dirkertrigeqlem1 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dirkertrigeqlem1		Structured version Visualization version GIF version

Theorem dirkertrigeqlem1 44801

Description: Sum of an even number of alternating cos values. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Assertion**
Ref	Expression
dirkertrigeqlem1	⊢ (𝐾 ∈ ℕ → Σ𝑛 ∈ (1...(2 · 𝐾))(cos‘(𝑛 · π)) = 0)

Distinct variable group: 𝑛,𝐾

Proof of Theorem dirkertrigeqlem1 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7414	. . . . 5 ⊢ (𝑥 = 1 → (2 · 𝑥) = (2 · 1))
2	1	oveq2d 7422	. . . 4 ⊢ (𝑥 = 1 → (1...(2 · 𝑥)) = (1...(2 · 1)))
3	2	sumeq1d 15644	. . 3 ⊢ (𝑥 = 1 → Σ𝑛 ∈ (1...(2 · 𝑥))(cos‘(𝑛 · π)) = Σ𝑛 ∈ (1...(2 · 1))(cos‘(𝑛 · π)))
4	3	eqeq1d 2735	. 2 ⊢ (𝑥 = 1 → (Σ𝑛 ∈ (1...(2 · 𝑥))(cos‘(𝑛 · π)) = 0 ↔ Σ𝑛 ∈ (1...(2 · 1))(cos‘(𝑛 · π)) = 0))
5		oveq2 7414	. . . . 5 ⊢ (𝑥 = 𝑦 → (2 · 𝑥) = (2 · 𝑦))
6	5	oveq2d 7422	. . . 4 ⊢ (𝑥 = 𝑦 → (1...(2 · 𝑥)) = (1...(2 · 𝑦)))
7	6	sumeq1d 15644	. . 3 ⊢ (𝑥 = 𝑦 → Σ𝑛 ∈ (1...(2 · 𝑥))(cos‘(𝑛 · π)) = Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)))
8	7	eqeq1d 2735	. 2 ⊢ (𝑥 = 𝑦 → (Σ𝑛 ∈ (1...(2 · 𝑥))(cos‘(𝑛 · π)) = 0 ↔ Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) = 0))
9		oveq2 7414	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (2 · 𝑥) = (2 · (𝑦 + 1)))
10	9	oveq2d 7422	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → (1...(2 · 𝑥)) = (1...(2 · (𝑦 + 1))))
11	10	sumeq1d 15644	. . 3 ⊢ (𝑥 = (𝑦 + 1) → Σ𝑛 ∈ (1...(2 · 𝑥))(cos‘(𝑛 · π)) = Σ𝑛 ∈ (1...(2 · (𝑦 + 1)))(cos‘(𝑛 · π)))
12	11	eqeq1d 2735	. 2 ⊢ (𝑥 = (𝑦 + 1) → (Σ𝑛 ∈ (1...(2 · 𝑥))(cos‘(𝑛 · π)) = 0 ↔ Σ𝑛 ∈ (1...(2 · (𝑦 + 1)))(cos‘(𝑛 · π)) = 0))
13		oveq2 7414	. . . . 5 ⊢ (𝑥 = 𝐾 → (2 · 𝑥) = (2 · 𝐾))
14	13	oveq2d 7422	. . . 4 ⊢ (𝑥 = 𝐾 → (1...(2 · 𝑥)) = (1...(2 · 𝐾)))
15	14	sumeq1d 15644	. . 3 ⊢ (𝑥 = 𝐾 → Σ𝑛 ∈ (1...(2 · 𝑥))(cos‘(𝑛 · π)) = Σ𝑛 ∈ (1...(2 · 𝐾))(cos‘(𝑛 · π)))
16	15	eqeq1d 2735	. 2 ⊢ (𝑥 = 𝐾 → (Σ𝑛 ∈ (1...(2 · 𝑥))(cos‘(𝑛 · π)) = 0 ↔ Σ𝑛 ∈ (1...(2 · 𝐾))(cos‘(𝑛 · π)) = 0))
17		ax-1cn 11165	. . . . . 6 ⊢ 1 ∈ ℂ
18	17	2timesi 12347	. . . . 5 ⊢ (2 · 1) = (1 + 1)
19	18	oveq2i 7417	. . . 4 ⊢ (1...(2 · 1)) = (1...(1 + 1))
20	19	sumeq1i 15641	. . 3 ⊢ Σ𝑛 ∈ (1...(2 · 1))(cos‘(𝑛 · π)) = Σ𝑛 ∈ (1...(1 + 1))(cos‘(𝑛 · π))
21		1z 12589	. . . . . . . 8 ⊢ 1 ∈ ℤ
22		uzid 12834	. . . . . . . 8 ⊢ (1 ∈ ℤ → 1 ∈ (ℤ_≥‘1))
23	21, 22	ax-mp 5	. . . . . . 7 ⊢ 1 ∈ (ℤ_≥‘1)
24	23	a1i 11	. . . . . 6 ⊢ (⊤ → 1 ∈ (ℤ_≥‘1))
25		elfzelz 13498	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(1 + 1)) → 𝑛 ∈ ℤ)
26	25	zcnd 12664	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(1 + 1)) → 𝑛 ∈ ℂ)
27	26	adantl 483	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑛 ∈ (1...(1 + 1))) → 𝑛 ∈ ℂ)
28		picn 25961	. . . . . . . . 9 ⊢ π ∈ ℂ
