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Theorem cshwshashlem1 15640

Description: If cyclically shifting a word of length being a prime number not consisting of identical symbols by at least one position (and not by as many positions as the length of the word), the result will not be the word itself. (Contributed by AV, 19-May-2018.) (Revised by AV, 8-Jun-2018.) (Revised by AV, 10-Nov-2018.)

**Hypothesis**
Ref	Expression
cshwshash.0	⊢ (𝜑 → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ))

**Assertion**
Ref	Expression
cshwshashlem1	⊢ ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) ∧ 𝐿 ∈ (1..^(#‘𝑊))) → (𝑊 cyclShift 𝐿) ≠ 𝑊)

Distinct variable groups: 𝑖,𝐿 𝑖,𝑉 𝑖,𝑊 𝜑,𝑖

**Proof of Theorem cshwshashlem1**
Step	Hyp	Ref	Expression
1		df-ne 2782	. . . . . . 7 ⊢ ((𝑊‘𝑖) ≠ (𝑊‘0) ↔ ¬ (𝑊‘𝑖) = (𝑊‘0))
2	1	rexbii 3023	. . . . . 6 ⊢ (∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) ↔ ∃𝑖 ∈ (0..^(#‘𝑊)) ¬ (𝑊‘𝑖) = (𝑊‘0))
3		rexnal 2978	. . . . . 6 ⊢ (∃𝑖 ∈ (0..^(#‘𝑊)) ¬ (𝑊‘𝑖) = (𝑊‘0) ↔ ¬ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
4	2, 3	bitri 263	. . . . 5 ⊢ (∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) ↔ ¬ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
5		simpll 786	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(#‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → 𝜑)
6		fzo0ss1 12367	. . . . . . . . . . . . . 14 ⊢ (1..^(#‘𝑊)) ⊆ (0..^(#‘𝑊))
7		fzossfz 12357	. . . . . . . . . . . . . 14 ⊢ (0..^(#‘𝑊)) ⊆ (0...(#‘𝑊))
8	6, 7	sstri 3577	. . . . . . . . . . . . 13 ⊢ (1..^(#‘𝑊)) ⊆ (0...(#‘𝑊))
9	8	sseli 3564	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ (1..^(#‘𝑊)) → 𝐿 ∈ (0...(#‘𝑊)))
10	9	ad2antlr 759	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(#‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → 𝐿 ∈ (0...(#‘𝑊)))
11		simpr 476	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(#‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝑊 cyclShift 𝐿) = 𝑊)
12		cshwshash.0	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ))
13		simpll 786	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(#‘𝑊))) → 𝑊 ∈ Word 𝑉)
14		simpr 476	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → (#‘𝑊) ∈ ℙ)
15	14	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(#‘𝑊))) → (#‘𝑊) ∈ ℙ)
16		elfzelz 12213	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐿 ∈ (0...(#‘𝑊)) → 𝐿 ∈ ℤ)
17	16	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(#‘𝑊))) → 𝐿 ∈ ℤ)
18		cshwsidrepswmod0 15639	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ ∧ 𝐿 ∈ ℤ) → ((𝑊 cyclShift 𝐿) = 𝑊 → ((𝐿 mod (#‘𝑊)) = 0 ∨ 𝑊 = ((𝑊‘0) repeatS (#‘𝑊)))))
19	13, 15, 17, 18	syl3anc 1318	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(#‘𝑊))) → ((𝑊 cyclShift 𝐿) = 𝑊 → ((𝐿 mod (#‘𝑊)) = 0 ∨ 𝑊 = ((𝑊‘0) repeatS (#‘𝑊)))))
20	19	ex 449	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → (𝐿 ∈ (0...(#‘𝑊)) → ((𝑊 cyclShift 𝐿) = 𝑊 → ((𝐿 mod (#‘𝑊)) = 0 ∨ 𝑊 = ((𝑊‘0) repeatS (#‘𝑊))))))
21	20	3imp 1249	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(#‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → ((𝐿 mod (#‘𝑊)) = 0 ∨ 𝑊 = ((𝑊‘0) repeatS (#‘𝑊))))
22		olc 398	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐿 = (#‘𝑊) → (𝐿 = 0 ∨ 𝐿 = (#‘𝑊)))
23	22	a1d 25	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐿 = (#‘𝑊) → (((𝐿 mod (#‘𝑊)) = 0 ∧ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(#‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (𝐿 = 0 ∨ 𝐿 = (#‘𝑊))))
24		fzofzim 12382	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐿 ≠ (#‘𝑊) ∧ 𝐿 ∈ (0...(#‘𝑊))) → 𝐿 ∈ (0..^(#‘𝑊)))
25		zmodidfzoimp 12562	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐿 ∈ (0..^(#‘𝑊)) → (𝐿 mod (#‘𝑊)) = 𝐿)
26		eqtr2 2630	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝐿 mod (#‘𝑊)) = 𝐿 ∧ (𝐿 mod (#‘𝑊)) = 0) → 𝐿 = 0)
27	26	a1d 25	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝐿 mod (#‘𝑊)) = 𝐿 ∧ (𝐿 mod (#‘𝑊)) = 0) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → 𝐿 = 0))
