Theorem cshwsidrepsw 15638

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 476	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → (#‘𝑊) ∈ ℙ)
2	1	adantr 480	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (#‘𝑊) ∈ ℙ)
3		simp1 1054	. . . . . . . . 9 ⊢ ((𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊) → 𝐿 ∈ ℤ)
4	3	adantl 481	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → 𝐿 ∈ ℤ)
5		simpr2 1061	. . . . . . . 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 1235	. . . . . . 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 480	. . . . . 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 476	. . . . . 6 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) ∧ 𝑖 ∈ (0..^(#‘𝑊))) → 𝑖 ∈ (0..^(#‘𝑊)))
9		modprmn0modprm0 15350	. . . . . 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 63	. . . . 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 12380	. . . . . . . . . 10 ⊢ (𝑗 ∈ (0..^(#‘𝑊)) → 𝑗 ∈ ℕ0)
12	11	ad2antrr 758	. . . . . . . . 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 472	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → 𝑊 ∈ Word 𝑉)
14	13, 3	anim12i 588	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ))
15	14	adantr 480	. . . . . . . . . . 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 481	. . . . . . . . . 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 1062	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (𝑊 cyclShift 𝐿) = 𝑊)
18	17	anim1i 590	. . . . . . . . . . 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 481	. . . . . . . . . 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 13418	. . . . . . . . . 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 63	. . . . . . . . 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 6556	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (𝑘 · 𝐿) = (𝑗 · 𝐿))
23	22	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝑖 + (𝑘 · 𝐿)) = (𝑖 + (𝑗 · 𝐿)))
24	23	oveq1d 6564	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → ((𝑖 + (𝑘 · 𝐿)) mod (#‘𝑊)) = ((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊)))
25	24	fveq2d 6107	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → (𝑊‘((𝑖 + (𝑘 · 𝐿)) mod (#‘𝑊))) = (𝑊‘((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊))))
26	25	eqeq2d 2620	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → ((𝑊‘𝑖) = (𝑊‘((𝑖 + (𝑘 · 𝐿)) mod (#‘𝑊))) ↔ (𝑊‘𝑖) = (𝑊‘((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊)))))
27	26	rspcva 3280	. . . . . . . . 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 691	. . . . . . . 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 6103	. . . . . . . . . 10 ⊢ (((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊)) = 0 → (𝑊‘((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊))) = (𝑊‘0))
30	29	adantl 481	. . . . . . . . 9 ⊢ ((𝑗 ∈ (0..^(#‘𝑊)) ∧ ((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊)) = 0) → (𝑊‘((𝑖 + (𝑗 · 𝐿)) mod (#‘𝑊))) = (𝑊‘0))
31	30	adantr 480	. . . . . . . 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 2644	. . . . . . 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 449	. . . . . 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 3010	. . . . 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 37	. . . 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 2949	. . 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 13378	. . . 4 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
38	37	ad2antrr 758	. . 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 246	. 2 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ (𝐿 ∈ ℤ ∧ (𝐿 mod (#‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → 𝑊 = ((𝑊‘0) repeatS (#‘𝑊)))
40	39	ex 449	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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ‘cfv 5804  (class class class)co 6549  0cc0 9815   + caddc 9818   · cmul 9820  ℕ₀cn0 11169  ℤcz 11254  ..^cfzo 12334   mod cmo 12530  #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-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-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-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-dvds 14822  df-gcd 15055  df-prm 15224  df-phi 15309

This theorem is referenced by:  cshwsidrepswmod0  15639