29	28	a1i 11	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑛 ∈ (1...(1 + 1))) → π ∈ ℂ)
30	27, 29	mulcld 11231	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ (1...(1 + 1))) → (𝑛 · π) ∈ ℂ)
31	30	coscld 16071	. . . . . 6 ⊢ ((⊤ ∧ 𝑛 ∈ (1...(1 + 1))) → (cos‘(𝑛 · π)) ∈ ℂ)
32		id 22	. . . . . . . 8 ⊢ (𝑛 = (1 + 1) → 𝑛 = (1 + 1))
33		1p1e2 12334	. . . . . . . 8 ⊢ (1 + 1) = 2
34	32, 33	eqtrdi 2789	. . . . . . 7 ⊢ (𝑛 = (1 + 1) → 𝑛 = 2)
35	34	fvoveq1d 7428	. . . . . 6 ⊢ (𝑛 = (1 + 1) → (cos‘(𝑛 · π)) = (cos‘(2 · π)))
36	24, 31, 35	fsump1 15699	. . . . 5 ⊢ (⊤ → Σ𝑛 ∈ (1...(1 + 1))(cos‘(𝑛 · π)) = (Σ𝑛 ∈ (1...1)(cos‘(𝑛 · π)) + (cos‘(2 · π))))
37	36	mptru 1549	. . . 4 ⊢ Σ𝑛 ∈ (1...(1 + 1))(cos‘(𝑛 · π)) = (Σ𝑛 ∈ (1...1)(cos‘(𝑛 · π)) + (cos‘(2 · π)))
38		coscl 16067	. . . . . . . 8 ⊢ (π ∈ ℂ → (cos‘π) ∈ ℂ)
39	28, 38	ax-mp 5	. . . . . . 7 ⊢ (cos‘π) ∈ ℂ
40		oveq1 7413	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (𝑛 · π) = (1 · π))
41	28	mullidi 11216	. . . . . . . . . 10 ⊢ (1 · π) = π
42	40, 41	eqtrdi 2789	. . . . . . . . 9 ⊢ (𝑛 = 1 → (𝑛 · π) = π)
43	42	fveq2d 6893	. . . . . . . 8 ⊢ (𝑛 = 1 → (cos‘(𝑛 · π)) = (cos‘π))
44	43	fsum1 15690	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ (cos‘π) ∈ ℂ) → Σ𝑛 ∈ (1...1)(cos‘(𝑛 · π)) = (cos‘π))
45	21, 39, 44	mp2an 691	. . . . . 6 ⊢ Σ𝑛 ∈ (1...1)(cos‘(𝑛 · π)) = (cos‘π)
46		cospi 25974	. . . . . 6 ⊢ (cos‘π) = -1
47	45, 46	eqtri 2761	. . . . 5 ⊢ Σ𝑛 ∈ (1...1)(cos‘(𝑛 · π)) = -1
48		cos2pi 25978	. . . . 5 ⊢ (cos‘(2 · π)) = 1
49	47, 48	oveq12i 7418	. . . 4 ⊢ (Σ𝑛 ∈ (1...1)(cos‘(𝑛 · π)) + (cos‘(2 · π))) = (-1 + 1)
50		neg1cn 12323	. . . . 5 ⊢ -1 ∈ ℂ
51		1pneg1e0 12328	. . . . 5 ⊢ (1 + -1) = 0
52	17, 50, 51	addcomli 11403	. . . 4 ⊢ (-1 + 1) = 0
53	37, 49, 52	3eqtri 2765	. . 3 ⊢ Σ𝑛 ∈ (1...(1 + 1))(cos‘(𝑛 · π)) = 0
54	20, 53	eqtri 2761	. 2 ⊢ Σ𝑛 ∈ (1...(2 · 1))(cos‘(𝑛 · π)) = 0
55	18	oveq2i 7417	. . . . . . . 8 ⊢ ((2 · 𝑦) + (2 · 1)) = ((2 · 𝑦) + (1 + 1))
56		2cnd 12287	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → 2 ∈ ℂ)
57		nncn 12217	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
58	17	a1i 11	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → 1 ∈ ℂ)
59	56, 57, 58	adddid 11235	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ → (2 · (𝑦 + 1)) = ((2 · 𝑦) + (2 · 1)))
60	56, 57	mulcld 11231	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → (2 · 𝑦) ∈ ℂ)
61	60, 58, 58	addassd 11233	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ → (((2 · 𝑦) + 1) + 1) = ((2 · 𝑦) + (1 + 1)))
62	55, 59, 61	3eqtr4a 2799	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → (2 · (𝑦 + 1)) = (((2 · 𝑦) + 1) + 1))
63	62	oveq2d 7422	. . . . . 6 ⊢ (𝑦 ∈ ℕ → (1...(2 · (𝑦 + 1))) = (1...(((2 · 𝑦) + 1) + 1)))