28	27	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐿 mod (#‘𝑊)) = 𝐿 → ((𝐿 mod (#‘𝑊)) = 0 → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → 𝐿 = 0)))
29	25, 28	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐿 ∈ (0..^(#‘𝑊)) → ((𝐿 mod (#‘𝑊)) = 0 → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → 𝐿 = 0)))
30	24, 29	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐿 ≠ (#‘𝑊) ∧ 𝐿 ∈ (0...(#‘𝑊))) → ((𝐿 mod (#‘𝑊)) = 0 → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → 𝐿 = 0)))
31	30	expcom 450	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐿 ∈ (0...(#‘𝑊)) → (𝐿 ≠ (#‘𝑊) → ((𝐿 mod (#‘𝑊)) = 0 → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → 𝐿 = 0))))
32	31	com24 93	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐿 ∈ (0...(#‘𝑊)) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → ((𝐿 mod (#‘𝑊)) = 0 → (𝐿 ≠ (#‘𝑊) → 𝐿 = 0))))
33	32	impcom 445	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(#‘𝑊))) → ((𝐿 mod (#‘𝑊)) = 0 → (𝐿 ≠ (#‘𝑊) → 𝐿 = 0)))
34	33	3adant3 1074	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(#‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → ((𝐿 mod (#‘𝑊)) = 0 → (𝐿 ≠ (#‘𝑊) → 𝐿 = 0)))
35	34	impcom 445	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐿 mod (#‘𝑊)) = 0 ∧ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(#‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (𝐿 ≠ (#‘𝑊) → 𝐿 = 0))
36	35	impcom 445	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐿 ≠ (#‘𝑊) ∧ ((𝐿 mod (#‘𝑊)) = 0 ∧ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(#‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊))) → 𝐿 = 0)
37	36	orcd 406	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ≠ (#‘𝑊) ∧ ((𝐿 mod (#‘𝑊)) = 0 ∧ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(#‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊))) → (𝐿 = 0 ∨ 𝐿 = (#‘𝑊)))
38	37	ex 449	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐿 ≠ (#‘𝑊) → (((𝐿 mod (#‘𝑊)) = 0 ∧ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(#‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (𝐿 = 0 ∨ 𝐿 = (#‘𝑊))))
39	23, 38	pm2.61ine 2865	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐿 mod (#‘𝑊)) = 0 ∧ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(#‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (𝐿 = 0 ∨ 𝐿 = (#‘𝑊)))
40	39	orcd 406	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐿 mod (#‘𝑊)) = 0 ∧ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(#‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → ((𝐿 = 0 ∨ 𝐿 = (#‘𝑊)) ∨ 𝑊 = ((𝑊‘0) repeatS (#‘𝑊))))
41		df-3or 1032	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 = 0 ∨ 𝐿 = (#‘𝑊) ∨ 𝑊 = ((𝑊‘0) repeatS (#‘𝑊))) ↔ ((𝐿 = 0 ∨ 𝐿 = (#‘𝑊)) ∨ 𝑊 = ((𝑊‘0) repeatS (#‘𝑊))))
42	40, 41	sylibr 223	. . . . . . . . . . . . . . . 16 ⊢ (((𝐿 mod (#‘𝑊)) = 0 ∧ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(#‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (𝐿 = 0 ∨ 𝐿 = (#‘𝑊) ∨ 𝑊 = ((𝑊‘0) repeatS (#‘𝑊))))
43	42	ex 449	. . . . . . . . . . . . . . 15 ⊢ ((𝐿 mod (#‘𝑊)) = 0 → (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(#‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = 0 ∨ 𝐿 = (#‘𝑊) ∨ 𝑊 = ((𝑊‘0) repeatS (#‘𝑊)))))
44		3mix3 1225	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) → (𝐿 = 0 ∨ 𝐿 = (#‘𝑊) ∨ 𝑊 = ((𝑊‘0) repeatS (#‘𝑊))))
45	44	a1d 25	. . . . . . . . . . . . . . 15 ⊢ (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) → (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(#‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = 0 ∨ 𝐿 = (#‘𝑊) ∨ 𝑊 = ((𝑊‘0) repeatS (#‘𝑊)))))
46	43, 45	jaoi 393	. . . . . . . . . . . . . 14 ⊢ (((𝐿 mod (#‘𝑊)) = 0 ∨ 𝑊 = ((𝑊‘0) repeatS (#‘𝑊))) → (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(#‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = 0 ∨ 𝐿 = (#‘𝑊) ∨ 𝑊 = ((𝑊‘0) repeatS (#‘𝑊)))))
