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Theorem atantayl2 27272

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 11263	. . . . . . . . . . . . . . . 16 ⊢ i ∈ ℂ
3	2	negcli 11630	. . . . . . . . . . . . . . 15 ⊢ -i ∈ ℂ
4	3	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → -i ∈ ℂ)
5		nnnn0 12617	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
6	5	ad2antlr 740	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 𝑛 ∈ ℕ₀)
7	4, 6	expcld 14293	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → ( -i↑𝑛) ∈ ℂ)
8		sqneg 14262	. . . . . . . . . . . . . . . . 17 ⊢ (i ∈ ℂ → ( -i↑2) = (i↑2))
9	2, 8	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ( -i↑2) = (i↑2)
10	9	oveq1i 7431	. . . . . . . . . . . . . . 15 ⊢ (( -i↑2)↑(𝑛 / 2)) = ((i↑2)↑(𝑛 / 2))
11		ine0 11755	. . . . . . . . . . . . . . . . . 18 ⊢ i ≠ 0
12	2, 11	negne0i 11637	. . . . . . . . . . . . . . . . 17 ⊢ -i ≠ 0
13	12	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → -i ≠ 0)
14		2z 12732	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℤ
15	14	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 2 ∈ ℤ)
16		2ne0 12453	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ≠ 0
17		nnz 12718	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
18	17	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
19		dvdsval2 16431	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ ℤ ∧ 2 ≠ 0 ∧ 𝑛 ∈ ℤ) → (2 ∥ 𝑛 ↔ (𝑛 / 2) ∈ ℤ))
20	14, 16, 18, 19	mp3an12i 1494	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (2 ∥ 𝑛 ↔ (𝑛 / 2) ∈ ℤ))
21	20	biimpa 482	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (𝑛 / 2) ∈ ℤ)
22		expmulz 14255	. . . . . . . . . . . . . . . 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 852	. . . . . . . . . . . . . . 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 14255	. . . . . . . . . . . . . . . 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 852	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (i↑(2 · (𝑛 / 2))) = ((i↑2)↑(𝑛 / 2)))
28	10, 23, 27	3eqtr4a 2821	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → ( -i↑(2 · (𝑛 / 2))) = (i↑(2 · (𝑛 / 2))))
29		nncn 12347	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
30	29	ad2antlr 740	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 𝑛 ∈ ℂ)
31		2cnd 12425	. . . . . . . . . . . . . . . 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 12099	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (2 · (𝑛 / 2)) = 𝑛)
34	33	oveq2d 7437	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → ( -i↑(2 · (𝑛 / 2))) = ( -i↑𝑛))
35	33	oveq2d 7437	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (i↑(2 · (𝑛 / 2))) = (i↑𝑛))
36	28, 34, 35	3eqtr3d 2803	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → ( -i↑𝑛) = (i↑𝑛))
37	7, 36	subeq0bd 11746	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (( -i↑𝑛) − (i↑𝑛)) = 0)
38	37	oveq2d 7437	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (i · (( -i↑𝑛) − (i↑𝑛))) = (i · 0))
39		it0e0 12573	. . . . . . . . . . 11 ⊢ (i · 0) = 0
40	38, 39	eqtrdi 2811	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (i · (( -i↑𝑛) − (i↑𝑛))) = 0)
41	40	oveq1d 7436	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → ((i · (( -i↑𝑛) − (i↑𝑛))) / 2) = (0 / 2))
42		2cn 12422	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
43	42, 16	div0i 12055	. . . . . . . . 9 ⊢ (0 / 2) = 0
44	41, 43	eqtrdi 2811	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → ((i · (( -i↑𝑛) − (i↑𝑛))) / 2) = 0)