64	63	sumeq1d 15644	. . . . 5 ⊢ (𝑦 ∈ ℕ → Σ𝑛 ∈ (1...(2 · (𝑦 + 1)))(cos‘(𝑛 · π)) = Σ𝑛 ∈ (1...(((2 · 𝑦) + 1) + 1))(cos‘(𝑛 · π)))
65	64	adantr 482	. . . 4 ⊢ ((𝑦 ∈ ℕ ∧ Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) = 0) → Σ𝑛 ∈ (1...(2 · (𝑦 + 1)))(cos‘(𝑛 · π)) = Σ𝑛 ∈ (1...(((2 · 𝑦) + 1) + 1))(cos‘(𝑛 · π)))
66		1red 11212	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ → 1 ∈ ℝ)
67		2re 12283	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
68	67	a1i 11	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 2 ∈ ℝ)
69		nnre 12216	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
70	68, 69	remulcld 11241	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → (2 · 𝑦) ∈ ℝ)
71	70, 66	readdcld 11240	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ → ((2 · 𝑦) + 1) ∈ ℝ)
72		2rp 12976	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ⁺
73	72	a1i 11	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 2 ∈ ℝ⁺)
74		nnrp 12982	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ⁺)
75	73, 74	rpmulcld 13029	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → (2 · 𝑦) ∈ ℝ⁺)
76	66, 75	ltaddrp2d 13047	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ → 1 < ((2 · 𝑦) + 1))
77	66, 71, 76	ltled 11359	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → 1 ≤ ((2 · 𝑦) + 1))
78		2z 12591	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
79	78	a1i 11	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 2 ∈ ℤ)
80		nnz 12576	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
81	79, 80	zmulcld 12669	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → (2 · 𝑦) ∈ ℤ)
82	81	peano2zd 12666	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ → ((2 · 𝑦) + 1) ∈ ℤ)
83		eluz 12833	. . . . . . . 8 ⊢ ((1 ∈ ℤ ∧ ((2 · 𝑦) + 1) ∈ ℤ) → (((2 · 𝑦) + 1) ∈ (ℤ_≥‘1) ↔ 1 ≤ ((2 · 𝑦) + 1)))
84	21, 82, 83	sylancr 588	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → (((2 · 𝑦) + 1) ∈ (ℤ_≥‘1) ↔ 1 ≤ ((2 · 𝑦) + 1)))
85	77, 84	mpbird 257	. . . . . 6 ⊢ (𝑦 ∈ ℕ → ((2 · 𝑦) + 1) ∈ (ℤ_≥‘1))
86		elfzelz 13498	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(((2 · 𝑦) + 1) + 1)) → 𝑛 ∈ ℤ)
87	86	zcnd 12664	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(((2 · 𝑦) + 1) + 1)) → 𝑛 ∈ ℂ)
88	28	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(((2 · 𝑦) + 1) + 1)) → π ∈ ℂ)
89	87, 88	mulcld 11231	. . . . . . . 8 ⊢ (𝑛 ∈ (1...(((2 · 𝑦) + 1) + 1)) → (𝑛 · π) ∈ ℂ)
90	89	coscld 16071	. . . . . . 7 ⊢ (𝑛 ∈ (1...(((2 · 𝑦) + 1) + 1)) → (cos‘(𝑛 · π)) ∈ ℂ)
91	90	adantl 483	. . . . . 6 ⊢ ((𝑦 ∈ ℕ ∧ 𝑛 ∈ (1...(((2 · 𝑦) + 1) + 1))) → (cos‘(𝑛 · π)) ∈ ℂ)
92		fvoveq1 7429	. . . . . 6 ⊢ (𝑛 = (((2 · 𝑦) + 1) + 1) → (cos‘(𝑛 · π)) = (cos‘((((2 · 𝑦) + 1) + 1) · π)))
