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Theorem cshwshashlem1 16979

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 2940	. . . . . . 7 ⊢ ((𝑊‘𝑖) ≠ (𝑊‘0) ↔ ¬ (𝑊‘𝑖) = (𝑊‘0))
2	1	rexbii 3093	. . . . . 6 ⊢ (∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) ↔ ∃𝑖 ∈ (0..^(♯‘𝑊)) ¬ (𝑊‘𝑖) = (𝑊‘0))
3		rexnal 3099	. . . . . 6 ⊢ (∃𝑖 ∈ (0..^(♯‘𝑊)) ¬ (𝑊‘𝑖) = (𝑊‘0) ↔ ¬ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
4	2, 3	bitri 274	. . . . 5 ⊢ (∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) ↔ ¬ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
5		simpll 765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(♯‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → 𝜑)
6		fzo0ss1 13612	. . . . . . . . . . . . . 14 ⊢ (1..^(♯‘𝑊)) ⊆ (0..^(♯‘𝑊))
7		fzossfz 13601	. . . . . . . . . . . . . 14 ⊢ (0..^(♯‘𝑊)) ⊆ (0...(♯‘𝑊))
8	6, 7	sstri 3956	. . . . . . . . . . . . 13 ⊢ (1..^(♯‘𝑊)) ⊆ (0...(♯‘𝑊))
9	8	sseli 3943	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ (1..^(♯‘𝑊)) → 𝐿 ∈ (0...(♯‘𝑊)))
10	9	ad2antlr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(♯‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → 𝐿 ∈ (0...(♯‘𝑊)))
11		simpr 485	. . . . . . . . . . 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 765	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → 𝑊 ∈ Word 𝑉)
14		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → (♯‘𝑊) ∈ ℙ)
15	14	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (♯‘𝑊) ∈ ℙ)
16		elfzelz 13451	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → 𝐿 ∈ ℤ)
17	16	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → 𝐿 ∈ ℤ)
18		cshwsidrepswmod0 16978	. . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . 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 1111	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → ((𝐿 mod (♯‘𝑊)) = 0 ∨ 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊))))
22		olc 866	. . . . . . . . . . . . . . . . . . . 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 13629	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐿 ≠ (♯‘𝑊) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → 𝐿 ∈ (0..^(♯‘𝑊)))
25		zmodidfzoimp 13816	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐿 ∈ (0..^(♯‘𝑊)) → (𝐿 mod (♯‘𝑊)) = 𝐿)
26		eqtr2 2755	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 414	. . . . . . . . . . . . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐿 mod (♯‘𝑊)) = 0 ∧ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (𝐿 ≠ (♯‘𝑊) → 𝐿 = 0))
36	35	impcom 408	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐿 ≠ (♯‘𝑊) ∧ ((𝐿 mod (♯‘𝑊)) = 0 ∧ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊))) → 𝐿 = 0)
37	36	orcd 871	. . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . 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 3024	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐿 mod (♯‘𝑊)) = 0 ∧ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (𝐿 = 0 ∨ 𝐿 = (♯‘𝑊)))
40	39	orcd 871	. . . . . . . . . . . . . . . . 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 1088	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 = 0 ∨ 𝐿 = (♯‘𝑊) ∨ 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊))) ↔ ((𝐿 = 0 ∨ 𝐿 = (♯‘𝑊)) ∨ 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊))))
42	40, 41	sylibr 233	. . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . 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 1332	. . . . . . . . . . . . . . . 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 855	. . . . . . . . . . . . . 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 1330	. . . . . . . . . . . . . 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 1331	. . . . . . . . . . . . . 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 14680	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 = ((𝑊‘0) repeatS (♯‘𝑊)) ↔ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
54	53	adantr 481	. . . . . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) ∧ 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊))) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
58	57	3mix3d 1338	. . . . . . . . . . . . . 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 414	. . . . . . . . . . . . 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 1427	. . . . . . . . . . . 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 1371	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(♯‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = 0 ∨ 𝐿 = (♯‘𝑊) ∨ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
63		elfzo1 13632	. . . . . . . . . . . . . 14 ⊢ (𝐿 ∈ (1..^(♯‘𝑊)) ↔ (𝐿 ∈ ℕ ∧ (♯‘𝑊) ∈ ℕ ∧ 𝐿 < (♯‘𝑊)))
64		nnne0 12196	. . . . . . . . . . . . . . . 16 ⊢ (𝐿 ∈ ℕ → 𝐿 ≠ 0)
65		df-ne 2940	. . . . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . 13 ⊢ (𝐿 ∈ (1..^(♯‘𝑊)) → (𝐿 = 0 → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
71	70	ad2antlr 725	. . . . . . . . . . . 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 12169	. . . . . . . . . . . . . . . . 17 ⊢ (𝐿 ∈ ℕ → 𝐿 ∈ ℝ)
74		ltne 11261	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ ℝ ∧ 𝐿 < (♯‘𝑊)) → (♯‘𝑊) ≠ 𝐿)
75	73, 74	sylan 580	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℕ ∧ 𝐿 < (♯‘𝑊)) → (♯‘𝑊) ≠ 𝐿)
76		df-ne 2940	. . . . . . . . . . . . . . . . 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 241	. . . . . . . . . . . . . . . . 17 ⊢ (¬ (♯‘𝑊) = 𝐿 → (𝐿 = (♯‘𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
80	76, 79	sylbi 216	. . . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . 13 ⊢ (𝐿 ∈ (1..^(♯‘𝑊)) → (𝐿 = (♯‘𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
84	83	ad2antlr 725	. . . . . . . . . . . 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 1427	. . . . . . . . . 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 420	. . . . . . 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 241	. . . 4 ⊢ (𝜑 → (∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) → (𝐿 ∈ (1..^(♯‘𝑊)) → ((𝑊 cyclShift 𝐿) = 𝑊 → (𝑊 cyclShift 𝐿) ≠ 𝑊))))
94	93	3imp 1111	. . 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 3024	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 205 ∧ wa 396 ∨ wo 845 ∨ w3o 1086 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 class class class wbr 5110 ‘cfv 6501 (class class class)co 7362 ℝcr 11059 0cc0 11060 1c1 11061 < clt 11198 ℕcn 12162 ℤcz 12508 ...cfz 13434 ..^cfzo 13577 mod cmo 13784 ♯chash 14240 Word cword 14414 repeatS creps 14668 cyclShift ccsh 14688 ℙcprime 16558

