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Theorem cshwshashlem1 17156

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 2959	. . . . . . 7 ⊢ ((𝑊‘𝑖) ≠ (𝑊‘0) ↔ ¬ (𝑊‘𝑖) = (𝑊‘0))
2	1	rexbii 3112	. . . . . 6 ⊢ (∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) ↔ ∃𝑖 ∈ (0..^(♯‘𝑊)) ¬ (𝑊‘𝑖) = (𝑊‘0))
3		rexnal 3117	. . . . . 6 ⊢ (∃𝑖 ∈ (0..^(♯‘𝑊)) ¬ (𝑊‘𝑖) = (𝑊‘0) ↔ ¬ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
4	2, 3	bitri 278	. . . . 5 ⊢ (∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) ↔ ¬ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
5		simpll 778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(♯‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → 𝜑)
6		fzo0ss1 13720	. . . . . . . . . . . . . 14 ⊢ (1..^(♯‘𝑊)) ⊆ (0..^(♯‘𝑊))
7		fzossfz 13709	. . . . . . . . . . . . . 14 ⊢ (0..^(♯‘𝑊)) ⊆ (0...(♯‘𝑊))
8	6, 7	sstri 3947	. . . . . . . . . . . . 13 ⊢ (1..^(♯‘𝑊)) ⊆ (0...(♯‘𝑊))
9	8	sseli 3934	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ (1..^(♯‘𝑊)) → 𝐿 ∈ (0...(♯‘𝑊)))
10	9	ad2antlr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(♯‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → 𝐿 ∈ (0...(♯‘𝑊)))
11		simpr 489	. . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → 𝑊 ∈ Word 𝑉)
14		simpr 489	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → (♯‘𝑊) ∈ ℙ)
15	14	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (♯‘𝑊) ∈ ℙ)
16		elfzelz 13553	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → 𝐿 ∈ ℤ)
17	16	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → 𝐿 ∈ ℤ)
18		cshwsidrepswmod0 17155	. . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → ((𝐿 mod (♯‘𝑊)) = 0 ∨ 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊))))
22		olc 881	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐿 = (♯‘𝑊) → (𝐿 = 0 ∨ 𝐿 = (♯‘𝑊)))
23	22	a1d 26	. . . . . . . . . . . . . . . . . . 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 13740	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐿 ≠ (♯‘𝑊) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → 𝐿 ∈ (0..^(♯‘𝑊)))
25		zmodidfzoimp 13936	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐿 ∈ (0..^(♯‘𝑊)) → (𝐿 mod (♯‘𝑊)) = 𝐿)
26		eqtr2 2784	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝐿 mod (♯‘𝑊)) = 𝐿 ∧ (𝐿 mod (♯‘𝑊)) = 0) → 𝐿 = 0)
27	26	a1d 26	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝐿 mod (♯‘𝑊)) = 𝐿 ∧ (𝐿 mod (♯‘𝑊)) = 0) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → 𝐿 = 0))
28	27	ex 417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐿 mod (♯‘𝑊)) = 𝐿 → ((𝐿 mod (♯‘𝑊)) = 0 → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → 𝐿 = 0)))
29	25, 28	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐿 ∈ (0..^(♯‘𝑊)) → ((𝐿 mod (♯‘𝑊)) = 0 → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → 𝐿 = 0)))
30	24, 29	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐿 ≠ (♯‘𝑊) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((𝐿 mod (♯‘𝑊)) = 0 → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → 𝐿 = 0)))
31	30	expcom 418	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → (𝐿 ≠ (♯‘𝑊) → ((𝐿 mod (♯‘𝑊)) = 0 → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → 𝐿 = 0))))
32	31	com24 96	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → ((𝐿 mod (♯‘𝑊)) = 0 → (𝐿 ≠ (♯‘𝑊) → 𝐿 = 0))))
33	32	impcom 412	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((𝐿 mod (♯‘𝑊)) = 0 → (𝐿 ≠ (♯‘𝑊) → 𝐿 = 0)))
34	33	3adant3 1150	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → ((𝐿 mod (♯‘𝑊)) = 0 → (𝐿 ≠ (♯‘𝑊) → 𝐿 = 0)))
35	34	impcom 412	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐿 mod (♯‘𝑊)) = 0 ∧ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (𝐿 ≠ (♯‘𝑊) → 𝐿 = 0))
36	35	impcom 412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐿 ≠ (♯‘𝑊) ∧ ((𝐿 mod (♯‘𝑊)) = 0 ∧ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊))) → 𝐿 = 0)
37	36	orcd 886	. . . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . . . 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 3041	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐿 mod (♯‘𝑊)) = 0 ∧ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (𝐿 = 0 ∨ 𝐿 = (♯‘𝑊)))
