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Theorem cshwshashlem2 15641

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 6556	. . . . . . . 8 ⊢ ((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) → ((𝑊 cyclShift 𝐿) cyclShift ((#‘𝑊) − 𝐿)) = ((𝑊 cyclShift 𝐾) cyclShift ((#‘𝑊) − 𝐿)))
2	1	eqcomd 2616	. . . . . . 7 ⊢ ((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) → ((𝑊 cyclShift 𝐾) cyclShift ((#‘𝑊) − 𝐿)) = ((𝑊 cyclShift 𝐿) cyclShift ((#‘𝑊) − 𝐿)))
3	2	ad2antrr 758	. . . . . 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 474	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ Word 𝑉)
6	5	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → 𝑊 ∈ Word 𝑉)
7	6	adantl 481	. . . . . . . 8 ⊢ (((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) ∧ (𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0))) → 𝑊 ∈ Word 𝑉)
8	7	adantr 480	. . . . . . 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 12354	. . . . . . . . 9 ⊢ (𝐾 ∈ (0..^(#‘𝑊)) → 𝐾 ∈ (0...(#‘𝑊)))
10	9	3ad2ant2 1076	. . . . . . . 8 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → 𝐾 ∈ (0...(#‘𝑊)))
11	10	adantl 481	. . . . . . 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 12354	. . . . . . . . . 10 ⊢ (𝐿 ∈ (0..^(#‘𝑊)) → 𝐿 ∈ (0...(#‘𝑊)))
13		fznn0sub2 12315	. . . . . . . . . 10 ⊢ (𝐿 ∈ (0...(#‘𝑊)) → ((#‘𝑊) − 𝐿) ∈ (0...(#‘𝑊)))
14	12, 13	syl 17	. . . . . . . . 9 ⊢ (𝐿 ∈ (0..^(#‘𝑊)) → ((#‘𝑊) − 𝐿) ∈ (0...(#‘𝑊)))
15	14	3ad2ant1 1075	. . . . . . . 8 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → ((#‘𝑊) − 𝐿) ∈ (0...(#‘𝑊)))
16	15	adantl 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) − 𝐿) ∈ (0...(#‘𝑊)))
17		elfzo0 12376	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ (0..^(#‘𝑊)) ↔ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)))
18		elfzoelz 12339	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ (0..^(#‘𝑊)) → 𝐾 ∈ ℤ)
19		zre 11258	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℝ)
20	19	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ ℤ ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ)) → 𝐾 ∈ ℝ)
21		nnre 10904	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘𝑊) ∈ ℕ → (#‘𝑊) ∈ ℝ)
22		nn0re 11178	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐿 ∈ ℕ₀ → 𝐿 ∈ ℝ)
23		resubcl 10224	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((#‘𝑊) ∈ ℝ ∧ 𝐿 ∈ ℝ) → ((#‘𝑊) − 𝐿) ∈ ℝ)
24	21, 22, 23	syl2anr 494	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ) → ((#‘𝑊) − 𝐿) ∈ ℝ)
25	24	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ ℤ ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ)) → ((#‘𝑊) − 𝐿) ∈ ℝ)
26	20, 25	readdcld 9948	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ ℤ ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ)) → (𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℝ)
27	21	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ) → (#‘𝑊) ∈ ℝ)
28	27	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ ℤ ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ)) → (#‘𝑊) ∈ ℝ)
29	26, 28	jca 553	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ ℤ ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ)) → ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℝ ∧ (#‘𝑊) ∈ ℝ))
