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Theorem cshwshashlem1 17004

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 2929	. . . . . . 7 ⊢ ((𝑊‘𝑖) ≠ (𝑊‘0) ↔ ¬ (𝑊‘𝑖) = (𝑊‘0))
2	1	rexbii 3079	. . . . . 6 ⊢ (∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) ↔ ∃𝑖 ∈ (0..^(♯‘𝑊)) ¬ (𝑊‘𝑖) = (𝑊‘0))
3		rexnal 3084	. . . . . 6 ⊢ (∃𝑖 ∈ (0..^(♯‘𝑊)) ¬ (𝑊‘𝑖) = (𝑊‘0) ↔ ¬ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
4	2, 3	bitri 275	. . . . 5 ⊢ (∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) ↔ ¬ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
5		simpll 766	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(♯‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → 𝜑)
6		fzo0ss1 13586	. . . . . . . . . . . . . 14 ⊢ (1..^(♯‘𝑊)) ⊆ (0..^(♯‘𝑊))
7		fzossfz 13575	. . . . . . . . . . . . . 14 ⊢ (0..^(♯‘𝑊)) ⊆ (0...(♯‘𝑊))
8	6, 7	sstri 3944	. . . . . . . . . . . . 13 ⊢ (1..^(♯‘𝑊)) ⊆ (0...(♯‘𝑊))
9	8	sseli 3930	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ (1..^(♯‘𝑊)) → 𝐿 ∈ (0...(♯‘𝑊)))
10	9	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(♯‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → 𝐿 ∈ (0...(♯‘𝑊)))
11		simpr 484	. . . . . . . . . . 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 766	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → 𝑊 ∈ Word 𝑉)
14		simpr 484	. . . . . . . . . . . . . . . . . 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 13421	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → 𝐿 ∈ ℤ)
17	16	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → 𝐿 ∈ ℤ)
18		cshwsidrepswmod0 17003	. . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . 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 1110	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → ((𝐿 mod (♯‘𝑊)) = 0 ∨ 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊))))
22		olc 868	. . . . . . . . . . . . . . . . . . . 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 13606	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐿 ≠ (♯‘𝑊) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → 𝐿 ∈ (0..^(♯‘𝑊)))
25		zmodidfzoimp 13802	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐿 ∈ (0..^(♯‘𝑊)) → (𝐿 mod (♯‘𝑊)) = 𝐿)
26		eqtr2 2752	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → (𝐿 ≠ (♯‘𝑊) → ((𝐿 mod (♯‘𝑊)) = 0 → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → 𝐿 = 0))))
32	31	com24 95	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → ((𝐿 mod (♯‘𝑊)) = 0 → (𝐿 ≠ (♯‘𝑊) → 𝐿 = 0))))
33	32	impcom 407	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((𝐿 mod (♯‘𝑊)) = 0 → (𝐿 ≠ (♯‘𝑊) → 𝐿 = 0)))
34	33	3adant3 1132	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → ((𝐿 mod (♯‘𝑊)) = 0 → (𝐿 ≠ (♯‘𝑊) → 𝐿 = 0)))
35	34	impcom 407	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐿 mod (♯‘𝑊)) = 0 ∧ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (𝐿 ≠ (♯‘𝑊) → 𝐿 = 0))
36	35	impcom 407	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐿 ≠ (♯‘𝑊) ∧ ((𝐿 mod (♯‘𝑊)) = 0 ∧ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊))) → 𝐿 = 0)
37	36	orcd 873	. . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . 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 3011	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐿 mod (♯‘𝑊)) = 0 ∧ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (𝐿 = 0 ∨ 𝐿 = (♯‘𝑊)))
40	39	orcd 873	. . . . . . . . . . . . . . . . 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 1087	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 = 0 ∨ 𝐿 = (♯‘𝑊) ∨ 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊))) ↔ ((𝐿 = 0 ∨ 𝐿 = (♯‘𝑊)) ∨ 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊))))
42	40, 41	sylibr 234	. . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . 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 1333	. . . . . . . . . . . . . . . 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 857	. . . . . . . . . . . . . 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 38	. . . . . . . . . . . . 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 1163	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = 0 ∨ 𝐿 = (♯‘𝑊) ∨ 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊))))
49		3mix1 1331	. . . . . . . . . . . . . 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 1332	. . . . . . . . . . . . . 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 14684	. . . . . . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝑊 = ((𝑊‘0) repeatS (♯‘𝑊)) ↔ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
57	56	biimpa 476	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) ∧ 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊))) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
