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Theorem atantayl2 26288

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 11110	. . . . . . . . . . . . . . . 16 ⊢ i ∈ ℂ
3	2	negcli 11469	. . . . . . . . . . . . . . 15 ⊢ -i ∈ ℂ
4	3	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → -i ∈ ℂ)
5		nnnn0 12420	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
6	5	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 𝑛 ∈ ℕ₀)
7	4, 6	expcld 14051	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (-i↑𝑛) ∈ ℂ)
8		sqneg 14021	. . . . . . . . . . . . . . . . 17 ⊢ (i ∈ ℂ → (-i↑2) = (i↑2))
9	2, 8	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (-i↑2) = (i↑2)
10	9	oveq1i 7367	. . . . . . . . . . . . . . 15 ⊢ ((-i↑2)↑(𝑛 / 2)) = ((i↑2)↑(𝑛 / 2))
11		ine0 11590	. . . . . . . . . . . . . . . . . 18 ⊢ i ≠ 0
12	2, 11	negne0i 11476	. . . . . . . . . . . . . . . . 17 ⊢ -i ≠ 0
13	12	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → -i ≠ 0)
14		2z 12535	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℤ
15	14	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 2 ∈ ℤ)
16		2ne0 12257	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ≠ 0
17		nnz 12520	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
18	17	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
19		dvdsval2 16139	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ ℤ ∧ 2 ≠ 0 ∧ 𝑛 ∈ ℤ) → (2 ∥ 𝑛 ↔ (𝑛 / 2) ∈ ℤ))
20	14, 16, 18, 19	mp3an12i 1465	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (2 ∥ 𝑛 ↔ (𝑛 / 2) ∈ ℤ))
21	20	biimpa 477	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (𝑛 / 2) ∈ ℤ)
22		expmulz 14014	. . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . 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 14014	. . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (i↑(2 · (𝑛 / 2))) = ((i↑2)↑(𝑛 / 2)))
28	10, 23, 27	3eqtr4a 2802	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (-i↑(2 · (𝑛 / 2))) = (i↑(2 · (𝑛 / 2))))
29		nncn 12161	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
30	29	ad2antlr 725	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 𝑛 ∈ ℂ)
31		2cnd 12231	. . . . . . . . . . . . . . . 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 11933	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (2 · (𝑛 / 2)) = 𝑛)
34	33	oveq2d 7373	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (-i↑(2 · (𝑛 / 2))) = (-i↑𝑛))
35	33	oveq2d 7373	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (i↑(2 · (𝑛 / 2))) = (i↑𝑛))
36	28, 34, 35	3eqtr3d 2784	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (-i↑𝑛) = (i↑𝑛))
37	7, 36	subeq0bd 11581	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → ((-i↑𝑛) − (i↑𝑛)) = 0)
38	37	oveq2d 7373	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (i · ((-i↑𝑛) − (i↑𝑛))) = (i · 0))
39		it0e0 12375	. . . . . . . . . . 11 ⊢ (i · 0) = 0
40	38, 39	eqtrdi 2792	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (i · ((-i↑𝑛) − (i↑𝑛))) = 0)
41	40	oveq1d 7372	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → ((i · ((-i↑𝑛) − (i↑𝑛))) / 2) = (0 / 2))
42		2cn 12228	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
43	42, 16	div0i 11889	. . . . . . . . 9 ⊢ (0 / 2) = 0
44	41, 43	eqtrdi 2792	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → ((i · ((-i↑𝑛) − (i↑𝑛))) / 2) = 0)
45	44	oveq1d 7372	. . . . . . 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 773	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 𝐴 ∈ ℂ)
47	46, 6	expcld 14051	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (𝐴↑𝑛) ∈ ℂ)
48		nnne0 12187	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
49	48	ad2antlr 725	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 𝑛 ≠ 0)
50	47, 30, 49	divcld 11931	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → ((𝐴↑𝑛) / 𝑛) ∈ ℂ)
51	50	mul02d 11353	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (0 · ((𝐴↑𝑛) / 𝑛)) = 0)
52	45, 51	eqtr2d 2777	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 0 = (((i · ((-i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛)))
53		2cnd 12231	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → 2 ∈ ℂ)
54		ax-1cn 11109	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
