Theorem cshwsidrepsw 15800

Description: If cyclically shifting a word of length being a prime number by a number of positions which is not divisible by the prime number results in the word itself, the word is a "repeated symbol word". (Contributed by AV, 18-May-2018.) (Revised by AV, 10-Nov-2018.)

Assertion

Ref	Expression
cshwsidrepsw	⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → ((𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊) → 𝑊 = ((𝑊‘0) repeatS (#‘𝑊))))

Proof of Theorem cshwsidrepsw

Dummy variables 𝑖 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → (#‘𝑊) ∈ ℙ)
2	1	adantr 481	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (#‘𝑊) ∈ ℙ)
3		simp1 1061	. . . . . . . . 9 ⊢ ((𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊) → 𝐿 ∈ ℤ)
4	3	adantl 482	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → 𝐿 ∈ ℤ)
5		simpr2 1068	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (𝐿 mod (#‘𝑊)) ≠ 0)
6	2, 4, 5	3jca 1242	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → ((#‘𝑊) ∈ ℙ ∧ 𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0))
7	6	adantr 481	. . . . . 6 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → ((#‘𝑊) ∈ ℙ ∧ 𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0))
8		simpr 477	. . . . . 6 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → 𝑖 ∈ (0..^(#‘𝑊)))
9		modprmn0modprm0 15512	. . . . . 6 ⊢ (((#‘𝑊) ∈ ℙ ∧ 𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0) → (𝑖 ∈ (0..^(#‘𝑊)) → ∃𝑗 ∈ (0..^(#‘𝑊))((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊)) = 0))
10	7, 8, 9	sylc 65	. . . . 5 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → ∃𝑗 ∈ (0..^(#‘𝑊))((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊)) = 0)
11		elfzonn0 12512	. . . . . . . . . 10 ⊢ (𝑗 ∈ (0..^(#‘𝑊)) → 𝑗 ∈ ℕ0)
12	11	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝑗 ∈ (0..^(#‘𝑊)) ∧ ((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊)) = 0) ∧ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(#‘𝑊)))) → 𝑗 ∈ ℕ0)
13		simpl 473	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → 𝑊 ∈ Word 𝑉)
14	13, 3	anim12i 590	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ))
15	14	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ))
16	15	adantl 482	. . . . . . . . . 10 ⊢ (((𝑗 ∈ (0..^(#‘𝑊)) ∧ ((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊)) = 0) ∧ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(#‘𝑊)))) → (𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ))
17		simpr3 1069	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (𝑊 cyclShift 𝐿) = 𝑊)
18	17	anim1i 592	. . . . . . . . . . 11 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝑖 ∈ (0..^(#‘𝑊))))
19	18	adantl 482	. . . . . . . . . 10 ⊢ (((𝑗 ∈ (0..^(#‘𝑊)) ∧ ((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊)) = 0) ∧ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(#‘𝑊)))) → ((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝑖 ∈ (0..^(#‘𝑊))))
20		cshweqrep 13567	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) → (((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝑖 ∈ (0..^(#‘𝑊))) → ∀𝑘 ∈ ℕ0 (𝑊‘𝑖) = (𝑊‘((𝑖 + (𝑘 · 𝐿)) mod (#‘𝑊)))))
21	16, 19, 20	sylc 65	. . . . . . . . 9 ⊢ (((𝑗 ∈ (0..^(#‘𝑊)) ∧ ((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊)) = 0) ∧ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(#‘𝑊)))) → ∀𝑘 ∈ ℕ0 (𝑊‘𝑖) = (𝑊‘((𝑖 + (𝑘 · 𝐿)) mod (#‘𝑊))))
22		oveq1 6657	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (𝑘 · 𝐿) = (𝑗 · 𝐿))
23	22	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑖 + (𝑘 · 𝐿)) = (𝑖 + (𝑗 · 𝐿)))
24	23	oveq1d 6665	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → ((𝑖 + (𝑘 · 𝐿)) mod (#‘𝑊)) = ((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊)))
25	24	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → (𝑊‘((𝑖 + (𝑘 · 𝐿)) mod (#‘𝑊))) = (𝑊‘((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊))))
26	25	eqeq2d 2632	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → ((𝑊‘𝑖) = (𝑊‘((𝑖 + (𝑘 · 𝐿)) mod (#‘𝑊))) ↔ (𝑊‘𝑖) = (𝑊‘((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊)))))
27	26	rspcva 3307	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 (𝑊‘𝑖) = (𝑊‘((𝑖 + (𝑘 · 𝐿)) mod (#‘𝑊)))) → (𝑊‘𝑖) = (𝑊‘((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊))))
28	12, 21, 27	syl2anc 693	. . . . . . . 8 ⊢ (((𝑗 ∈ (0..^(#‘𝑊)) ∧ ((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊)) = 0) ∧ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(#‘𝑊)))) → (𝑊‘𝑖) = (𝑊‘((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊))))
29		fveq2 6191	. . . . . . . . . 10 ⊢ (((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊)) = 0 → (𝑊‘((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊))) = (𝑊‘0))
30	29	adantl 482	. . . . . . . . 9 ⊢ ((𝑗 ∈ (0..^(#‘𝑊)) ∧ ((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊)) = 0) → (𝑊‘((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊))) = (𝑊‘0))
31	30	adantr 481	. . . . . . . 8 ⊢ (((𝑗 ∈ (0..^(#‘𝑊)) ∧ ((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊)) = 0) ∧ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(#‘𝑊)))) → (𝑊‘((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊))) = (𝑊‘0))
32	28, 31	eqtrd 2656	. . . . . . 7 ⊢ (((𝑗 ∈ (0..^(#‘𝑊)) ∧ ((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊)) = 0) ∧ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(#‘𝑊)))) → (𝑊‘𝑖) = (𝑊‘0))
33	32	ex 450	. . . . . 6 ⊢ ((𝑗 ∈ (0..^(#‘𝑊)) ∧ ((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊)) = 0) → ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (𝑊‘𝑖) = (𝑊‘0)))
34	33	rexlimiva 3028	. . . . 5 ⊢ (∃𝑗 ∈ (0..^(#‘𝑊))((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊)) = 0 → ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (𝑊‘𝑖) = (𝑊‘0)))
35	10, 34	mpcom 38	. . . 4 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → (𝑊‘𝑖) = (𝑊‘0))
36	35	ralrimiva 2966	. . 3 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
37		repswsymballbi 13527	. . . 4 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
38	37	ad2antrr 762	. . 3 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
39	36, 38	mpbird 247	. 2 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → 𝑊 = ((𝑊‘0) repeatS (#‘𝑊)))
40	39	ex 450	1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → ((𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊) → 𝑊 = ((𝑊‘0) repeatS (#‘𝑊))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  ‘cfv 5888  (class class class)co 6650  0cc0 9936   + caddc 9939   · cmul 9941  ℕ₀cn0 11292  ℤcz 11377  ..^cfzo 12465   mod cmo 12668  #chash 13117  Word cword 13291   repeatS creps 13298   cyclShift ccsh 13534  ℙcprime 15385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-concat 13301  df-substr 13303  df-reps 13306  df-csh 13535  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-phi 15471

This theorem is referenced by:  cshwsidrepswmod0  15801