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Theorem cshwshashlem2 15727

Description: If cyclically shifting a word of length being a prime number and not of identical symbols by different numbers of positions, the resulting words are different. (Contributed by Alexander van der Vekens, 19-May-2018.) (Revised by Alexander van der Vekens, 8-Jun-2018.)

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

**Assertion**
Ref	Expression
cshwshashlem2	⊢ ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾)))

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

**Proof of Theorem cshwshashlem2**
Step	Hyp	Ref	Expression
1		oveq1 6611	. . . . . . . 8 ⊢ ((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) → ((𝑊 cyclShift 𝐿) cyclShift ((#‘𝑊) − 𝐿)) = ((𝑊 cyclShift 𝐾) cyclShift ((#‘𝑊) − 𝐿)))
2	1	eqcomd 2627	. . . . . . 7 ⊢ ((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) → ((𝑊 cyclShift 𝐾) cyclShift ((#‘𝑊) − 𝐿)) = ((𝑊 cyclShift 𝐿) cyclShift ((#‘𝑊) − 𝐿)))
3	2	ad2antrr 761	. . . . . 6 ⊢ ((((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) ∧ (𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0))) ∧ (𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿)) → ((𝑊 cyclShift 𝐾) cyclShift ((#‘𝑊) − 𝐿)) = ((𝑊 cyclShift 𝐿) cyclShift ((#‘𝑊) − 𝐿)))
4		cshwshash.0	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) ∈ ℙ))
5	4	simpld 475	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ Word 𝑉)
6	5	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → 𝑊 ∈ Word 𝑉)
7	6	adantl 482	. . . . . . . 8 ⊢ (((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) ∧ (𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0))) → 𝑊 ∈ Word 𝑉)
8	7	adantr 481	. . . . . . 7 ⊢ ((((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) ∧ (𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0))) ∧ (𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿)) → 𝑊 ∈ Word 𝑉)
9		elfzofz 12426	. . . . . . . . 9 ⊢ (𝐾 ∈ (0..^(#‘𝑊)) → 𝐾 ∈ (0...(#‘𝑊)))
10	9	3ad2ant2 1081	. . . . . . . 8 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → 𝐾 ∈ (0...(#‘𝑊)))
11	10	adantl 482	. . . . . . 7 ⊢ ((((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) ∧ (𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0))) ∧ (𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿)) → 𝐾 ∈ (0...(#‘𝑊)))
12		elfzofz 12426	. . . . . . . . . 10 ⊢ (𝐿 ∈ (0..^(#‘𝑊)) → 𝐿 ∈ (0...(#‘𝑊)))
13		fznn0sub2 12387	. . . . . . . . . 10 ⊢ (𝐿 ∈ (0...(#‘𝑊)) → ((#‘𝑊) − 𝐿) ∈ (0...(#‘𝑊)))
14	12, 13	syl 17	. . . . . . . . 9 ⊢ (𝐿 ∈ (0..^(#‘𝑊)) → ((#‘𝑊) − 𝐿) ∈ (0...(#‘𝑊)))
15	14	3ad2ant1 1080	. . . . . . . 8 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → ((#‘𝑊) − 𝐿) ∈ (0...(#‘𝑊)))
16	15	adantl 482	. . . . . . 7 ⊢ ((((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) ∧ (𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0))) ∧ (𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿)) → ((#‘𝑊) − 𝐿) ∈ (0...(#‘𝑊)))
17		elfzo0 12449	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ (0..^(#‘𝑊)) ↔ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)))
18		zre 11325	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℝ)
19	18	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ ℤ ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ)) → 𝐾 ∈ ℝ)
20		nnre 10971	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝑊) ∈ ℕ → (#‘𝑊) ∈ ℝ)
21		nn0re 11245	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐿 ∈ ℕ₀ → 𝐿 ∈ ℝ)
22		resubcl 10289	. . . . . . . . . . . . . . . . . . 19 ⊢ (((#‘𝑊) ∈ ℝ ∧ 𝐿 ∈ ℝ) → ((#‘𝑊) − 𝐿) ∈ ℝ)
23	20, 21, 22	syl2anr 495	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ) → ((#‘𝑊) − 𝐿) ∈ ℝ)
24	23	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ ℤ ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ)) → ((#‘𝑊) − 𝐿) ∈ ℝ)
25	19, 24	readdcld 10013	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ ℤ ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ)) → (𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℝ)
