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Theorem atantayl2 25993

Description: The Taylor series for arctan(𝐴). (Contributed by Mario Carneiro, 1-Apr-2015.)

**Hypothesis**
Ref	Expression
atantayl2.1	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, ((-1↑((𝑛 − 1) / 2)) · ((𝐴↑𝑛) / 𝑛))))

**Assertion**
Ref	Expression
atantayl2	⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , 𝐹) ⇝ (arctan‘𝐴))

Distinct variable group: 𝐴,𝑛 Allowed substitution hint: 𝐹(𝑛)

**Proof of Theorem atantayl2**
Step	Hyp	Ref	Expression
1		atantayl2.1	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, ((-1↑((𝑛 − 1) / 2)) · ((𝐴↑𝑛) / 𝑛))))
2		ax-icn 10861	. . . . . . . . . . . . . . . 16 ⊢ i ∈ ℂ
3	2	negcli 11219	. . . . . . . . . . . . . . 15 ⊢ -i ∈ ℂ
4	3	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → -i ∈ ℂ)
5		nnnn0 12170	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
6	5	ad2antlr 723	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 𝑛 ∈ ℕ₀)
7	4, 6	expcld 13792	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (-i↑𝑛) ∈ ℂ)
8		sqneg 13764	. . . . . . . . . . . . . . . . 17 ⊢ (i ∈ ℂ → (-i↑2) = (i↑2))
9	2, 8	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (-i↑2) = (i↑2)
10	9	oveq1i 7265	. . . . . . . . . . . . . . 15 ⊢ ((-i↑2)↑(𝑛 / 2)) = ((i↑2)↑(𝑛 / 2))
11		ine0 11340	. . . . . . . . . . . . . . . . . 18 ⊢ i ≠ 0
12	2, 11	negne0i 11226	. . . . . . . . . . . . . . . . 17 ⊢ -i ≠ 0
13	12	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → -i ≠ 0)
14		2z 12282	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℤ
15	14	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 2 ∈ ℤ)
16		2ne0 12007	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ≠ 0
17		nnz 12272	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
18	17	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
19		dvdsval2 15894	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ ℤ ∧ 2 ≠ 0 ∧ 𝑛 ∈ ℤ) → (2 ∥ 𝑛 ↔ (𝑛 / 2) ∈ ℤ))
20	14, 16, 18, 19	mp3an12i 1463	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (2 ∥ 𝑛 ↔ (𝑛 / 2) ∈ ℤ))
21	20	biimpa 476	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (𝑛 / 2) ∈ ℤ)
22		expmulz 13757	. . . . . . . . . . . . . . . 16 ⊢ (((-i ∈ ℂ ∧ -i ≠ 0) ∧ (2 ∈ ℤ ∧ (𝑛 / 2) ∈ ℤ)) → (-i↑(2 · (𝑛 / 2))) = ((-i↑2)↑(𝑛 / 2)))
23	4, 13, 15, 21, 22	syl22anc 835	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (-i↑(2 · (𝑛 / 2))) = ((-i↑2)↑(𝑛 / 2)))
24	2	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → i ∈ ℂ)
25	11	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → i ≠ 0)
26		expmulz 13757	. . . . . . . . . . . . . . . 16 ⊢ (((i ∈ ℂ ∧ i ≠ 0) ∧ (2 ∈ ℤ ∧ (𝑛 / 2) ∈ ℤ)) → (i↑(2 · (𝑛 / 2))) = ((i↑2)↑(𝑛 / 2)))
27	24, 25, 15, 21, 26	syl22anc 835	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (i↑(2 · (𝑛 / 2))) = ((i↑2)↑(𝑛 / 2)))
28	10, 23, 27	3eqtr4a 2805	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (-i↑(2 · (𝑛 / 2))) = (i↑(2 · (𝑛 / 2))))
29		nncn 11911	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
30	29	ad2antlr 723	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 𝑛 ∈ ℂ)
31		2cnd 11981	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 2 ∈ ℂ)
32	16	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 2 ≠ 0)