93	85, 91, 92	fsump1 15699	. . . . 5 ⊢ (𝑦 ∈ ℕ → Σ𝑛 ∈ (1...(((2 · 𝑦) + 1) + 1))(cos‘(𝑛 · π)) = (Σ𝑛 ∈ (1...((2 · 𝑦) + 1))(cos‘(𝑛 · π)) + (cos‘((((2 · 𝑦) + 1) + 1) · π))))
94	93	adantr 482	. . . 4 ⊢ ((𝑦 ∈ ℕ ∧ Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) = 0) → Σ𝑛 ∈ (1...(((2 · 𝑦) + 1) + 1))(cos‘(𝑛 · π)) = (Σ𝑛 ∈ (1...((2 · 𝑦) + 1))(cos‘(𝑛 · π)) + (cos‘((((2 · 𝑦) + 1) + 1) · π))))
95		1lt2 12380	. . . . . . . . . . . 12 ⊢ 1 < 2
96	95	a1i 11	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → 1 < 2)
97		2t1e2 12372	. . . . . . . . . . . 12 ⊢ (2 · 1) = 2
98		nnge1 12237	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ → 1 ≤ 𝑦)
99	66, 69, 73	lemul2d 13057	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ → (1 ≤ 𝑦 ↔ (2 · 1) ≤ (2 · 𝑦)))
100	98, 99	mpbid 231	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ → (2 · 1) ≤ (2 · 𝑦))
101	97, 100	eqbrtrrid 5184	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → 2 ≤ (2 · 𝑦))
102	66, 68, 70, 96, 101	ltletrd 11371	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 1 < (2 · 𝑦))
103	66, 70, 102	ltled 11359	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → 1 ≤ (2 · 𝑦))
104		eluz 12833	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ (2 · 𝑦) ∈ ℤ) → ((2 · 𝑦) ∈ (ℤ_≥‘1) ↔ 1 ≤ (2 · 𝑦)))
105	21, 81, 104	sylancr 588	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → ((2 · 𝑦) ∈ (ℤ_≥‘1) ↔ 1 ≤ (2 · 𝑦)))
106	103, 105	mpbird 257	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ → (2 · 𝑦) ∈ (ℤ_≥‘1))
107		elfzelz 13498	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...((2 · 𝑦) + 1)) → 𝑛 ∈ ℤ)
108	107	zcnd 12664	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...((2 · 𝑦) + 1)) → 𝑛 ∈ ℂ)
109	28	a1i 11	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...((2 · 𝑦) + 1)) → π ∈ ℂ)
110	108, 109	mulcld 11231	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...((2 · 𝑦) + 1)) → (𝑛 · π) ∈ ℂ)
111	110	coscld 16071	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...((2 · 𝑦) + 1)) → (cos‘(𝑛 · π)) ∈ ℂ)
112	111	adantl 483	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝑛 ∈ (1...((2 · 𝑦) + 1))) → (cos‘(𝑛 · π)) ∈ ℂ)
113		fvoveq1 7429	. . . . . . . 8 ⊢ (𝑛 = ((2 · 𝑦) + 1) → (cos‘(𝑛 · π)) = (cos‘(((2 · 𝑦) + 1) · π)))
114	106, 112, 113	fsump1 15699	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → Σ𝑛 ∈ (1...((2 · 𝑦) + 1))(cos‘(𝑛 · π)) = (Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) + (cos‘(((2 · 𝑦) + 1) · π))))
115	33, 97	eqtr4i 2764	. . . . . . . . . . . 12 ⊢ (1 + 1) = (2 · 1)
116	115	a1i 11	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → (1 + 1) = (2 · 1))
117	116	oveq2d 7422	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → ((2 · 𝑦) + (1 + 1)) = ((2 · 𝑦) + (2 · 1)))