26	20	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ) → (#‘𝑊) ∈ ℝ)
27	26	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ ℤ ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ)) → (#‘𝑊) ∈ ℝ)
28	25, 27	jca 554	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ ℤ ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ)) → ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℝ ∧ (#‘𝑊) ∈ ℝ))
29	28	ex 450	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℤ → ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ) → ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℝ ∧ (#‘𝑊) ∈ ℝ)))
30		elfzoelz 12411	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (0..^(#‘𝑊)) → 𝐾 ∈ ℤ)
31	29, 30	syl11 33	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ) → (𝐾 ∈ (0..^(#‘𝑊)) → ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℝ ∧ (#‘𝑊) ∈ ℝ)))
32	31	3adant3 1079	. . . . . . . . . . . 12 ⊢ ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)) → (𝐾 ∈ (0..^(#‘𝑊)) → ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℝ ∧ (#‘𝑊) ∈ ℝ)))
33	17, 32	sylbi 207	. . . . . . . . . . 11 ⊢ (𝐿 ∈ (0..^(#‘𝑊)) → (𝐾 ∈ (0..^(#‘𝑊)) → ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℝ ∧ (#‘𝑊) ∈ ℝ)))
34	33	imp 445	. . . . . . . . . 10 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊))) → ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℝ ∧ (#‘𝑊) ∈ ℝ))
35	34	3adant3 1079	. . . . . . . . 9 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℝ ∧ (#‘𝑊) ∈ ℝ))
36		fzonmapblen 12454	. . . . . . . . 9 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → (𝐾 + ((#‘𝑊) − 𝐿)) < (#‘𝑊))
37		ltle 10070	. . . . . . . . 9 ⊢ (((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℝ ∧ (#‘𝑊) ∈ ℝ) → ((𝐾 + ((#‘𝑊) − 𝐿)) < (#‘𝑊) → (𝐾 + ((#‘𝑊) − 𝐿)) ≤ (#‘𝑊)))
38	35, 36, 37	sylc 65	. . . . . . . 8 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → (𝐾 + ((#‘𝑊) − 𝐿)) ≤ (#‘𝑊))
39	38	adantl 482	. . . . . . 7 ⊢ ((((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) ∧ (𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0))) ∧ (𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿)) → (𝐾 + ((#‘𝑊) − 𝐿)) ≤ (#‘𝑊))
40		simpl 473	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝐾 ∈ (0...(#‘𝑊)) ∧ ((#‘𝑊) − 𝐿) ∈ (0...(#‘𝑊)) ∧ (𝐾 + ((#‘𝑊) − 𝐿)) ≤ (#‘𝑊))) → 𝑊 ∈ Word 𝑉)
41		elfzelz 12284	. . . . . . . . . 10 ⊢ (𝐾 ∈ (0...(#‘𝑊)) → 𝐾 ∈ ℤ)
42	41	3ad2ant1 1080	. . . . . . . . 9 ⊢ ((𝐾 ∈ (0...(#‘𝑊)) ∧ ((#‘𝑊) − 𝐿) ∈ (0...(#‘𝑊)) ∧ (𝐾 + ((#‘𝑊) − 𝐿)) ≤ (#‘𝑊)) → 𝐾 ∈ ℤ)
43	42	adantl 482	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝐾 ∈ (0...(#‘𝑊)) ∧ ((#‘𝑊) − 𝐿) ∈ (0...(#‘𝑊)) ∧ (𝐾 + ((#‘𝑊) − 𝐿)) ≤ (#‘𝑊))) → 𝐾 ∈ ℤ)
44		elfzelz 12284	. . . . . . . . . 10 ⊢ (((#‘𝑊) − 𝐿) ∈ (0...(#‘𝑊)) → ((#‘𝑊) − 𝐿) ∈ ℤ)
45	44	3ad2ant2 1081	. . . . . . . . 9 ⊢ ((𝐾 ∈ (0...(#‘𝑊)) ∧ ((#‘𝑊) − 𝐿) ∈ (0...(#‘𝑊)) ∧ (𝐾 + ((#‘𝑊) − 𝐿)) ≤ (#‘𝑊)) → ((#‘𝑊) − 𝐿) ∈ ℤ)
46	45	adantl 482	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝐾 ∈ (0...(#‘𝑊)) ∧ ((#‘𝑊) − 𝐿) ∈ (0...(#‘𝑊)) ∧ (𝐾 + ((#‘𝑊) − 𝐿)) ≤ (#‘𝑊))) → ((#‘𝑊) − 𝐿) ∈ ℤ)
47		2cshw 13496	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐾 ∈ ℤ ∧ ((#‘𝑊) − 𝐿) ∈ ℤ) → ((𝑊 cyclShift 𝐾) cyclShift ((#‘𝑊) − 𝐿)) = (𝑊 cyclShift (𝐾 + ((#‘𝑊) − 𝐿))))
48	40, 43, 46, 47	syl3anc 1323	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝐾 ∈ (0...(#‘𝑊)) ∧ ((#‘𝑊) − 𝐿) ∈ (0...(#‘𝑊)) ∧ (𝐾 + ((#‘𝑊) − 𝐿)) ≤ (#‘𝑊))) → ((𝑊 cyclShift 𝐾) cyclShift ((#‘𝑊) − 𝐿)) = (𝑊 cyclShift (𝐾 + ((#‘𝑊) − 𝐿))))
49	8, 11, 16, 39, 48	syl13anc 1325	. . . . . 6 ⊢ ((((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) ∧ (𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0))) ∧ (𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿)) → ((𝑊 cyclShift 𝐾) cyclShift ((#‘𝑊) − 𝐿)) = (𝑊 cyclShift (𝐾 + ((#‘𝑊) − 𝐿))))
50	12	3ad2ant1 1080	. . . . . . 7 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → 𝐿 ∈ (0...(#‘𝑊)))
51		elfzelz 12284	. . . . . . . 8 ⊢ (𝐿 ∈ (0...(#‘𝑊)) → 𝐿 ∈ ℤ)
52		2cshwid 13497	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) → ((𝑊 cyclShift 𝐿) cyclShift ((#‘𝑊) − 𝐿)) = 𝑊)