47	21, 46	mpcom 37	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(#‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = 0 ∨ 𝐿 = (#‘𝑊) ∨ 𝑊 = ((𝑊‘0) repeatS (#‘𝑊))))
48	12, 47	syl3an1 1351	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐿 ∈ (0...(#‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = 0 ∨ 𝐿 = (#‘𝑊) ∨ 𝑊 = ((𝑊‘0) repeatS (#‘𝑊))))
49		3mix1 1223	. . . . . . . . . . . . . 14 ⊢ (𝐿 = 0 → (𝐿 = 0 ∨ 𝐿 = (#‘𝑊) ∨ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
50	49	a1d 25	. . . . . . . . . . . . 13 ⊢ (𝐿 = 0 → ((𝜑 ∧ 𝐿 ∈ (0...(#‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = 0 ∨ 𝐿 = (#‘𝑊) ∨ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))))
51		3mix2 1224	. . . . . . . . . . . . . 14 ⊢ (𝐿 = (#‘𝑊) → (𝐿 = 0 ∨ 𝐿 = (#‘𝑊) ∨ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
52	51	a1d 25	. . . . . . . . . . . . 13 ⊢ (𝐿 = (#‘𝑊) → ((𝜑 ∧ 𝐿 ∈ (0...(#‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = 0 ∨ 𝐿 = (#‘𝑊) ∨ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))))
53		repswsymballbi 13378	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
54	53	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ) → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
55	12, 54	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
56	55	3ad2ant1 1075	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐿 ∈ (0...(#‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
57	56	biimpa 500	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐿 ∈ (0...(#‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) ∧ 𝑊 = ((𝑊‘0) repeatS (#‘𝑊))) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
58	57	3mix3d 1231	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐿 ∈ (0...(#‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) ∧ 𝑊 = ((𝑊‘0) repeatS (#‘𝑊))) → (𝐿 = 0 ∨ 𝐿 = (#‘𝑊) ∨ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
59	58	expcom 450	. . . . . . . . . . . . 13 ⊢ (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) → ((𝜑 ∧ 𝐿 ∈ (0...(#‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = 0 ∨ 𝐿 = (#‘𝑊) ∨ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))))
60	50, 52, 59	3jaoi 1383	. . . . . . . . . . . 12 ⊢ ((𝐿 = 0 ∨ 𝐿 = (#‘𝑊) ∨ 𝑊 = ((𝑊‘0) repeatS (#‘𝑊))) → ((𝜑 ∧ 𝐿 ∈ (0...(#‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = 0 ∨ 𝐿 = (#‘𝑊) ∨ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))))
61	48, 60	mpcom 37	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐿 ∈ (0...(#‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = 0 ∨ 𝐿 = (#‘𝑊) ∨ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
62	5, 10, 11, 61	syl3anc 1318	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(#‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = 0 ∨ 𝐿 = (#‘𝑊) ∨ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
63		elfzo1 12385	. . . . . . . . . . . . . 14 ⊢ (𝐿 ∈ (1..^(#‘𝑊)) ↔ (𝐿 ∈ ℕ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)))
64		nnne0 10930	. . . . . . . . . . . . . . . 16 ⊢ (𝐿 ∈ ℕ → 𝐿 ≠ 0)
65		df-ne 2782	. . . . . . . . . . . . . . . . 17 ⊢ (𝐿 ≠ 0 ↔ ¬ 𝐿 = 0)
66		pm2.21 119	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝐿 = 0 → (𝐿 = 0 → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
67	65, 66	sylbi 206	. . . . . . . . . . . . . . . 16 ⊢ (𝐿 ≠ 0 → (𝐿 = 0 → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
68	64, 67	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐿 ∈ ℕ → (𝐿 = 0 → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
69	68	3ad2ant1 1075	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ∈ ℕ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)) → (𝐿 = 0 → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
70	63, 69	sylbi 206	. . . . . . . . . . . . 13 ⊢ (𝐿 ∈ (1..^(#‘𝑊)) → (𝐿 = 0 → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
71	70	ad2antlr 759	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(#‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = 0 → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