118	117, 61, 59	3eqtr4d 2783	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → (((2 · 𝑦) + 1) + 1) = (2 · (𝑦 + 1)))
119	118	fvoveq1d 7428	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ → (cos‘((((2 · 𝑦) + 1) + 1) · π)) = (cos‘((2 · (𝑦 + 1)) · π)))
120	57, 58	addcld 11230	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℂ)
121	28	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ → π ∈ ℂ)
122	56, 120, 121	mulassd 11234	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ → ((2 · (𝑦 + 1)) · π) = (2 · ((𝑦 + 1) · π)))
123	122	oveq1d 7421	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → (((2 · (𝑦 + 1)) · π) / (2 · π)) = ((2 · ((𝑦 + 1) · π)) / (2 · π)))
124	120, 121	mulcld 11231	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) · π) ∈ ℂ)
125		0re 11213	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
126		pipos 25962	. . . . . . . . . . . . . 14 ⊢ 0 < π
127	125, 126	gtneii 11323	. . . . . . . . . . . . 13 ⊢ π ≠ 0
128	127	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ → π ≠ 0)
129	73	rpne0d 13018	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ → 2 ≠ 0)
130	124, 121, 56, 128, 129	divcan5d 12013	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → ((2 · ((𝑦 + 1) · π)) / (2 · π)) = (((𝑦 + 1) · π) / π))
131	120, 121, 128	divcan4d 11993	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → (((𝑦 + 1) · π) / π) = (𝑦 + 1))
132	123, 130, 131	3eqtrd 2777	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (((2 · (𝑦 + 1)) · π) / (2 · π)) = (𝑦 + 1))
133	80	peano2zd 12666	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℤ)
134	132, 133	eqeltrd 2834	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → (((2 · (𝑦 + 1)) · π) / (2 · π)) ∈ ℤ)
135		peano2cn 11383	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℂ → (𝑦 + 1) ∈ ℂ)
136	57, 135	syl 17	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℂ)
137	56, 136	mulcld 11231	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → (2 · (𝑦 + 1)) ∈ ℂ)
138	137, 121	mulcld 11231	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → ((2 · (𝑦 + 1)) · π) ∈ ℂ)
139		coseq1 26026	. . . . . . . . . 10 ⊢ (((2 · (𝑦 + 1)) · π) ∈ ℂ → ((cos‘((2 · (𝑦 + 1)) · π)) = 1 ↔ (((2 · (𝑦 + 1)) · π) / (2 · π)) ∈ ℤ))
140	138, 139	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → ((cos‘((2 · (𝑦 + 1)) · π)) = 1 ↔ (((2 · (𝑦 + 1)) · π) / (2 · π)) ∈ ℤ))
141	134, 140	mpbird 257	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ → (cos‘((2 · (𝑦 + 1)) · π)) = 1)
142	119, 141	eqtrd 2773	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → (cos‘((((2 · 𝑦) + 1) + 1) · π)) = 1)
143	114, 142	oveq12d 7424	. . . . . 6 ⊢ (𝑦 ∈ ℕ → (Σ𝑛 ∈ (1...((2 · 𝑦) + 1))(cos‘(𝑛 · π)) + (cos‘((((2 · 𝑦) + 1) + 1) · π))) = ((Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) + (cos‘(((2 · 𝑦) + 1) · π))) + 1))