30	29	ex 449	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ ℤ → ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ) → ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℝ ∧ (#‘𝑊) ∈ ℝ)))
31	18, 30	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (0..^(#‘𝑊)) → ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ) → ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℝ ∧ (#‘𝑊) ∈ ℝ)))
32	31	com12 32	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ) → (𝐾 ∈ (0..^(#‘𝑊)) → ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℝ ∧ (#‘𝑊) ∈ ℝ)))
33	32	3adant3 1074	. . . . . . . . . . . 12 ⊢ ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)) → (𝐾 ∈ (0..^(#‘𝑊)) → ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℝ ∧ (#‘𝑊) ∈ ℝ)))
34	17, 33	sylbi 206	. . . . . . . . . . 11 ⊢ (𝐿 ∈ (0..^(#‘𝑊)) → (𝐾 ∈ (0..^(#‘𝑊)) → ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℝ ∧ (#‘𝑊) ∈ ℝ)))
35	34	imp 444	. . . . . . . . . 10 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊))) → ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℝ ∧ (#‘𝑊) ∈ ℝ))
36	35	3adant3 1074	. . . . . . . . 9 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℝ ∧ (#‘𝑊) ∈ ℝ))
37		fzonmapblen 12381	. . . . . . . . 9 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → (𝐾 + ((#‘𝑊) − 𝐿)) < (#‘𝑊))
38		ltle 10005	. . . . . . . . 9 ⊢ (((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℝ ∧ (#‘𝑊) ∈ ℝ) → ((𝐾 + ((#‘𝑊) − 𝐿)) < (#‘𝑊) → (𝐾 + ((#‘𝑊) − 𝐿)) ≤ (#‘𝑊)))
39	36, 37, 38	sylc 63	. . . . . . . 8 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → (𝐾 + ((#‘𝑊) − 𝐿)) ≤ (#‘𝑊))
40	39	adantl 481	. . . . . . 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))
41		simpl 472	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝐾 ∈ (0...(#‘𝑊)) ∧ ((#‘𝑊) − 𝐿) ∈ (0...(#‘𝑊)) ∧ (𝐾 + ((#‘𝑊) − 𝐿)) ≤ (#‘𝑊))) → 𝑊 ∈ Word 𝑉)
42		elfzelz 12213	. . . . . . . . . 10 ⊢ (𝐾 ∈ (0...(#‘𝑊)) → 𝐾 ∈ ℤ)
43	42	3ad2ant1 1075	. . . . . . . . 9 ⊢ ((𝐾 ∈ (0...(#‘𝑊)) ∧ ((#‘𝑊) − 𝐿) ∈ (0...(#‘𝑊)) ∧ (𝐾 + ((#‘𝑊) − 𝐿)) ≤ (#‘𝑊)) → 𝐾 ∈ ℤ)
44	43	adantl 481	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝐾 ∈ (0...(#‘𝑊)) ∧ ((#‘𝑊) − 𝐿) ∈ (0...(#‘𝑊)) ∧ (𝐾 + ((#‘𝑊) − 𝐿)) ≤ (#‘𝑊))) → 𝐾 ∈ ℤ)
45		elfzelz 12213	. . . . . . . . . 10 ⊢ (((#‘𝑊) − 𝐿) ∈ (0...(#‘𝑊)) → ((#‘𝑊) − 𝐿) ∈ ℤ)
46	45	3ad2ant2 1076	. . . . . . . . 9 ⊢ ((𝐾 ∈ (0...(#‘𝑊)) ∧ ((#‘𝑊) − 𝐿) ∈ (0...(#‘𝑊)) ∧ (𝐾 + ((#‘𝑊) − 𝐿)) ≤ (#‘𝑊)) → ((#‘𝑊) − 𝐿) ∈ ℤ)
47	46	adantl 481	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝐾 ∈ (0...(#‘𝑊)) ∧ ((#‘𝑊) − 𝐿) ∈ (0...(#‘𝑊)) ∧ (𝐾 + ((#‘𝑊) − 𝐿)) ≤ (#‘𝑊))) → ((#‘𝑊) − 𝐿) ∈ ℤ)
48		2cshw 13410	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐾 ∈ ℤ ∧ ((#‘𝑊) − 𝐿) ∈ ℤ) → ((𝑊 cyclShift 𝐾) cyclShift ((#‘𝑊) − 𝐿)) = (𝑊 cyclShift (𝐾 + ((#‘𝑊) − 𝐿))))
49	41, 44, 47, 48	syl3anc 1318	. . . . . . 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))))
50	8, 11, 16, 40, 49	syl13anc 1320	. . . . . 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))))
51	12	3ad2ant1 1075	. . . . . . 7 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → 𝐿 ∈ (0...(#‘𝑊)))
52		elfzelz 12213	. . . . . . . 8 ⊢ (𝐿 ∈ (0...(#‘𝑊)) → 𝐿 ∈ ℤ)
53		2cshwid 13411	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ ℤ) → ((𝑊 cyclShift 𝐿) cyclShift ((#‘𝑊) − 𝐿)) = 𝑊)
54	52, 53	sylan2 490	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(#‘𝑊))) → ((𝑊 cyclShift 𝐿) cyclShift ((#‘𝑊) − 𝐿)) = 𝑊)