45	44	oveq1d 7436	. . . . . . 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 787	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 𝐴 ∈ ℂ)
47	46, 6	expcld 14293	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (𝐴↑𝑛) ∈ ℂ)
48		nnne0 12376	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
49	48	ad2antlr 740	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 𝑛 ≠ 0)
50	47, 30, 49	divcld 12097	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → ((𝐴↑𝑛) / 𝑛) ∈ ℂ)
51	50	mul02d 11512	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (0 · ((𝐴↑𝑛) / 𝑛)) = 0)
52	45, 51	eqtr2d 2796	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 0 = (((i · (( -i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛)))
53		2cnd 12425	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → 2 ∈ ℂ)
54		ax-1cn 11262	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
55	54	negcli 11630	. . . . . . . . . 10 ⊢ -1 ∈ ℂ
56	55	a1i 11	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → -1 ∈ ℂ)
57		neg1ne0 12311	. . . . . . . . . 10 ⊢ -1 ≠ 0
58	57	a1i 11	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → -1 ≠ 0)
59	29	ad2antlr 740	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → 𝑛 ∈ ℂ)
60		peano2cn 11486	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℂ → (𝑛 + 1) ∈ ℂ)
61	59, 60	syl 18	. . . . . . . . . . . . 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 12136	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (((𝑛 + 1) − 2) / 2) = (((𝑛 + 1) / 2) − (2 / 2)))
64		2div2e1 12487	. . . . . . . . . . . . 13 ⊢ (2 / 2) = 1
65	64	oveq2i 7432	. . . . . . . . . . . 12 ⊢ (((𝑛 + 1) / 2) − (2 / 2)) = (((𝑛 + 1) / 2) − 1)
66	63, 65	eqtrdi 2811	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (((𝑛 + 1) − 2) / 2) = (((𝑛 + 1) / 2) − 1))
67		df-2 12409	. . . . . . . . . . . . . 14 ⊢ 2 = (1 + 1)
68	67	oveq2i 7432	. . . . . . . . . . . . 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 11711	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((𝑛 + 1) − (1 + 1)) = (𝑛 − 1))
71	68, 70	eqtrid 2807	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((𝑛 + 1) − 2) = (𝑛 − 1))
72	71	oveq1d 7436	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (((𝑛 + 1) − 2) / 2) = ((𝑛 − 1) / 2))
73	66, 72	eqtr3d 2797	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (((𝑛 + 1) / 2) − 1) = ((𝑛 − 1) / 2))
74	20	notbid 321	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (¬ 2 ∥ 𝑛 ↔ ¬ (𝑛 / 2) ∈ ℤ))
75		zeo 12789	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℤ → ((𝑛 / 2) ∈ ℤ ∨ ((𝑛 + 1) / 2) ∈ ℤ))
76	18, 75	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → ((𝑛 / 2) ∈ ℤ ∨ ((𝑛 + 1) / 2) ∈ ℤ))
77	76	ord 878	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (¬ (𝑛 / 2) ∈ ℤ → ((𝑛 + 1) / 2) ∈ ℤ))
78	74, 77	sylbid 243	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → (¬ 2 ∥ 𝑛 → ((𝑛 + 1) / 2) ∈ ℤ))
79	78	imp 412	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((𝑛 + 1) / 2) ∈ ℤ)
80		peano2zm 12743	. . . . . . . . . . 11 ⊢ (((𝑛 + 1) / 2) ∈ ℤ → (((𝑛 + 1) / 2) − 1) ∈ ℤ)
81	79, 80	syl 18	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (((𝑛 + 1) / 2) − 1) ∈ ℤ)
82	73, 81	eqeltrrd 2861	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((𝑛 − 1) / 2) ∈ ℤ)
83	56, 58, 82	expclzd 14298	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ( -1↑((𝑛 − 1) / 2)) ∈ ℂ)
84	83	2timesd 12593	. . . . . . . . 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 11560	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑛 − 1) ∈ ℂ)
86	59, 54, 85	sylancl 598	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (𝑛 − 1) ∈ ℂ)