144	143	adantr 482	. . . . 5 ⊢ ((𝑦 ∈ ℕ ∧ Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) = 0) → (Σ𝑛 ∈ (1...((2 · 𝑦) + 1))(cos‘(𝑛 · π)) + (cos‘((((2 · 𝑦) + 1) + 1) · π))) = ((Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) + (cos‘(((2 · 𝑦) + 1) · π))) + 1))
145		simpr 486	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) = 0) → Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) = 0)
146	60, 58, 121	adddird 11236	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ → (((2 · 𝑦) + 1) · π) = (((2 · 𝑦) · π) + (1 · π)))
147	60, 121	mulcld 11231	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ → ((2 · 𝑦) · π) ∈ ℂ)
148	41, 121	eqeltrid 2838	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ → (1 · π) ∈ ℂ)
149	147, 148	addcomd 11413	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ → (((2 · 𝑦) · π) + (1 · π)) = ((1 · π) + ((2 · 𝑦) · π)))
150	41	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ → (1 · π) = π)
151	56, 57	mulcomd 11232	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ → (2 · 𝑦) = (𝑦 · 2))
152	151	oveq1d 7421	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℕ → ((2 · 𝑦) · π) = ((𝑦 · 2) · π))
153	57, 56, 121	mulassd 11234	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℕ → ((𝑦 · 2) · π) = (𝑦 · (2 · π)))
154	152, 153	eqtrd 2773	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ → ((2 · 𝑦) · π) = (𝑦 · (2 · π)))
155	150, 154	oveq12d 7424	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ → ((1 · π) + ((2 · 𝑦) · π)) = (π + (𝑦 · (2 · π))))
156	146, 149, 155	3eqtrd 2777	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → (((2 · 𝑦) + 1) · π) = (π + (𝑦 · (2 · π))))
157	156	fveq2d 6893	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (cos‘(((2 · 𝑦) + 1) · π)) = (cos‘(π + (𝑦 · (2 · π)))))
158		cosper 25984	. . . . . . . . . . 11 ⊢ ((π ∈ ℂ ∧ 𝑦 ∈ ℤ) → (cos‘(π + (𝑦 · (2 · π)))) = (cos‘π))
159	28, 80, 158	sylancr 588	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (cos‘(π + (𝑦 · (2 · π)))) = (cos‘π))
160	46	a1i 11	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (cos‘π) = -1)
161	157, 159, 160	3eqtrd 2777	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → (cos‘(((2 · 𝑦) + 1) · π)) = -1)
162	161	adantr 482	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) = 0) → (cos‘(((2 · 𝑦) + 1) · π)) = -1)
163	145, 162	oveq12d 7424	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) = 0) → (Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) + (cos‘(((2 · 𝑦) + 1) · π))) = (0 + -1))
164	163	oveq1d 7421	. . . . . 6 ⊢ ((𝑦 ∈ ℕ ∧ Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) = 0) → ((Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) + (cos‘(((2 · 𝑦) + 1) · π))) + 1) = ((0 + -1) + 1))