33	30, 31, 32	divcan2d 11683	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (2 · (𝑛 / 2)) = 𝑛)
34	33	oveq2d 7271	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (-i↑(2 · (𝑛 / 2))) = (-i↑𝑛))
35	33	oveq2d 7271	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (i↑(2 · (𝑛 / 2))) = (i↑𝑛))
36	28, 34, 35	3eqtr3d 2786	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (-i↑𝑛) = (i↑𝑛))
37	7, 36	subeq0bd 11331	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → ((-i↑𝑛) − (i↑𝑛)) = 0)
38	37	oveq2d 7271	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (i · ((-i↑𝑛) − (i↑𝑛))) = (i · 0))
39		it0e0 12125	. . . . . . . . . . 11 ⊢ (i · 0) = 0
40	38, 39	eqtrdi 2795	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (i · ((-i↑𝑛) − (i↑𝑛))) = 0)
41	40	oveq1d 7270	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → ((i · ((-i↑𝑛) − (i↑𝑛))) / 2) = (0 / 2))
42		2cn 11978	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
43	42, 16	div0i 11639	. . . . . . . . 9 ⊢ (0 / 2) = 0
44	41, 43	eqtrdi 2795	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → ((i · ((-i↑𝑛) − (i↑𝑛))) / 2) = 0)
45	44	oveq1d 7270	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (((i · ((-i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛)) = (0 · ((𝐴↑𝑛) / 𝑛)))
46		simplll 771	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 𝐴 ∈ ℂ)
47	46, 6	expcld 13792	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (𝐴↑𝑛) ∈ ℂ)
48		nnne0 11937	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
49	48	ad2antlr 723	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 𝑛 ≠ 0)
50	47, 30, 49	divcld 11681	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → ((𝐴↑𝑛) / 𝑛) ∈ ℂ)
51	50	mul02d 11103	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (0 · ((𝐴↑𝑛) / 𝑛)) = 0)
52	45, 51	eqtr2d 2779	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 0 = (((i · ((-i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛)))
53		2cnd 11981	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → 2 ∈ ℂ)
54		ax-1cn 10860	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
55	54	negcli 11219	. . . . . . . . . 10 ⊢ -1 ∈ ℂ
56	55	a1i 11	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → -1 ∈ ℂ)
57		neg1ne0 12019	. . . . . . . . . 10 ⊢ -1 ≠ 0
58	57	a1i 11	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → -1 ≠ 0)
59	29	ad2antlr 723	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → 𝑛 ∈ ℂ)
60		peano2cn 11077	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℂ → (𝑛 + 1) ∈ ℂ)
61	59, 60	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (𝑛 + 1) ∈ ℂ)
62	16	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → 2 ≠ 0)
63	61, 53, 53, 62	divsubdird 11720	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (((𝑛 + 1) − 2) / 2) = (((𝑛 + 1) / 2) − (2 / 2)))
64		2div2e1 12044	. . . . . . . . . . . . 13 ⊢ (2 / 2) = 1
65	64	oveq2i 7266	. . . . . . . . . . . 12 ⊢ (((𝑛 + 1) / 2) − (2 / 2)) = (((𝑛 + 1) / 2) − 1)
66	63, 65	eqtrdi 2795	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (((𝑛 + 1) − 2) / 2) = (((𝑛 + 1) / 2) − 1))
67		df-2 11966	. . . . . . . . . . . . . 14 ⊢ 2 = (1 + 1)
68	67	oveq2i 7266	. . . . . . . . . . . . 13 ⊢ ((𝑛 + 1) − 2) = ((𝑛 + 1) − (1 + 1))
69	54	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → 1 ∈ ℂ)
