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Theorem cshwshashlem2 15803

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 6657	. . . . . . . 8 ⊢ ((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) → ((𝑊 cyclShift 𝐿) cyclShift ((#‘𝑊) − 𝐿)) = ((𝑊 cyclShift 𝐾) cyclShift ((#‘𝑊) − 𝐿)))
2	1	eqcomd 2628	. . . . . . 7 ⊢ ((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) → ((𝑊 cyclShift 𝐾) cyclShift ((#‘𝑊) − 𝐿)) = ((𝑊 cyclShift 𝐿) cyclShift ((#‘𝑊) − 𝐿)))
3	2	ad2antrr 762	. . . . . 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 12485	. . . . . . . . 9 ⊢ (𝐾 ∈ (0..^(#‘𝑊)) → 𝐾 ∈ (0...(#‘𝑊)))
10	9	3ad2ant2 1083	. . . . . . . 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 12485	. . . . . . . . . 10 ⊢ (𝐿 ∈ (0..^(#‘𝑊)) → 𝐿 ∈ (0...(#‘𝑊)))
13		fznn0sub2 12446	. . . . . . . . . 10 ⊢ (𝐿 ∈ (0...(#‘𝑊)) → ((#‘𝑊) − 𝐿) ∈ (0...(#‘𝑊)))
14	12, 13	syl 17	. . . . . . . . 9 ⊢ (𝐿 ∈ (0..^(#‘𝑊)) → ((#‘𝑊) − 𝐿) ∈ (0...(#‘𝑊)))
15	14	3ad2ant1 1082	. . . . . . . 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 12508	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ (0..^(#‘𝑊)) ↔ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)))
18		zre 11381	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℝ)
19	18	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ ℤ ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ)) → 𝐾 ∈ ℝ)
20		nnre 11027	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝑊) ∈ ℕ → (#‘𝑊) ∈ ℝ)
21		nn0re 11301	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐿 ∈ ℕ₀ → 𝐿 ∈ ℝ)
22		resubcl 10345	. . . . . . . . . . . . . . . . . . 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 10069	. . . . . . . . . . . . . . . 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 12470	. . . . . . . . . . . . . 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 1081	. . . . . . . . . . . 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 1081	. . . . . . . . 9 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℝ ∧ (#‘𝑊) ∈ ℝ))
36		fzonmapblen 12513	. . . . . . . . 9 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → (𝐾 + ((#‘𝑊) − 𝐿)) < (#‘𝑊))
37		ltle 10126	. . . . . . . . 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 12342	. . . . . . . . . 10 ⊢ (𝐾 ∈ (0...(#‘𝑊)) → 𝐾 ∈ ℤ)
42	41	3ad2ant1 1082	. . . . . . . . 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 12342	. . . . . . . . . 10 ⊢ (((#‘𝑊) − 𝐿) ∈ (0...(#‘𝑊)) → ((#‘𝑊) − 𝐿) ∈ ℤ)
45	44	3ad2ant2 1083	. . . . . . . . 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 13559	. . . . . . . 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 1326	. . . . . . 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 1328	. . . . . 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 1082	. . . . . . 7 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → 𝐿 ∈ (0...(#‘𝑊)))
51		elfzelz 12342	. . . . . . . 8 ⊢ (𝐿 ∈ (0...(#‘𝑊)) → 𝐿 ∈ ℤ)
52		2cshwid 13560	. . . . . . . 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 2664	. . . . 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 800	. . . . . 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 801	. . . . . 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 1058	. . . . . . . . . . . 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 11399	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑊) ∈ ℕ → (#‘𝑊) ∈ ℤ)
61		nn0z 11400	. . . . . . . . . . . . . . 15 ⊢ (𝐿 ∈ ℕ₀ → 𝐿 ∈ ℤ)
62		zsubcl 11419	. . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . 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 11417	. . . . . . . . . . . 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 1081	. . . . . . . . 9 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → (𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℤ)
70		elfzo0 12508	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (0..^(#‘𝑊)) ↔ (𝐾 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐾 < (#‘𝑊)))
71		elnn0z 11390	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ ℕ₀ ↔ (𝐾 ∈ ℤ ∧ 0 ≤ 𝐾))
72	18	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾) ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊))) → 𝐾 ∈ ℝ)
73	23	3adant3 1081	. . . . . . . . . . . . . . . . . . 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 792	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾) ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊))) → 0 ≤ 𝐾)
76		posdif 10521	. . . . . . . . . . . . . . . . . . . . 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 1432	. . . . . . . . . . . . . . . . . . 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 10601	. . . . . . . . . . . . . . . . 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 1082	. . . . . . . . . . . . . 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 1081	. . . . . . . . 9 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → 0 < (𝐾 + ((#‘𝑊) − 𝐿)))
89		elnnz 11387	. . . . . . . . 9 ⊢ ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℕ ↔ ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℤ ∧ 0 < (𝐾 + ((#‘𝑊) − 𝐿))))
90	69, 88, 89	sylanbrc 698	. . . . . . . 8 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → (𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℕ)
91	17	simp2bi 1077	. . . . . . . . 9 ⊢ (𝐿 ∈ (0..^(#‘𝑊)) → (#‘𝑊) ∈ ℕ)
92	91	3ad2ant1 1082	. . . . . . . 8 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → (#‘𝑊) ∈ ℕ)
93		elfzo1 12517	. . . . . . . 8 ⊢ ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ (1..^(#‘𝑊)) ↔ ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℕ ∧ (#‘𝑊) ∈ ℕ ∧ (𝐾 + ((#‘𝑊) − 𝐿)) < (#‘𝑊)))
94	90, 92, 36, 93	syl3anbrc 1246	. . . . . . 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 15802	. . . . . 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 1326	. . . . 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 2878	. . . 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 2877	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 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∃wrex 2913 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 < clt 10074 ≤ cle 10075 − cmin 10266 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 cyclShift ccsh 13534 ℙcprime 15385

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-word 13299 df-concat 13301 df-substr 13303 df-reps 13306 df-csh 13535 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-gcd 15217 df-prm 15386 df-phi 15471

This theorem is referenced by: cshwshashlem3 15804

W3C validator