58	57	3mix3d 1339	. . . . . . . . . . . . . 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 413	. . . . . . . . . . . . 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 1430	. . . . . . . . . . . 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 38	. . . . . . . . . . 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 1373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(♯‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = 0 ∨ 𝐿 = (♯‘𝑊) ∨ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
63		elfzo1 13609	. . . . . . . . . . . . . 14 ⊢ (𝐿 ∈ (1..^(♯‘𝑊)) ↔ (𝐿 ∈ ℕ ∧ (♯‘𝑊) ∈ ℕ ∧ 𝐿 < (♯‘𝑊)))
64		nnne0 12156	. . . . . . . . . . . . . . . 16 ⊢ (𝐿 ∈ ℕ → 𝐿 ≠ 0)
65		df-ne 2929	. . . . . . . . . . . . . . . . 17 ⊢ (𝐿 ≠ 0 ↔ ¬ 𝐿 = 0)
66		pm2.21 123	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝐿 = 0 → (𝐿 = 0 → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
67	65, 66	sylbi 217	. . . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ∈ ℕ ∧ (♯‘𝑊) ∈ ℕ ∧ 𝐿 < (♯‘𝑊)) → (𝐿 = 0 → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
70	63, 69	sylbi 217	. . . . . . . . . . . . 13 ⊢ (𝐿 ∈ (1..^(♯‘𝑊)) → (𝐿 = 0 → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
71	70	ad2antlr 727	. . . . . . . . . . . 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 12129	. . . . . . . . . . . . . . . . 17 ⊢ (𝐿 ∈ ℕ → 𝐿 ∈ ℝ)
74		ltne 11207	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ ℝ ∧ 𝐿 < (♯‘𝑊)) → (♯‘𝑊) ≠ 𝐿)
75	73, 74	sylan 580	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℕ ∧ 𝐿 < (♯‘𝑊)) → (♯‘𝑊) ≠ 𝐿)
76		df-ne 2929	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑊) ≠ 𝐿 ↔ ¬ (♯‘𝑊) = 𝐿)
77		eqcom 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐿 = (♯‘𝑊) ↔ (♯‘𝑊) = 𝐿)
78		pm2.21 123	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (♯‘𝑊) = 𝐿 → ((♯‘𝑊) = 𝐿 → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
79	77, 78	biimtrid 242	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (♯‘𝑊) = 𝐿 → (𝐿 = (♯‘𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
80	76, 79	sylbi 217	. . . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ∈ ℕ ∧ (♯‘𝑊) ∈ ℕ ∧ 𝐿 < (♯‘𝑊)) → (𝐿 = (♯‘𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
83	63, 82	sylbi 217	. . . . . . . . . . . . 13 ⊢ (𝐿 ∈ (1..^(♯‘𝑊)) → (𝐿 = (♯‘𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
84	83	ad2antlr 727	. . . . . . . . . . . 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 1430	. . . . . . . . . 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 38	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(♯‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
89	88	pm2.24d 151	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(♯‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (¬ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0) → (𝑊 cyclShift 𝐿) ≠ 𝑊))
90	89	exp31 419	. . . . . . 7 ⊢ (𝜑 → (𝐿 ∈ (1..^(♯‘𝑊)) → ((𝑊 cyclShift 𝐿) = 𝑊 → (¬ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0) → (𝑊 cyclShift 𝐿) ≠ 𝑊))))
91	90	com34 91	. . . . . 6 ⊢ (𝜑 → (𝐿 ∈ (1..^(♯‘𝑊)) → (¬ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0) → ((𝑊 cyclShift 𝐿) = 𝑊 → (𝑊 cyclShift 𝐿) ≠ 𝑊))))
92	91	com23 86	. . . . 5 ⊢ (𝜑 → (¬ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0) → (𝐿 ∈ (1..^(♯‘𝑊)) → ((𝑊 cyclShift 𝐿) = 𝑊 → (𝑊 cyclShift 𝐿) ≠ 𝑊))))
93	4, 92	biimtrid 242	. . . 4 ⊢ (𝜑 → (∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) → (𝐿 ∈ (1..^(♯‘𝑊)) → ((𝑊 cyclShift 𝐿) = 𝑊 → (𝑊 cyclShift 𝐿) ≠ 𝑊))))
94	93	3imp 1110	. . 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 3011	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 206 ∧ wa 395 ∨ wo 847 ∨ w3o 1085 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∃wrex 3056 class class class wbr 5091 ‘cfv 6481 (class class class)co 7346 ℝcr 11002 0cc0 11003 1c1 11004 < clt 11143 ℕcn 12122 ℤcz 12465 ...cfz 13404 ..^cfzo 13551 mod cmo 13770 ♯chash 14234 Word cword 14417 repeatS creps 14672 cyclShift ccsh 14692 ℙcprime 16579

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080 ax-pre-sup 11081

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-inf 9327 df-dju 9791 df-card 9829 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-div 11772 df-nn 12123 df-2 12185 df-3 12186 df-n0 12379 df-xnn0 12452 df-z 12466 df-uz 12730 df-rp 12888 df-fz 13405 df-fzo 13552 df-fl 13693 df-mod 13771 df-seq 13906 df-exp 13966 df-hash 14235 df-word 14418 df-concat 14475 df-substr 14546 df-pfx 14576 df-reps 14673 df-csh 14693 df-cj 15003 df-re 15004 df-im 15005 df-sqrt 15139 df-abs 15140 df-dvds 16161 df-gcd 16403 df-prm 16580 df-phi 16674

This theorem is referenced by: cshwshashlem2 17005

W3C validator