55	7, 51, 54	syl2an 493	. . . . . 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)
56	3, 50, 55	3eqtr3d 2652	. . . . 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)
57		simplrl 796	. . . . . 6 ⊢ ((((𝑊 cyclShift 𝐿) = (𝑊 cyclShift 𝐾) ∧ (𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0))) ∧ (𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿)) → 𝜑)
58		simplrr 797	. . . . . 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))
59		3simpa 1051	. . . . . . . . . . . 12 ⊢ ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)) → (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ))
60	17, 59	sylbi 206	. . . . . . . . . . 11 ⊢ (𝐿 ∈ (0..^(#‘𝑊)) → (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ))
61		nnz 11276	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑊) ∈ ℕ → (#‘𝑊) ∈ ℤ)
62		nn0z 11277	. . . . . . . . . . . . . . 15 ⊢ (𝐿 ∈ ℕ₀ → 𝐿 ∈ ℤ)
63		zsubcl 11296	. . . . . . . . . . . . . . 15 ⊢ (((#‘𝑊) ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((#‘𝑊) − 𝐿) ∈ ℤ)
64	61, 62, 63	syl2anr 494	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ) → ((#‘𝑊) − 𝐿) ∈ ℤ)
65	64	anim2i 591	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℤ ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ)) → (𝐾 ∈ ℤ ∧ ((#‘𝑊) − 𝐿) ∈ ℤ))
66	65	ancoms 468	. . . . . . . . . . . 12 ⊢ (((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ) ∧ 𝐾 ∈ ℤ) → (𝐾 ∈ ℤ ∧ ((#‘𝑊) − 𝐿) ∈ ℤ))
67		zaddcl 11294	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℤ ∧ ((#‘𝑊) − 𝐿) ∈ ℤ) → (𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℤ)
68	66, 67	syl 17	. . . . . . . . . . 11 ⊢ (((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ) ∧ 𝐾 ∈ ℤ) → (𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℤ)
69	60, 18, 68	syl2an 493	. . . . . . . . . 10 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊))) → (𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℤ)
70	69	3adant3 1074	. . . . . . . . 9 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → (𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℤ)
71		elfzo0 12376	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (0..^(#‘𝑊)) ↔ (𝐾 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐾 < (#‘𝑊)))
72		elnn0z 11267	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ ℕ₀ ↔ (𝐾 ∈ ℤ ∧ 0 ≤ 𝐾))
73	19	ad2antrr 758	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾) ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊))) → 𝐾 ∈ ℝ)
74	24	3adant3 1074	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)) → ((#‘𝑊) − 𝐿) ∈ ℝ)
75	74	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾) ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊))) → ((#‘𝑊) − 𝐿) ∈ ℝ)
76		simplr 788	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾) ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊))) → 0 ≤ 𝐾)
77		posdif 10400	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐿 ∈ ℝ ∧ (#‘𝑊) ∈ ℝ) → (𝐿 < (#‘𝑊) ↔ 0 < ((#‘𝑊) − 𝐿)))
78	22, 21, 77	syl2an 493	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ) → (𝐿 < (#‘𝑊) ↔ 0 < ((#‘𝑊) − 𝐿)))
79	78	biimp3a 1424	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)) → 0 < ((#‘𝑊) − 𝐿))
80	79	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾) ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊))) → 0 < ((#‘𝑊) − 𝐿))