70	59, 69, 69	pnpcan2d 11300	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((𝑛 + 1) − (1 + 1)) = (𝑛 − 1))
71	68, 70	syl5eq 2791	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((𝑛 + 1) − 2) = (𝑛 − 1))
72	71	oveq1d 7270	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (((𝑛 + 1) − 2) / 2) = ((𝑛 − 1) / 2))
73	66, 72	eqtr3d 2780	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (((𝑛 + 1) / 2) − 1) = ((𝑛 − 1) / 2))
74	20	notbid 317	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (¬ 2 ∥ 𝑛 ↔ ¬ (𝑛 / 2) ∈ ℤ))
75		zeo 12336	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℤ → ((𝑛 / 2) ∈ ℤ ∨ ((𝑛 + 1) / 2) ∈ ℤ))
76	18, 75	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → ((𝑛 / 2) ∈ ℤ ∨ ((𝑛 + 1) / 2) ∈ ℤ))
77	76	ord 860	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (¬ (𝑛 / 2) ∈ ℤ → ((𝑛 + 1) / 2) ∈ ℤ))
78	74, 77	sylbid 239	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (¬ 2 ∥ 𝑛 → ((𝑛 + 1) / 2) ∈ ℤ))
79	78	imp 406	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((𝑛 + 1) / 2) ∈ ℤ)
80		peano2zm 12293	. . . . . . . . . . 11 ⊢ (((𝑛 + 1) / 2) ∈ ℤ → (((𝑛 + 1) / 2) − 1) ∈ ℤ)
81	79, 80	syl 17	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (((𝑛 + 1) / 2) − 1) ∈ ℤ)
82	73, 81	eqeltrrd 2840	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((𝑛 − 1) / 2) ∈ ℤ)
83	56, 58, 82	expclzd 13797	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (-1↑((𝑛 − 1) / 2)) ∈ ℂ)
84	83	2timesd 12146	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (2 · (-1↑((𝑛 − 1) / 2))) = ((-1↑((𝑛 − 1) / 2)) + (-1↑((𝑛 − 1) / 2))))
85		subcl 11150	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑛 − 1) ∈ ℂ)
86	59, 54, 85	sylancl 585	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (𝑛 − 1) ∈ ℂ)
87	86, 53, 62	divcan2d 11683	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (2 · ((𝑛 − 1) / 2)) = (𝑛 − 1))
88	87	oveq2d 7271	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (-i↑(2 · ((𝑛 − 1) / 2))) = (-i↑(𝑛 − 1)))
89	3	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → -i ∈ ℂ)
90	12	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → -i ≠ 0)
91	17	ad2antlr 723	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → 𝑛 ∈ ℤ)
92	89, 90, 91	expm1d 13802	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (-i↑(𝑛 − 1)) = ((-i↑𝑛) / -i))
93	88, 92	eqtrd 2778	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (-i↑(2 · ((𝑛 − 1) / 2))) = ((-i↑𝑛) / -i))
94	14	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → 2 ∈ ℤ)
95		expmulz 13757	. . . . . . . . . . . . 13 ⊢ (((-i ∈ ℂ ∧ -i ≠ 0) ∧ (2 ∈ ℤ ∧ ((𝑛 − 1) / 2) ∈ ℤ)) → (-i↑(2 · ((𝑛 − 1) / 2))) = ((-i↑2)↑((𝑛 − 1) / 2)))
96	89, 90, 94, 82, 95	syl22anc 835	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (-i↑(2 · ((𝑛 − 1) / 2))) = ((-i↑2)↑((𝑛 − 1) / 2)))
97	5	ad2antlr 723	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → 𝑛 ∈ ℕ₀)
98		expcl 13728	. . . . . . . . . . . . . 14 ⊢ ((-i ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → (-i↑𝑛) ∈ ℂ)
99	3, 97, 98	sylancr 586	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (-i↑𝑛) ∈ ℂ)
100	99, 89, 90	divrec2d 11685	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((-i↑𝑛) / -i) = ((1 / -i) · (-i↑𝑛)))
101	93, 96, 100	3eqtr3d 2786	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((-i↑2)↑((𝑛 − 1) / 2)) = ((1 / -i) · (-i↑𝑛)))