53	51, 52	sylan2 491	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(#‘𝑊))) → ((𝑊 cyclShift 𝐿) cyclShift ((#‘𝑊) − 𝐿)) = 𝑊)
54	7, 50, 53	syl2an 494	. . . . . 6 ⊢ ((((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) ∧ (𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0))) ∧ (𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿)) → ((𝑊 cyclShift 𝐿) cyclShift ((#‘𝑊) − 𝐿)) = 𝑊)
55	3, 49, 54	3eqtr3d 2663	. . . . 5 ⊢ ((((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) ∧ (𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0))) ∧ (𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿)) → (𝑊 cyclShift (𝐾 + ((#‘𝑊) − 𝐿))) = 𝑊)
56		simplrl 799	. . . . . 6 ⊢ ((((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) ∧ (𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0))) ∧ (𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿)) → 𝜑)
57		simplrr 800	. . . . . 6 ⊢ ((((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) ∧ (𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0))) ∧ (𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿)) → ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0))
58		3simpa 1056	. . . . . . . . . . . 12 ⊢ ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)) → (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ))
59	17, 58	sylbi 207	. . . . . . . . . . 11 ⊢ (𝐿 ∈ (0..^(#‘𝑊)) → (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ))
60		nnz 11343	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑊) ∈ ℕ → (#‘𝑊) ∈ ℤ)
61		nn0z 11344	. . . . . . . . . . . . . . 15 ⊢ (𝐿 ∈ ℕ₀ → 𝐿 ∈ ℤ)
62		zsubcl 11363	. . . . . . . . . . . . . . 15 ⊢ (((#‘𝑊) ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((#‘𝑊) − 𝐿) ∈ ℤ)
63	60, 61, 62	syl2anr 495	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ) → ((#‘𝑊) − 𝐿) ∈ ℤ)
64	63	anim2i 592	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℤ ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ)) → (𝐾 ∈ ℤ ∧ ((#‘𝑊) − 𝐿) ∈ ℤ))
65	64	ancoms 469	. . . . . . . . . . . 12 ⊢ (((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ) ∧ 𝐾 ∈ ℤ) → (𝐾 ∈ ℤ ∧ ((#‘𝑊) − 𝐿) ∈ ℤ))
66		zaddcl 11361	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℤ ∧ ((#‘𝑊) − 𝐿) ∈ ℤ) → (𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℤ)
67	65, 66	syl 17	. . . . . . . . . . 11 ⊢ (((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ) ∧ 𝐾 ∈ ℤ) → (𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℤ)
68	59, 30, 67	syl2an 494	. . . . . . . . . 10 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊))) → (𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℤ)
69	68	3adant3 1079	. . . . . . . . 9 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → (𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℤ)
70		elfzo0 12449	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (0..^(#‘𝑊)) ↔ (𝐾 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐾 < (#‘𝑊)))
71		elnn0z 11334	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ ℕ₀ ↔ (𝐾 ∈ ℤ ∧ 0 ≤ 𝐾))
72	18	ad2antrr 761	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾) ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊))) → 𝐾 ∈ ℝ)
73	23	3adant3 1079	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)) → ((#‘𝑊) − 𝐿) ∈ ℝ)
74	73	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾) ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊))) → ((#‘𝑊) − 𝐿) ∈ ℝ)
75		simplr 791	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾) ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊))) → 0 ≤ 𝐾)
76		posdif 10465	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐿 ∈ ℝ ∧ (#‘𝑊) ∈ ℝ) → (𝐿 < (#‘𝑊) ↔ 0 < ((#‘𝑊) − 𝐿)))
77	21, 20, 76	syl2an 494	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ) → (𝐿 < (#‘𝑊) ↔ 0 < ((#‘𝑊) − 𝐿)))
78	77	biimp3a 1429	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)) → 0 < ((#‘𝑊) − 𝐿))
79	78	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾) ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊))) → 0 < ((#‘𝑊) − 𝐿))