165	50	addlidi 11399	. . . . . . . 8 ⊢ (0 + -1) = -1
166	165	oveq1i 7416	. . . . . . 7 ⊢ ((0 + -1) + 1) = (-1 + 1)
167	166, 52	eqtri 2761	. . . . . 6 ⊢ ((0 + -1) + 1) = 0
168	164, 167	eqtrdi 2789	. . . . 5 ⊢ ((𝑦 ∈ ℕ ∧ Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) = 0) → ((Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) + (cos‘(((2 · 𝑦) + 1) · π))) + 1) = 0)
169	144, 168	eqtrd 2773	. . . 4 ⊢ ((𝑦 ∈ ℕ ∧ Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) = 0) → (Σ𝑛 ∈ (1...((2 · 𝑦) + 1))(cos‘(𝑛 · π)) + (cos‘((((2 · 𝑦) + 1) + 1) · π))) = 0)
170	65, 94, 169	3eqtrd 2777	. . 3 ⊢ ((𝑦 ∈ ℕ ∧ Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) = 0) → Σ𝑛 ∈ (1...(2 · (𝑦 + 1)))(cos‘(𝑛 · π)) = 0)
171	170	ex 414	. 2 ⊢ (𝑦 ∈ ℕ → (Σ𝑛 ∈ (1...(2 · 𝑦))(cos‘(𝑛 · π)) = 0 → Σ𝑛 ∈ (1...(2 · (𝑦 + 1)))(cos‘(𝑛 · π)) = 0))
172	4, 8, 12, 16, 54, 171	nnind 12227	1 ⊢ (𝐾 ∈ ℕ → Σ𝑛 ∈ (1...(2 · 𝐾))(cos‘(𝑛 · π)) = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 ≠ wne 2941 class class class wbr 5148 ‘cfv 6541 (class class class)co 7406 ℂcc 11105 ℝcr 11106 0cc0 11107 1c1 11108 + caddc 11110 · cmul 11112 < clt 11245 ≤ cle 11246 -cneg 11442 / cdiv 11868 ℕcn 12209 2c2 12264 ℤcz 12555 ℤ_≥cuz 12819 ℝ⁺crp 12971 ...cfz 13481 Σcsu 15629 cosccos 16005 πcpi 16007

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-supp 8144 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-2o 8464 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ioc 13326 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15011 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-limsup 15412 df-clim 15429 df-rlim 15430 df-sum 15630 df-ef 16008 df-sin 16010 df-cos 16011 df-pi 16013 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-starv 17209 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-hom 17218 df-cco 17219 df-rest 17365 df-topn 17366 df-0g 17384 df-gsum 17385 df-topgen 17386 df-pt 17387 df-prds 17390 df-xrs 17445 df-qtop 17450 df-imas 17451 df-xps 17453 df-mre 17527 df-mrc 17528 df-acs 17530 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-submnd 18669 df-mulg 18946 df-cntz 19176 df-cmn 19645 df-psmet 20929 df-xmet 20930 df-met 20931 df-bl 20932 df-mopn 20933 df-fbas 20934 df-fg 20935 df-cnfld 20938 df-top 22388 df-topon 22405 df-topsp 22427 df-bases 22441 df-cld 22515 df-ntr 22516 df-cls 22517 df-nei 22594 df-lp 22632 df-perf 22633 df-cn 22723 df-cnp 22724 df-haus 22811 df-tx 23058 df-hmeo 23251 df-fil 23342 df-fm 23434 df-flim 23435 df-flf 23436 df-xms 23818 df-ms 23819 df-tms 23820 df-cncf 24386 df-limc 25375 df-dv 25376

This theorem is referenced by: dirkertrigeqlem3 44803

W3C validator