72	71	com12 32	. . . . . . . . . . 11 ⊢ (𝐿 = 0 → (((𝜑 ∧ 𝐿 ∈ (1..^(#‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
73		nnre 10904	. . . . . . . . . . . . . . . . 17 ⊢ (𝐿 ∈ ℕ → 𝐿 ∈ ℝ)
74		ltne 10013	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ ℝ ∧ 𝐿 < (#‘𝑊)) → (#‘𝑊) ≠ 𝐿)
75	73, 74	sylan 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℕ ∧ 𝐿 < (#‘𝑊)) → (#‘𝑊) ≠ 𝐿)
76		df-ne 2782	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝑊) ≠ 𝐿 ↔ ¬ (#‘𝑊) = 𝐿)
77		eqcom 2617	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐿 = (#‘𝑊) ↔ (#‘𝑊) = 𝐿)
78		pm2.21 119	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (#‘𝑊) = 𝐿 → ((#‘𝑊) = 𝐿 → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
79	77, 78	syl5bi 231	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (#‘𝑊) = 𝐿 → (𝐿 = (#‘𝑊) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
80	76, 79	sylbi 206	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑊) ≠ 𝐿 → (𝐿 = (#‘𝑊) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
81	75, 80	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐿 ∈ ℕ ∧ 𝐿 < (#‘𝑊)) → (𝐿 = (#‘𝑊) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
82	81	3adant2 1073	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ∈ ℕ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)) → (𝐿 = (#‘𝑊) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
83	63, 82	sylbi 206	. . . . . . . . . . . . 13 ⊢ (𝐿 ∈ (1..^(#‘𝑊)) → (𝐿 = (#‘𝑊) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
84	83	ad2antlr 759	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(#‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = (#‘𝑊) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
85	84	com12 32	. . . . . . . . . . 11 ⊢ (𝐿 = (#‘𝑊) → (((𝜑 ∧ 𝐿 ∈ (1..^(#‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
86		ax-1 6	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0) → (((𝜑 ∧ 𝐿 ∈ (1..^(#‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
87	72, 85, 86	3jaoi 1383	. . . . . . . . . 10 ⊢ ((𝐿 = 0 ∨ 𝐿 = (#‘𝑊) ∨ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)) → (((𝜑 ∧ 𝐿 ∈ (1..^(#‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
88	62, 87	mpcom 37	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(#‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
89	88	pm2.24d 146	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(#‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (¬ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0) → (𝑊 cyclShift 𝐿) ≠ 𝑊))
90	89	exp31 628	. . . . . . 7 ⊢ (𝜑 → (𝐿 ∈ (1..^(#‘𝑊)) → ((𝑊 cyclShift 𝐿) = 𝑊 → (¬ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0) → (𝑊 cyclShift 𝐿) ≠ 𝑊))))
91	90	com34 89	. . . . . 6 ⊢ (𝜑 → (𝐿 ∈ (1..^(#‘𝑊)) → (¬ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0) → ((𝑊 cyclShift 𝐿) = 𝑊 → (𝑊 cyclShift 𝐿) ≠ 𝑊))))
92	91	com23 84	. . . . 5 ⊢ (𝜑 → (¬ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0) → (𝐿 ∈ (1..^(#‘𝑊)) → ((𝑊 cyclShift 𝐿) = 𝑊 → (𝑊 cyclShift 𝐿) ≠ 𝑊))))
93	4, 92	syl5bi 231	. . . 4 ⊢ (𝜑 → (∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) → (𝐿 ∈ (1..^(#‘𝑊)) → ((𝑊 cyclShift 𝐿) = 𝑊 → (𝑊 cyclShift 𝐿) ≠ 𝑊))))
94	93	3imp 1249	. . 3 ⊢ ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) ∧ 𝐿 ∈ (1..^(#‘𝑊))) → ((𝑊 cyclShift 𝐿) = 𝑊 → (𝑊 cyclShift 𝐿) ≠ 𝑊))
95	94	com12 32	. 2 ⊢ ((𝑊 cyclShift 𝐿) = 𝑊 → ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) ∧ 𝐿 ∈ (1..^(#‘𝑊))) → (𝑊 cyclShift 𝐿) ≠ 𝑊))
96		ax-1 6	. 2 ⊢ ((𝑊 cyclShift 𝐿) ≠ 𝑊 → ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) ∧ 𝐿 ∈ (1..^(#‘𝑊))) → (𝑊 cyclShift 𝐿) ≠ 𝑊))
97	95, 96	pm2.61ine 2865	1 ⊢ ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) ∧ 𝐿 ∈ (1..^(#‘𝑊))) → (𝑊 cyclShift 𝐿) ≠ 𝑊)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∨ w3o 1030 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 < clt 9953 ℕcn 10897 ℤcz 11254 ...cfz 12197 ..^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: cshwshashlem2 15641

W3C validator