55	54	negcli 11469	. . . . . . . . . 10 ⊢ -1 ∈ ℂ
56	55	a1i 11	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → -1 ∈ ℂ)
57		neg1ne0 12269	. . . . . . . . . 10 ⊢ -1 ≠ 0
58	57	a1i 11	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → -1 ≠ 0)
59	29	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → 𝑛 ∈ ℂ)
60		peano2cn 11327	. . . . . . . . . . . . . 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 11970	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (((𝑛 + 1) − 2) / 2) = (((𝑛 + 1) / 2) − (2 / 2)))
64		2div2e1 12294	. . . . . . . . . . . . 13 ⊢ (2 / 2) = 1
65	64	oveq2i 7368	. . . . . . . . . . . 12 ⊢ (((𝑛 + 1) / 2) − (2 / 2)) = (((𝑛 + 1) / 2) − 1)
66	63, 65	eqtrdi 2792	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (((𝑛 + 1) − 2) / 2) = (((𝑛 + 1) / 2) − 1))
67		df-2 12216	. . . . . . . . . . . . . 14 ⊢ 2 = (1 + 1)
68	67	oveq2i 7368	. . . . . . . . . . . . 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 11550	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((𝑛 + 1) − (1 + 1)) = (𝑛 − 1))
71	68, 70	eqtrid 2788	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((𝑛 + 1) − 2) = (𝑛 − 1))
72	71	oveq1d 7372	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (((𝑛 + 1) − 2) / 2) = ((𝑛 − 1) / 2))
73	66, 72	eqtr3d 2778	. . . . . . . . . 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 12589	. . . . . . . . . . . . . . 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 862	. . . . . . . . . . . . 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 407	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((𝑛 + 1) / 2) ∈ ℤ)
80		peano2zm 12546	. . . . . . . . . . 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 2839	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((𝑛 − 1) / 2) ∈ ℤ)
83	56, 58, 82	expclzd 14056	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (-1↑((𝑛 − 1) / 2)) ∈ ℂ)
84	83	2timesd 12396	. . . . . . . . 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 11400	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑛 − 1) ∈ ℂ)
86	59, 54, 85	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (𝑛 − 1) ∈ ℂ)
87	86, 53, 62	divcan2d 11933	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (2 · ((𝑛 − 1) / 2)) = (𝑛 − 1))
88	87	oveq2d 7373	. . . . . . . . . . . . 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 725	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → 𝑛 ∈ ℤ)
92	89, 90, 91	expm1d 14061	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (-i↑(𝑛 − 1)) = ((-i↑𝑛) / -i))
93	88, 92	eqtrd 2776	. . . . . . . . . . . 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 14014	. . . . . . . . . . . . 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 837	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (-i↑(2 · ((𝑛 − 1) / 2))) = ((-i↑2)↑((𝑛 − 1) / 2)))
97	5	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → 𝑛 ∈ ℕ₀)
98		expcl 13985	. . . . . . . . . . . . . 14 ⊢ ((-i ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → (-i↑𝑛) ∈ ℂ)
99	3, 97, 98	sylancr 587	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (-i↑𝑛) ∈ ℂ)
100	99, 89, 90	divrec2d 11935	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((-i↑𝑛) / -i) = ((1 / -i) · (-i↑𝑛)))
101	93, 96, 100	3eqtr3d 2784	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((-i↑2)↑((𝑛 − 1) / 2)) = ((1 / -i) · (-i↑𝑛)))
102		i2 14106	. . . . . . . . . . . . 13 ⊢ (i↑2) = -1
103	9, 102	eqtri 2764	. . . . . . . . . . . 12 ⊢ (-i↑2) = -1
104	103	oveq1i 7367	. . . . . . . . . . 11 ⊢ ((-i↑2)↑((𝑛 − 1) / 2)) = (-1↑((𝑛 − 1) / 2))
105		irec 14105	. . . . . . . . . . . . . 14 ⊢ (1 / i) = -i
106	105	negeqi 11394	. . . . . . . . . . . . 13 ⊢ -(1 / i) = --i
107		divneg2 11879	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℂ ∧ i ∈ ℂ ∧ i ≠ 0) → -(1 / i) = (1 / -i))
108	54, 2, 11, 107	mp3an 1461	. . . . . . . . . . . . 13 ⊢ -(1 / i) = (1 / -i)
109	2	negnegi 11471	. . . . . . . . . . . . 13 ⊢ --i = i
110	106, 108, 109	3eqtr3i 2772	. . . . . . . . . . . 12 ⊢ (1 / -i) = i
111	110	oveq1i 7367	. . . . . . . . . . 11 ⊢ ((1 / -i) · (-i↑𝑛)) = (i · (-i↑𝑛))