102		i2 13847	. . . . . . . . . . . . 13 ⊢ (i↑2) = -1
103	9, 102	eqtri 2766	. . . . . . . . . . . 12 ⊢ (-i↑2) = -1
104	103	oveq1i 7265	. . . . . . . . . . 11 ⊢ ((-i↑2)↑((𝑛 − 1) / 2)) = (-1↑((𝑛 − 1) / 2))
105		irec 13846	. . . . . . . . . . . . . 14 ⊢ (1 / i) = -i
106	105	negeqi 11144	. . . . . . . . . . . . 13 ⊢ -(1 / i) = --i
107		divneg2 11629	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℂ ∧ i ∈ ℂ ∧ i ≠ 0) → -(1 / i) = (1 / -i))
108	54, 2, 11, 107	mp3an 1459	. . . . . . . . . . . . 13 ⊢ -(1 / i) = (1 / -i)
109	2	negnegi 11221	. . . . . . . . . . . . 13 ⊢ --i = i
110	106, 108, 109	3eqtr3i 2774	. . . . . . . . . . . 12 ⊢ (1 / -i) = i
111	110	oveq1i 7265	. . . . . . . . . . 11 ⊢ ((1 / -i) · (-i↑𝑛)) = (i · (-i↑𝑛))
112	101, 104, 111	3eqtr3g 2802	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (-1↑((𝑛 − 1) / 2)) = (i · (-i↑𝑛)))
113	87	oveq2d 7271	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (i↑(2 · ((𝑛 − 1) / 2))) = (i↑(𝑛 − 1)))
114	2	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → i ∈ ℂ)
115	11	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → i ≠ 0)
116	114, 115, 91	expm1d 13802	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (i↑(𝑛 − 1)) = ((i↑𝑛) / i))
117	113, 116	eqtrd 2778	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (i↑(2 · ((𝑛 − 1) / 2))) = ((i↑𝑛) / i))
118		expmulz 13757	. . . . . . . . . . . . . 14 ⊢ (((i ∈ ℂ ∧ i ≠ 0) ∧ (2 ∈ ℤ ∧ ((𝑛 − 1) / 2) ∈ ℤ)) → (i↑(2 · ((𝑛 − 1) / 2))) = ((i↑2)↑((𝑛 − 1) / 2)))
119	114, 115, 94, 82, 118	syl22anc 835	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (i↑(2 · ((𝑛 − 1) / 2))) = ((i↑2)↑((𝑛 − 1) / 2)))
120		expcl 13728	. . . . . . . . . . . . . . 15 ⊢ ((i ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → (i↑𝑛) ∈ ℂ)
121	2, 97, 120	sylancr 586	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (i↑𝑛) ∈ ℂ)
122	121, 114, 115	divrec2d 11685	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((i↑𝑛) / i) = ((1 / i) · (i↑𝑛)))
123	117, 119, 122	3eqtr3d 2786	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((i↑2)↑((𝑛 − 1) / 2)) = ((1 / i) · (i↑𝑛)))
124	102	oveq1i 7265	. . . . . . . . . . . 12 ⊢ ((i↑2)↑((𝑛 − 1) / 2)) = (-1↑((𝑛 − 1) / 2))
125	105	oveq1i 7265	. . . . . . . . . . . 12 ⊢ ((1 / i) · (i↑𝑛)) = (-i · (i↑𝑛))
126	123, 124, 125	3eqtr3g 2802	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (-1↑((𝑛 − 1) / 2)) = (-i · (i↑𝑛)))
127		mulneg1 11341	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ (i↑𝑛) ∈ ℂ) → (-i · (i↑𝑛)) = -(i · (i↑𝑛)))
128	2, 121, 127	sylancr 586	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (-i · (i↑𝑛)) = -(i · (i↑𝑛)))
129	126, 128	eqtrd 2778	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (-1↑((𝑛 − 1) / 2)) = -(i · (i↑𝑛)))
130	112, 129	oveq12d 7273	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((-1↑((𝑛 − 1) / 2)) + (-1↑((𝑛 − 1) / 2))) = ((i · (-i↑𝑛)) + -(i · (i↑𝑛))))
131		mulcl 10886	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ (-i↑𝑛) ∈ ℂ) → (i · (-i↑𝑛)) ∈ ℂ)
132	2, 99, 131	sylancr 586	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (i · (-i↑𝑛)) ∈ ℂ)
133		mulcl 10886	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ (i↑𝑛) ∈ ℂ) → (i · (i↑𝑛)) ∈ ℂ)