80	72, 74, 75, 79	addgegt0d 10545	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾) ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊))) → 0 < (𝐾 + ((#‘𝑊) − 𝐿)))
81	80	ex 450	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾) → ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)) → 0 < (𝐾 + ((#‘𝑊) − 𝐿))))
82	71, 81	sylbi 207	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ ℕ₀ → ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)) → 0 < (𝐾 + ((#‘𝑊) − 𝐿))))
83	82	3ad2ant1 1080	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐾 < (#‘𝑊)) → ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)) → 0 < (𝐾 + ((#‘𝑊) − 𝐿))))
84	70, 83	sylbi 207	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ (0..^(#‘𝑊)) → ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)) → 0 < (𝐾 + ((#‘𝑊) − 𝐿))))
85	84	com12 32	. . . . . . . . . . . 12 ⊢ ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)) → (𝐾 ∈ (0..^(#‘𝑊)) → 0 < (𝐾 + ((#‘𝑊) − 𝐿))))
86	17, 85	sylbi 207	. . . . . . . . . . 11 ⊢ (𝐿 ∈ (0..^(#‘𝑊)) → (𝐾 ∈ (0..^(#‘𝑊)) → 0 < (𝐾 + ((#‘𝑊) − 𝐿))))
87	86	imp 445	. . . . . . . . . 10 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊))) → 0 < (𝐾 + ((#‘𝑊) − 𝐿)))
88	87	3adant3 1079	. . . . . . . . 9 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → 0 < (𝐾 + ((#‘𝑊) − 𝐿)))
89		elnnz 11331	. . . . . . . . 9 ⊢ ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℕ ↔ ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℤ ∧ 0 < (𝐾 + ((#‘𝑊) − 𝐿))))
90	69, 88, 89	sylanbrc 697	. . . . . . . 8 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → (𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℕ)
91	17	simp2bi 1075	. . . . . . . . 9 ⊢ (𝐿 ∈ (0..^(#‘𝑊)) → (#‘𝑊) ∈ ℕ)
92	91	3ad2ant1 1080	. . . . . . . 8 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → (#‘𝑊) ∈ ℕ)
93		elfzo1 12458	. . . . . . . 8 ⊢ ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ (1..^(#‘𝑊)) ↔ ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℕ ∧ (#‘𝑊) ∈ ℕ ∧ (𝐾 + ((#‘𝑊) − 𝐿)) < (#‘𝑊)))
94	90, 92, 36, 93	syl3anbrc 1244	. . . . . . 7 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → (𝐾 + ((#‘𝑊) − 𝐿)) ∈ (1..^(#‘𝑊)))
95	94	adantl 482	. . . . . 6 ⊢ ((((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) ∧ (𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0))) ∧ (𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿)) → (𝐾 + ((#‘𝑊) − 𝐿)) ∈ (1..^(#‘𝑊)))
96	4	cshwshashlem1 15726	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) ∧ (𝐾 + ((#‘𝑊) − 𝐿)) ∈ (1..^(#‘𝑊))) → (𝑊 cyclShift (𝐾 + ((#‘𝑊) − 𝐿))) ≠ 𝑊)
97	56, 57, 95, 96	syl3anc 1323	. . . . 5 ⊢ ((((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) ∧ (𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0))) ∧ (𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿)) → (𝑊 cyclShift (𝐾 + ((#‘𝑊) − 𝐿))) ≠ 𝑊)
98	55, 97	pm2.21ddne 2874	. . . 4 ⊢ ((((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) ∧ (𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0))) ∧ (𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿)) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾))
99	98	ex 450	. . 3 ⊢ (((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) ∧ (𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0))) → ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾)))
100	99	ex 450	. 2 ⊢ ((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) → ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾))))
101		2a1 28	. 2 ⊢ ((𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾) → ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾))))
102	100, 101	pm2.61ine 2873	1 ⊢ ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1036 = wceq 1480 ∈ wcel 1987 ≠ wne 2790 ∃wrex 2908 class class class wbr 4613 ‘cfv 5847 (class class class)co 6604 ℝcr 9879 0cc0 9880 1c1 9881 + caddc 9883 < clt 10018 ≤ cle 10019 − cmin 10210 ℕcn 10964 ℕ₀cn0 11236 ℤcz 11321 ...cfz 12268 ..^cfzo 12406 #chash 13057 Word cword 13230 cyclShift ccsh 13471 ℙcprime 15309