40	39	orcd 886	. . . . . . . . . . . . . . . . 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 1104	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 = 0 ∨ 𝐿 = (♯‘𝑊) ∨ 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊))) ↔ ((𝐿 = 0 ∨ 𝐿 = (♯‘𝑊)) ∨ 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊))))
42	40, 41	sylibr 237	. . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . 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 1351	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 = ((𝑊‘0) repeatS (♯‘𝑊)) → (𝐿 = 0 ∨ 𝐿 = (♯‘𝑊) ∨ 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊))))
45	44	a1d 26	. . . . . . . . . . . . . . 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 870	. . . . . . . . . . . . . 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 39	. . . . . . . . . . . . 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 1181	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = 0 ∨ 𝐿 = (♯‘𝑊) ∨ 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊))))
49		3mix1 1349	. . . . . . . . . . . . . 14 ⊢ (𝐿 = 0 → (𝐿 = 0 ∨ 𝐿 = (♯‘𝑊) ∨ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
50	49	a1d 26	. . . . . . . . . . . . 13 ⊢ (𝐿 = 0 → ((𝜑 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = 0 ∨ 𝐿 = (♯‘𝑊) ∨ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))))
51		3mix2 1350	. . . . . . . . . . . . . 14 ⊢ (𝐿 = (♯‘𝑊) → (𝐿 = 0 ∨ 𝐿 = (♯‘𝑊) ∨ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
52	51	a1d 26	. . . . . . . . . . . . 13 ⊢ (𝐿 = (♯‘𝑊) → ((𝜑 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = 0 ∨ 𝐿 = (♯‘𝑊) ∨ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))))
53		repswsymballbi 14819	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 = ((𝑊‘0) repeatS (♯‘𝑊)) ↔ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
54	53	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → (𝑊 = ((𝑊‘0) repeatS (♯‘𝑊)) ↔ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
55	12, 54	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑊 = ((𝑊‘0) repeatS (♯‘𝑊)) ↔ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
56	55	3ad2ant1 1151	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝑊 = ((𝑊‘0) repeatS (♯‘𝑊)) ↔ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
57	56	biimpa 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) ∧ 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊))) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
58	57	3mix3d 1357	. . . . . . . . . . . . . 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 418	. . . . . . . . . . . . 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 1454	. . . . . . . . . . . 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 39	. . . . . . . . . . 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 1398	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(♯‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = 0 ∨ 𝐿 = (♯‘𝑊) ∨ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
63		elfzo1 13743	. . . . . . . . . . . . . 14 ⊢ (𝐿 ∈ (1..^(♯‘𝑊)) ↔ (𝐿 ∈ ℕ ∧ (♯‘𝑊) ∈ ℕ ∧ 𝐿 < (♯‘𝑊)))
64		nnne0 12271	. . . . . . . . . . . . . . . 16 ⊢ (𝐿 ∈ ℕ → 𝐿 ≠ 0)
65		df-ne 2959	. . . . . . . . . . . . . . . . 17 ⊢ (𝐿 ≠ 0 ↔ ¬ 𝐿 = 0)
66		pm2.21 124	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝐿 = 0 → (𝐿 = 0 → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
67	65, 66	sylbi 220	. . . . . . . . . . . . . . . 16 ⊢ (𝐿 ≠ 0 → (𝐿 = 0 → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
68	64, 67	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝐿 ∈ ℕ → (𝐿 = 0 → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
69	68	3ad2ant1 1151	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ∈ ℕ ∧ (♯‘𝑊) ∈ ℕ ∧ 𝐿 < (♯‘𝑊)) → (𝐿 = 0 → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
70	63, 69	sylbi 220	. . . . . . . . . . . . 13 ⊢ (𝐿 ∈ (1..^(♯‘𝑊)) → (𝐿 = 0 → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
71	70	ad2antlr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(♯‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = 0 → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
