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Theorem cshwshashlem1 16725

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 2943	. . . . . . 7 ⊢ ((𝑊‘𝑖) ≠ (𝑊‘0) ↔ ¬ (𝑊‘𝑖) = (𝑊‘0))
2	1	rexbii 3177	. . . . . 6 ⊢ (∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) ↔ ∃𝑖 ∈ (0..^(♯‘𝑊)) ¬ (𝑊‘𝑖) = (𝑊‘0))
3		rexnal 3165	. . . . . 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 763	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(♯‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → 𝜑)
6		fzo0ss1 13345	. . . . . . . . . . . . . 14 ⊢ (1..^(♯‘𝑊)) ⊆ (0..^(♯‘𝑊))
7		fzossfz 13334	. . . . . . . . . . . . . 14 ⊢ (0..^(♯‘𝑊)) ⊆ (0...(♯‘𝑊))
8	6, 7	sstri 3926	. . . . . . . . . . . . 13 ⊢ (1..^(♯‘𝑊)) ⊆ (0...(♯‘𝑊))
9	8	sseli 3913	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ (1..^(♯‘𝑊)) → 𝐿 ∈ (0...(♯‘𝑊)))
10	9	ad2antlr 723	. . . . . . . . . . 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 763	. . . . . . . . . . . . . . . . 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 13185	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐿 ∈ (0...(♯‘𝑊)) → 𝐿 ∈ ℤ)
17	16	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → 𝐿 ∈ ℤ)
18		cshwsidrepswmod0 16724	. . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . 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 1109	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → ((𝐿 mod (♯‘𝑊)) = 0 ∨ 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊))))
22		olc 864	. . . . . . . . . . . . . . . . . . . 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 13362	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐿 ≠ (♯‘𝑊) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → 𝐿 ∈ (0..^(♯‘𝑊)))
25		zmodidfzoimp 13549	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐿 ∈ (0..^(♯‘𝑊)) → (𝐿 mod (♯‘𝑊)) = 𝐿)
26		eqtr2 2762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1130	. . . . . . . . . . . . . . . . . . . . . . 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 869	. . . . . . . . . . . . . . . . . . . 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 3027	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐿 mod (♯‘𝑊)) = 0 ∧ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊)) → (𝐿 = 0 ∨ 𝐿 = (♯‘𝑊)))
40	39	orcd 869	. . . . . . . . . . . . . . . . 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 1086	. . . . . . . . . . . . . . . . 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 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 1330	. . . . . . . . . . . . . . . 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 853	. . . . . . . . . . . . . 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 1161	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = 0 ∨ 𝐿 = (♯‘𝑊) ∨ 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊))))
49		3mix1 1328	. . . . . . . . . . . . . 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 1329	. . . . . . . . . . . . . 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 14421	. . . . . . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . . . . 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 1336	. . . . . . . . . . . . . 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 1425	. . . . . . . . . . . 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 1369	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐿 ∈ (1..^(♯‘𝑊))) ∧ (𝑊 cyclShift 𝐿) = 𝑊) → (𝐿 = 0 ∨ 𝐿 = (♯‘𝑊) ∨ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
63		elfzo1 13365	. . . . . . . . . . . . . 14 ⊢ (𝐿 ∈ (1..^(♯‘𝑊)) ↔ (𝐿 ∈ ℕ ∧ (♯‘𝑊) ∈ ℕ ∧ 𝐿 < (♯‘𝑊)))
64		nnne0 11937	. . . . . . . . . . . . . . . 16 ⊢ (𝐿 ∈ ℕ → 𝐿 ≠ 0)
65		df-ne 2943	. . . . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . . 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 723	. . . . . . . . . . . 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 11910	. . . . . . . . . . . . . . . . 17 ⊢ (𝐿 ∈ ℕ → 𝐿 ∈ ℝ)
74		ltne 11002	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ ℝ ∧ 𝐿 < (♯‘𝑊)) → (♯‘𝑊) ≠ 𝐿)
75	73, 74	sylan 579	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℕ ∧ 𝐿 < (♯‘𝑊)) → (♯‘𝑊) ≠ 𝐿)
76		df-ne 2943	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑊) ≠ 𝐿 ↔ ¬ (♯‘𝑊) = 𝐿)
77		eqcom 2745	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐿 = (♯‘𝑊) ↔ (♯‘𝑊) = 𝐿)
78		pm2.21 123	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (♯‘𝑊) = 𝐿 → ((♯‘𝑊) = 𝐿 → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
79	77, 78	syl5bi 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 1129	. . . . . . . . . . . . . 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 723	. . . . . . . . . . . 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 1425	. . . . . . . . . 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	syl5bi 241	. . . 4 ⊢ (𝜑 → (∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) → (𝐿 ∈ (1..^(♯‘𝑊)) → ((𝑊 cyclShift 𝐿) = 𝑊 → (𝑊 cyclShift 𝐿) ≠ 𝑊))))
94	93	3imp 1109	. . 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 3027	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 395 ∨ wo 843 ∨ w3o 1084 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 ℝcr 10801 0cc0 10802 1c1 10803 < clt 10940 ℕcn 11903 ℤcz 12249 ...cfz 13168 ..^cfzo 13311 mod cmo 13517 ♯chash 13972 Word cword 14145 repeatS creps 14409 cyclShift ccsh 14429 ℙcprime 16304

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-xnn0 12236 df-z 12250 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-hash 13973 df-word 14146 df-concat 14202 df-substr 14282 df-pfx 14312 df-reps 14410 df-csh 14430 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-dvds 15892 df-gcd 16130 df-prm 16305 df-phi 16395

This theorem is referenced by: cshwshashlem2 16726

W3C validator