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1719 ax-4 1734 ax-5 1836 ax-6 1885 ax-7 1932 ax-8 1989 ax-9 1996 ax-10 2016 ax-11 2031 ax-12 2044 ax-13 2245 ax-ext 2601 ax-rep 4731 ax-sep 4741 ax-nul 4749 ax-pow 4803 ax-pr 4867 ax-un 6902 ax-cnex 9936 ax-resscn 9937 ax-1cn 9938 ax-icn 9939 ax-addcl 9940 ax-addrcl 9941 ax-mulcl 9942 ax-mulrcl 9943 ax-mulcom 9944 ax-addass 9945 ax-mulass 9946 ax-distr 9947 ax-i2m1 9948 ax-1ne0 9949 ax-1rid 9950 ax-rnegex 9951 ax-rrecex 9952 ax-cnre 9953 ax-pre-lttri 9954 ax-pre-lttrn 9955 ax-pre-ltadd 9956 ax-pre-mulgt0 9957 ax-pre-sup 9958

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1037 df-3an 1038 df-tru 1483 df-ex 1702 df-nf 1707 df-sb 1878 df-eu 2473 df-mo 2474 df-clab 2608 df-cleq 2614 df-clel 2617 df-nfc 2750 df-ne 2791 df-nel 2894 df-ral 2912 df-rex 2913 df-reu 2914 df-rmo 2915 df-rab 2916 df-v 3188 df-sbc 3418 df-csb 3515 df-dif 3558 df-un 3560 df-in 3562 df-ss 3569 df-pss 3571 df-nul 3892 df-if 4059 df-pw 4132 df-sn 4149 df-pr 4151 df-tp 4153 df-op 4155 df-uni 4403 df-int 4441 df-iun 4487 df-br 4614 df-opab 4674 df-mpt 4675 df-tr 4713 df-eprel 4985 df-id 4989 df-po 4995 df-so 4996 df-fr 5033 df-we 5035 df-xp 5080 df-rel 5081 df-cnv 5082 df-co 5083 df-dm 5084 df-rn 5085 df-res 5086 df-ima 5087 df-pred 5639 df-ord 5685 df-on 5686 df-lim 5687 df-suc 5688 df-iota 5810 df-fun 5849 df-fn 5850 df-f 5851 df-f1 5852 df-fo 5853 df-f1o 5854 df-fv 5855 df-riota 6565 df-ov 6607 df-oprab 6608 df-mpt2 6609 df-om 7013 df-1st 7113 df-2nd 7114 df-wrecs 7352 df-recs 7413 df-rdg 7451 df-1o 7505 df-2o 7506 df-oadd 7509 df-er 7687 df-map 7804 df-en 7900 df-dom 7901 df-sdom 7902 df-fin 7903 df-sup 8292 df-inf 8293 df-card 8709 df-cda 8934 df-pnf 10020 df-mnf 10021 df-xr 10022 df-ltxr 10023 df-le 10024 df-sub 10212 df-neg 10213 df-div 10629 df-nn 10965 df-2 11023 df-3 11024 df-n0 11237 df-xnn0 11308 df-z 11322 df-uz 11632 df-rp 11777 df-fz 12269 df-fzo 12407 df-fl 12533 df-mod 12609 df-seq 12742 df-exp 12801 df-hash 13058 df-word 13238 df-concat 13240 df-substr 13242 df-reps 13245 df-csh 13472 df-cj 13773 df-re 13774 df-im 13775 df-sqrt 13909 df-abs 13910 df-dvds 14908 df-gcd 15141 df-prm 15310 df-phi 15395

This theorem is referenced by: cshwshashlem3 15728

W3C validator