cshwshashnsame - Metamath Proof Explorer

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Theorem cshwshashnsame 17067

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.)

**Hypothesis**
Ref	Expression
cshwrepswhash1.m	⊢ 𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}

**Assertion**
Ref	Expression
cshwshashnsame	⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → (∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) → (♯‘𝑀) = (♯‘𝑊)))

Distinct variable groups: 𝑛,𝑉,𝑤 𝑛,𝑊,𝑤,𝑖 𝑖,𝑉 Allowed substitution hints: 𝑀(𝑤,𝑖,𝑛)

**Proof of Theorem cshwshashnsame**
Step	Hyp	Ref	Expression
1		cshwrepswhash1.m	. . . . . 6 ⊢ 𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}
2	1	cshwsiun 17063	. . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → 𝑀 = ∪ 𝑛 ∈ (0..^(♯‘𝑊)){(𝑊 cyclShift 𝑛)})
3	2	ad2antrr 724	. . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → 𝑀 = ∪ 𝑛 ∈ (0..^(♯‘𝑊)){(𝑊 cyclShift 𝑛)})
4	3	fveq2d 6894	. . 3 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → (♯‘𝑀) = (♯‘∪ 𝑛 ∈ (0..^(♯‘𝑊)){(𝑊 cyclShift 𝑛)}))
5		fzofi 13966	. . . . 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 9062	. . . . 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 17062	. . . 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 15795	. . 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 7446	. . . . . 6 ⊢ (𝑊 cyclShift 𝑛) ∈ V
13		hashsng 14355	. . . . . 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 15677	. . . 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 11234	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → 1 ∈ ℂ)
17		fsumconst 15763	. . . . . . 7 ⊢ (((0..^(♯‘𝑊)) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑛 ∈ (0..^(♯‘𝑊))1 = ((♯‘(0..^(♯‘𝑊))) · 1))
18	5, 16, 17	sylancr 585	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → Σ𝑛 ∈ (0..^(♯‘𝑊))1 = ((♯‘(0..^(♯‘𝑊))) · 1))
19		lencl 14510	. . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ₀)
20	19	adantr 479	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → (♯‘𝑊) ∈ ℕ₀)
21		hashfzo0 14416	. . . . . . . 8 ⊢ ((♯‘𝑊) ∈ ℕ₀ → (♯‘(0..^(♯‘𝑊))) = (♯‘𝑊))
22	20, 21	syl 17	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → (♯‘(0..^(♯‘𝑊))) = (♯‘𝑊))
23	22	oveq1d 7428	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → ((♯‘(0..^(♯‘𝑊))) · 1) = ((♯‘𝑊) · 1))
24		prmnn 16639	. . . . . . . . 9 ⊢ ((♯‘𝑊) ∈ ℙ → (♯‘𝑊) ∈ ℕ)
25	24	nnred 12252	. . . . . . . 8 ⊢ ((♯‘𝑊) ∈ ℙ → (♯‘𝑊) ∈ ℝ)
26	25	adantl 480	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → (♯‘𝑊) ∈ ℝ)
27		ax-1rid 11203	. . . . . . 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 2769	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → Σ𝑛 ∈ (0..^(♯‘𝑊))1 = (♯‘𝑊))
30	29	adantr 479	. . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → Σ𝑛 ∈ (0..^(♯‘𝑊))1 = (♯‘𝑊))
31	15, 30	eqtrd 2765	. . 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 2769	. 2 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → (♯‘𝑀) = (♯‘𝑊))
33	32	ex 411	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 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∃wrex 3060 {crab 3419 Vcvv 3463 {csn 4625 ∪ ciun 4992 ‘cfv 6543 (class class class)co 7413 Fincfn 8957 ℂcc 11131 ℝcr 11132 0cc0 11133 1c1 11134 · cmul 11138 ℕ₀cn0 12497 ..^cfzo 13654 ♯chash 14316 Word cword 14491 cyclShift ccsh 14765 Σcsu 15659 ℙcprime 16636

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-disj 5110 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-se 5629 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 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9460 df-inf 9461 df-oi 9528 df-dju 9919 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-xnn0 12570 df-z 12584 df-uz 12848 df-rp 13002 df-fz 13512 df-fzo 13655 df-fl 13784 df-mod 13862 df-seq 13994 df-exp 14054 df-hash 14317 df-word 14492 df-concat 14548 df-substr 14618 df-pfx 14648 df-reps 14746 df-csh 14766 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-clim 15459 df-sum 15660 df-dvds 16226 df-gcd 16464 df-prm 16637 df-phi 16729

This theorem is referenced by: cshwshash 17068 umgrhashecclwwlk 29927

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