ∈ (0..^(#‘𝑊)){(𝑊 cyclShift 𝑛)}) |
4 | 3 | fveq2d 6107 | . . 3 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → (#‘𝑀) = (#‘∪ 𝑛 ∈ (0..^(#‘𝑊)){(𝑊 cyclShift 𝑛)})) |
5 | fzofi 12635 | . . . . 5 ⊢ (0..^(#‘𝑊)) ∈ Fin | |
6 | 5 | a1i 11 | . . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → (0..^(#‘𝑊)) ∈ Fin) |
7 | snfi 7923 | . . . . 5 ⊢ {(𝑊 cyclShift 𝑛)} ∈ Fin | |
8 | 7 | a1i 11 | . . . 4 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) ∧ 𝑛 ∈ (0..^(#‘𝑊))) → {(𝑊 cyclShift 𝑛)} ∈ Fin) |
9 | id 22 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ)) | |
10 | 9 | cshwsdisj 15643 | . . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → Disj 𝑛 ∈ (0..^(#‘𝑊)){(𝑊 cyclShift 𝑛)}) |
11 | 6, 8, 10 | hashiun 14395 | . . 3 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → (#‘∪ 𝑛 ∈ (0..^(#‘𝑊)){(𝑊 cyclShift 𝑛)}) = Σ𝑛 ∈ (0..^(#‘𝑊))(#‘{(𝑊 cyclShift 𝑛)})) |
12 | ovex 6577 | . . . . . 6 ⊢ (𝑊 cyclShift 𝑛) ∈ V | |
13 | hashsng 13020 | . . . . . 6 ⊢ ((𝑊 cyclShift 𝑛) ∈ V → (#‘{(𝑊 cyclShift 𝑛)}) = 1) | |
14 | 12, 13 | mp1i 13 | . . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → (#‘{(𝑊 cyclShift 𝑛)}) = 1) |
15 | 14 | sumeq2sdv 14282 | . . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → Σ𝑛 ∈ (0..^(#‘𝑊))(#‘{(𝑊 cyclShift 𝑛)}) = Σ𝑛 ∈ (0..^(#‘𝑊))1) |
16 | 1cnd 9935 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → 1 ∈ ℂ) | |
17 | fsumconst 14364 | . . . . . . 7 ⊢ (((0..^(#‘𝑊)) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑛 ∈ (0..^(#‘𝑊))1 = ((#‘(0..^(#‘𝑊))) · 1)) | |
18 | 5, 16, 17 | sylancr 694 | . . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → Σ𝑛 ∈ (0..^(#‘𝑊))1 = ((#‘(0..^(#‘𝑊))) · 1)) |
19 | lencl 13179 | . . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ0) | |
20 | 19 | adantr 480 | . . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → (#‘𝑊) ∈ ℕ0) |
21 | hashfzo0 13077 | . . . . . . . 8 ⊢ ((#‘𝑊) ∈ ℕ0 → (#‘(0..^(#‘𝑊))) = (#‘𝑊)) | |
22 | 20, 21 | syl 17 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → (#‘(0..^(#‘𝑊))) = (#‘𝑊)) |
23 | 22 | oveq1d 6564 | . . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → ((#‘(0..^(#‘𝑊))) · 1) = ((#‘𝑊) · 1)) |
24 | prmnn 15226 | . . . . . . . . 9 ⊢ ((#‘𝑊) ∈ ℙ → (#‘𝑊) ∈ ℕ) | |
25 | 24 | nnred 10912 | . . . . . . . 8 ⊢ ((#‘𝑊) ∈ ℙ → (#‘𝑊) ∈ ℝ) |
26 | 25 | adantl 481 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → (#‘𝑊) ∈ ℝ) |
27 | ax-1rid 9885 | . . . . . . 7 ⊢ ((#‘𝑊) ∈ ℝ → ((#‘𝑊) · 1) = (#‘𝑊)) | |
28 | 26, 27 | syl 17 | . . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → ((#‘𝑊) · 1) = (#‘𝑊)) |
29 | 18, 23, 28 | 3eqtrd 2648 | . . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → Σ𝑛 ∈ (0..^(#‘𝑊))1 = (#‘𝑊)) |
30 | 29 | adantr 480 | . . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → Σ𝑛 ∈ (0..^(#‘𝑊))1 = (#‘𝑊)) |
31 | 15, 30 | eqtrd 2644 | . . 3 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → Σ𝑛 ∈ (0..^(#‘𝑊))(#‘{(𝑊 cyclShift 𝑛)}) = (#‘𝑊)) |
32 | 4, 11, 31 | 3eqtrd 2648 | . 2 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → (#‘𝑀) = (#‘𝑊)) |
33 | 32 | ex 449 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → (∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) → (#‘𝑀) = (#‘𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 {crab 2900 Vcvv 3173 {csn 4125 ∪ ciun 4455 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 · cmul 9820 ℕ0cn0 11169 ..^cfzo 12334 #chash 12979 Word cword 13146 cyclShift ccsh 13385 Σcsu 14264 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: cshwshash 15649 usghashecclwwlk 26362 umgrhashecclwwlk 41262 |
Copyright terms: Public domain | W3C validator |