112	101, 104, 111	3eqtr3g 2799	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (-1↑((𝑛 − 1) / 2)) = (i · (-i↑𝑛)))
113	87	oveq2d 7373	. . . . . . . . . . . . . 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 14061	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (i↑(𝑛 − 1)) = ((i↑𝑛) / i))
117	113, 116	eqtrd 2776	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (i↑(2 · ((𝑛 − 1) / 2))) = ((i↑𝑛) / i))
118		expmulz 14014	. . . . . . . . . . . . . 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 837	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (i↑(2 · ((𝑛 − 1) / 2))) = ((i↑2)↑((𝑛 − 1) / 2)))
120		expcl 13985	. . . . . . . . . . . . . . 15 ⊢ ((i ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → (i↑𝑛) ∈ ℂ)
121	2, 97, 120	sylancr 587	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (i↑𝑛) ∈ ℂ)
122	121, 114, 115	divrec2d 11935	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((i↑𝑛) / i) = ((1 / i) · (i↑𝑛)))
123	117, 119, 122	3eqtr3d 2784	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((i↑2)↑((𝑛 − 1) / 2)) = ((1 / i) · (i↑𝑛)))
124	102	oveq1i 7367	. . . . . . . . . . . 12 ⊢ ((i↑2)↑((𝑛 − 1) / 2)) = (-1↑((𝑛 − 1) / 2))
125	105	oveq1i 7367	. . . . . . . . . . . 12 ⊢ ((1 / i) · (i↑𝑛)) = (-i · (i↑𝑛))
126	123, 124, 125	3eqtr3g 2799	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (-1↑((𝑛 − 1) / 2)) = (-i · (i↑𝑛)))
127		mulneg1 11591	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ (i↑𝑛) ∈ ℂ) → (-i · (i↑𝑛)) = -(i · (i↑𝑛)))
128	2, 121, 127	sylancr 587	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (-i · (i↑𝑛)) = -(i · (i↑𝑛)))
129	126, 128	eqtrd 2776	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (-1↑((𝑛 − 1) / 2)) = -(i · (i↑𝑛)))
130	112, 129	oveq12d 7375	. . . . . . . . 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 11135	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ (-i↑𝑛) ∈ ℂ) → (i · (-i↑𝑛)) ∈ ℂ)
132	2, 99, 131	sylancr 587	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (i · (-i↑𝑛)) ∈ ℂ)
133		mulcl 11135	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ (i↑𝑛) ∈ ℂ) → (i · (i↑𝑛)) ∈ ℂ)
134	2, 121, 133	sylancr 587	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (i · (i↑𝑛)) ∈ ℂ)
135	132, 134	negsubd 11518	. . . . . . . . . 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 11611	. . . . . . . . . 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 2779	. . . . . . . . 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 2780	. . . . . . . 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 11987	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (-1↑((𝑛 − 1) / 2)) = ((i · ((-i↑𝑛) − (i↑𝑛))) / 2))
140	139	oveq1d 7372	. . . . . 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 4522	. . . . 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 5205	. . . 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	eqtrid 2788	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐹 = (𝑛 ∈ ℕ ↦ (((i · ((-i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛))))
144	143	seqeq3d 13914	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , 𝐹) = seq1( + , (𝑛 ∈ ℕ ↦ (((i · ((-i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛)))))
145		eqid 2736	. . 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 26287	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (((i · ((-i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛)))) ⇝ (arctan‘𝐴))
147	144, 146	eqbrtrd 5127	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 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ifcif 4486 class class class wbr 5105 ↦ cmpt 5188 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 0cc0 11051 1c1 11052 ici 11053 + caddc 11054 · cmul 11056 < clt 11189 − cmin 11385 -cneg 11386 / cdiv 11812 ℕcn 12153 2c2 12208 ℕ₀cn0 12413 ℤcz 12499 seqcseq 13906 ↑cexp 13967 abscabs 15119 ⇝ cli 15366 ∥ cdvds 16136 arctancatan 26214

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-ef 15950 df-sin 15952 df-cos 15953 df-tan 15954 df-pi 15955 df-dvds 16137 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-limc 25230 df-dv 25231 df-ulm 25736 df-log 25912 df-atan 26217

This theorem is referenced by: atantayl3 26289 leibpi 26292

W3C validator