72	71	com12 33	. . . . . . . . . . 11 ⊢ (𝐿 = 0 → (((𝜑 ∧ 𝐿 ∈ (1..^(♯‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
73		nnre 12241	. . . . . . . . . . . . . . . . 17 ⊢ (𝐿 ∈ ℕ → 𝐿 ∈ ℝ)
74		ltne 11308	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ ℝ ∧ 𝐿 < (♯‘𝑊)) → (♯‘𝑊) ≠ 𝐿)
75	73, 74	sylan 591	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℕ ∧ 𝐿 < (♯‘𝑊)) → (♯‘𝑊) ≠ 𝐿)
76		df-ne 2959	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑊) ≠ 𝐿 ↔ ¬ (♯‘𝑊) = 𝐿)
77		eqcom 2770	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐿 = (♯‘𝑊) ↔ (♯‘𝑊) = 𝐿)
78		pm2.21 124	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (♯‘𝑊) = 𝐿 → ((♯‘𝑊) = 𝐿 → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
79	77, 78	biimtrid 245	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (♯‘𝑊) = 𝐿 → (𝐿 = (♯‘𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
80	76, 79	sylbi 220	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑊) ≠ 𝐿 → (𝐿 = (♯‘𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
81	75, 80	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝐿 ∈ ℕ ∧ 𝐿 < (♯‘𝑊)) → (𝐿 = (♯‘𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
82	81	3adant2 1149	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ∈ ℕ ∧ (♯‘𝑊) ∈ ℕ ∧ 𝐿 < (♯‘𝑊)) → (𝐿 = (♯‘𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
83	63, 82	sylbi 220	. . . . . . . . . . . . 13 ⊢ (𝐿 ∈ (1..^(♯‘𝑊)) → (𝐿 = (♯‘𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
84	83	ad2antlr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(♯‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = (♯‘𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
85	84	com12 33	. . . . . . . . . . 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 1454	. . . . . . . . . 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 39	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(♯‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
89	88	pm2.24d 152	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(♯‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (¬ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0) → (𝑊 cyclShift 𝐿) ≠ 𝑊))
90	89	exp31 424	. . . . . . 7 ⊢ (𝜑 → (𝐿 ∈ (1..^(♯‘𝑊)) → ((𝑊 cyclShift 𝐿) = 𝑊 → (¬ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0) → (𝑊 cyclShift 𝐿) ≠ 𝑊))))
91	90	com34 92	. . . . . 6 ⊢ (𝜑 → (𝐿 ∈ (1..^(♯‘𝑊)) → (¬ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0) → ((𝑊 cyclShift 𝐿) = 𝑊 → (𝑊 cyclShift 𝐿) ≠ 𝑊))))
92	91	com23 87	. . . . 5 ⊢ (𝜑 → (¬ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0) → (𝐿 ∈ (1..^(♯‘𝑊)) → ((𝑊 cyclShift 𝐿) = 𝑊 → (𝑊 cyclShift 𝐿) ≠ 𝑊))))
93	4, 92	biimtrid 245	. . . 4 ⊢ (𝜑 → (∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) → (𝐿 ∈ (1..^(♯‘𝑊)) → ((𝑊 cyclShift 𝐿) = 𝑊 → (𝑊 cyclShift 𝐿) ≠ 𝑊))))
94	93	3imp 1128	. . 3 ⊢ ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) ∧ 𝐿 ∈ (1..^(♯‘𝑊))) → ((𝑊 cyclShift 𝐿) = 𝑊 → (𝑊 cyclShift 𝐿) ≠ 𝑊))
95	94	com12 33	. 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 3041	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 209 ∧ wa 400 ∨ wo 860 ∨ w3o 1102 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 class class class wbr 5110 ‘cfv 6538 (class class class)co 7412 ℝcr 11100 0cc0 11101 1c1 11102 < clt 11244 ℕcn 12234 ℤcz 12592 ...cfz 13536 ..^cfzo 13684 mod cmo 13904 ♯chash 14368 Word cword 14552 repeatS creps 14807 cyclShift ccsh 14827 ℙcprime 16730

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-2o 8455 df-oadd 8458 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-sup 9403 df-inf 9404 df-dju 9888 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-n0 12506 df-xnn0 12579 df-z 12593 df-uz 12864 df-rp 13018 df-fz 13537 df-fzo 13685 df-fl 13827 df-mod 13905 df-seq 14040 df-exp 14100 df-hash 14369 df-word 14553 df-concat 14610 df-substr 14681 df-pfx 14711 df-reps 14808 df-csh 14828 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-dvds 16312 df-gcd 16554 df-prm 16731 df-phi 16826

This theorem is referenced by: cshwshashlem2 17157

W3C validator