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11116 ax-resscn 11117 ax-1cn 11118 ax-icn 11119 ax-addcl 11120 ax-addrcl 11121 ax-mulcl 11122 ax-mulrcl 11123 ax-mulcom 11124 ax-addass 11125 ax-mulass 11126 ax-distr 11127 ax-i2m1 11128 ax-1ne0 11129 ax-1rid 11130 ax-rnegex 11131 ax-rrecex 11132 ax-cnre 11133 ax-pre-lttri 11134 ax-pre-lttrn 11135 ax-pre-ltadd 11136 ax-pre-mulgt0 11137 ax-pre-sup 11138

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3448 df-sbc 3743 df-csb 3859 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-oadd 8421 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9387 df-inf 9388 df-dju 9846 df-card 9884 df-pnf 11200 df-mnf 11201 df-xr 11202 df-ltxr 11203 df-le 11204 df-sub 11396 df-neg 11397 df-div 11822 df-nn 12163 df-2 12225 df-3 12226 df-n0 12423 df-xnn0 12495 df-z 12509 df-uz 12773 df-rp 12925 df-fz 13435 df-fzo 13578 df-fl 13707 df-mod 13785 df-seq 13917 df-exp 13978 df-hash 14241 df-word 14415 df-concat 14471 df-substr 14541 df-pfx 14571 df-reps 14669 df-csh 14689 df-cj 14996 df-re 14997 df-im 14998 df-sqrt 15132 df-abs 15133 df-dvds 16148 df-gcd 16386 df-prm 16559 df-phi 16649

This theorem is referenced by: cshwshashlem2 16980

W3C validator