81	73, 75, 76, 80	addgegt0d 10480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾) ∧ (𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊))) → 0 < (𝐾 + ((#‘𝑊) − 𝐿)))
82	81	ex 449	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾) → ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)) → 0 < (𝐾 + ((#‘𝑊) − 𝐿))))
83	72, 82	sylbi 206	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ ℕ₀ → ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)) → 0 < (𝐾 + ((#‘𝑊) − 𝐿))))
84	83	3ad2ant1 1075	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐾 < (#‘𝑊)) → ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)) → 0 < (𝐾 + ((#‘𝑊) − 𝐿))))
85	71, 84	sylbi 206	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ (0..^(#‘𝑊)) → ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)) → 0 < (𝐾 + ((#‘𝑊) − 𝐿))))
86	85	com12 32	. . . . . . . . . . . 12 ⊢ ((𝐿 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 𝐿 < (#‘𝑊)) → (𝐾 ∈ (0..^(#‘𝑊)) → 0 < (𝐾 + ((#‘𝑊) − 𝐿))))
87	17, 86	sylbi 206	. . . . . . . . . . 11 ⊢ (𝐿 ∈ (0..^(#‘𝑊)) → (𝐾 ∈ (0..^(#‘𝑊)) → 0 < (𝐾 + ((#‘𝑊) − 𝐿))))
88	87	imp 444	. . . . . . . . . 10 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊))) → 0 < (𝐾 + ((#‘𝑊) − 𝐿)))
89	88	3adant3 1074	. . . . . . . . 9 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → 0 < (𝐾 + ((#‘𝑊) − 𝐿)))
90		elnnz 11264	. . . . . . . . 9 ⊢ ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℕ ↔ ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℤ ∧ 0 < (𝐾 + ((#‘𝑊) − 𝐿))))
91	70, 89, 90	sylanbrc 695	. . . . . . . 8 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → (𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℕ)
92	17	simp2bi 1070	. . . . . . . . 9 ⊢ (𝐿 ∈ (0..^(#‘𝑊)) → (#‘𝑊) ∈ ℕ)
93	92	3ad2ant1 1075	. . . . . . . 8 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → (#‘𝑊) ∈ ℕ)
94		elfzo1 12385	. . . . . . . 8 ⊢ ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ (1..^(#‘𝑊)) ↔ ((𝐾 + ((#‘𝑊) − 𝐿)) ∈ ℕ ∧ (#‘𝑊) ∈ ℕ ∧ (𝐾 + ((#‘𝑊) − 𝐿)) < (#‘𝑊)))
95	91, 93, 37, 94	syl3anbrc 1239	. . . . . . 7 ⊢ ((𝐿 ∈ (0..^(#‘𝑊)) ∧ 𝐾 ∈ (0..^(#‘𝑊)) ∧ 𝐾 < 𝐿) → (𝐾 + ((#‘𝑊) − 𝐿)) ∈ (1..^(#‘𝑊)))
96	95	adantl 481	. . . . . 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)))
97	4	cshwshashlem1 15640	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0) ∧ (𝐾 + ((#‘𝑊) − 𝐿)) ∈ (1..^(#‘𝑊))) → (𝑊 cyclShift (𝐾 + ((#‘𝑊) − 𝐿))) ≠ 𝑊)
98	57, 58, 96, 97	syl3anc 1318	. . . . 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)
99	56, 98	pm2.21ddne 2866	. . . 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))
100	99	ex 449	. . 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)))
101	100	ex 449	. 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		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))))
103	101, 102	pm2.61ine 2865	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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 − cmin 10145 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 cyclShift ccsh 13385 ℙcprime 15223

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-hash 12980 df-word 13154 df-concat 13156 df-substr 13158 df-reps 13161 df-csh 13386 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-gcd 15055 df-prm 15224 df-phi 15309

This theorem is referenced by: cshwshashlem3 15642

W3C validator