134	2, 121, 133	sylancr 586	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (i · (i↑𝑛)) ∈ ℂ)
135	132, 134	negsubd 11268	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((i · (-i↑𝑛)) + -(i · (i↑𝑛))) = ((i · (-i↑𝑛)) − (i · (i↑𝑛))))
136	114, 99, 121	subdid 11361	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (i · ((-i↑𝑛) − (i↑𝑛))) = ((i · (-i↑𝑛)) − (i · (i↑𝑛))))
137	135, 136	eqtr4d 2781	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((i · (-i↑𝑛)) + -(i · (i↑𝑛))) = (i · ((-i↑𝑛) − (i↑𝑛))))
138	84, 130, 137	3eqtrd 2782	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (2 · (-1↑((𝑛 − 1) / 2))) = (i · ((-i↑𝑛) − (i↑𝑛))))
139	53, 83, 62, 138	mvllmuld 11737	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (-1↑((𝑛 − 1) / 2)) = ((i · ((-i↑𝑛) − (i↑𝑛))) / 2))
140	139	oveq1d 7270	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((-1↑((𝑛 − 1) / 2)) · ((𝐴↑𝑛) / 𝑛)) = (((i · ((-i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛)))
141	52, 140	ifeqda 4492	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → if(2 ∥ 𝑛, 0, ((-1↑((𝑛 − 1) / 2)) · ((𝐴↑𝑛) / 𝑛))) = (((i · ((-i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛)))
142	141	mpteq2dva 5170	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, ((-1↑((𝑛 − 1) / 2)) · ((𝐴↑𝑛) / 𝑛)))) = (𝑛 ∈ ℕ ↦ (((i · ((-i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛))))
143	1, 142	syl5eq 2791	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐹 = (𝑛 ∈ ℕ ↦ (((i · ((-i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛))))
144	143	seqeq3d 13657	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , 𝐹) = seq1( + , (𝑛 ∈ ℕ ↦ (((i · ((-i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛)))))
145		eqid 2738	. . 3 ⊢ (𝑛 ∈ ℕ ↦ (((i · ((-i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛))) = (𝑛 ∈ ℕ ↦ (((i · ((-i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛)))
146	145	atantayl 25992	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (((i · ((-i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛)))) ⇝ (arctan‘𝐴))
147	144, 146	eqbrtrd 5092	1 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , 𝐹) ⇝ (arctan‘𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ifcif 4456 class class class wbr 5070 ↦ cmpt 5153 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 0cc0 10802 1c1 10803 ici 10804 + caddc 10805 · cmul 10807 < clt 10940 − cmin 11135 -cneg 11136 / cdiv 11562 ℕcn 11903 2c2 11958 ℕ₀cn0 12163 ℤcz 12249 seqcseq 13649 ↑cexp 13710 abscabs 14873 ⇝ cli 15121 ∥ cdvds 15891 arctancatan 25919

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-inf2 9329 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 ax-addf 10881 ax-mulf 10882

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-iin 4924 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-se 5536 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-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 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-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-tan 15709 df-pi 15710 df-dvds 15892 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-cmp 22446 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936 df-ulm 25441 df-log 25617 df-atan 25922

This theorem is referenced by: atantayl3 25994 leibpi 25997

W3C validator