87	86, 53, 62	divcan2d 12099	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (2 · ((𝑛 − 1) / 2)) = (𝑛 − 1))
88	87	oveq2d 7437	. . . . . . . . . . . . 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 740	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → 𝑛 ∈ ℤ)
92	89, 90, 91	expm1d 14303	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ( -i↑(𝑛 − 1)) = (( -i↑𝑛) / -i))
93	88, 92	eqtrd 2795	. . . . . . . . . . . 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 14255	. . . . . . . . . . . . 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 852	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ( -i↑(2 · ((𝑛 − 1) / 2))) = (( -i↑2)↑((𝑛 − 1) / 2)))
97	5	ad2antlr 740	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → 𝑛 ∈ ℕ₀)
98		expcl 14226	. . . . . . . . . . . . . 14 ⊢ (( -i ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → ( -i↑𝑛) ∈ ℂ)
99	3, 97, 98	sylancr 599	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ( -i↑𝑛) ∈ ℂ)
100	99, 89, 90	divrec2d 12101	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (( -i↑𝑛) / -i) = ((1 / -i) · ( -i↑𝑛)))
101	93, 96, 100	3eqtr3d 2803	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (( -i↑2)↑((𝑛 − 1) / 2)) = ((1 / -i) · ( -i↑𝑛)))
102		i2 14350	. . . . . . . . . . . . 13 ⊢ (i↑2) = -1
103	9, 102	eqtri 2783	. . . . . . . . . . . 12 ⊢ ( -i↑2) = -1
104	103	oveq1i 7431	. . . . . . . . . . 11 ⊢ (( -i↑2)↑((𝑛 − 1) / 2)) = ( -1↑((𝑛 − 1) / 2))
105		irec 14349	. . . . . . . . . . . . . 14 ⊢ (1 / i) = -i
106	105	negeqi 11554	. . . . . . . . . . . . 13 ⊢ -(1 / i) = - -i
107		divneg2 12045	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℂ ∧ i ∈ ℂ ∧ i ≠ 0) → -(1 / i) = (1 / -i))
108	54, 2, 11, 107	mp3an 1490	. . . . . . . . . . . . 13 ⊢ -(1 / i) = (1 / -i)
109	2	negnegi 11632	. . . . . . . . . . . . 13 ⊢ - -i = i
110	106, 108, 109	3eqtr3i 2791	. . . . . . . . . . . 12 ⊢ (1 / -i) = i
111	110	oveq1i 7431	. . . . . . . . . . 11 ⊢ ((1 / -i) · ( -i↑𝑛)) = (i · ( -i↑𝑛))
112	101, 104, 111	3eqtr3g 2818	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ( -1↑((𝑛 − 1) / 2)) = (i · ( -i↑𝑛)))
113	87	oveq2d 7437	. . . . . . . . . . . . . 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 14303	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (i↑(𝑛 − 1)) = ((i↑𝑛) / i))
117	113, 116	eqtrd 2795	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (i↑(2 · ((𝑛 − 1) / 2))) = ((i↑𝑛) / i))
118		expmulz 14255	. . . . . . . . . . . . . 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 852	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (i↑(2 · ((𝑛 − 1) / 2))) = ((i↑2)↑((𝑛 − 1) / 2)))
120		expcl 14226	. . . . . . . . . . . . . . 15 ⊢ ((i ∈ ℂ ∧ 𝑛 ∈ ℕ₀) → (i↑𝑛) ∈ ℂ)
121	2, 97, 120	sylancr 599	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (i↑𝑛) ∈ ℂ)
122	121, 114, 115	divrec2d 12101	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((i↑𝑛) / i) = ((1 / i) · (i↑𝑛)))
123	117, 119, 122	3eqtr3d 2803	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((i↑2)↑((𝑛 − 1) / 2)) = ((1 / i) · (i↑𝑛)))
124	102	oveq1i 7431	. . . . . . . . . . . 12 ⊢ ((i↑2)↑((𝑛 − 1) / 2)) = ( -1↑((𝑛 − 1) / 2))
125	105	oveq1i 7431	. . . . . . . . . . . 12 ⊢ ((1 / i) · (i↑𝑛)) = ( -i · (i↑𝑛))
126	123, 124, 125	3eqtr3g 2818	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ( -1↑((𝑛 − 1) / 2)) = ( -i · (i↑𝑛)))
127		mulneg1 11756	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ (i↑𝑛) ∈ ℂ) → ( -i · (i↑𝑛)) = -(i · (i↑𝑛)))
128	2, 121, 127	sylancr 599	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ( -i · (i↑𝑛)) = -(i · (i↑𝑛)))
129	126, 128	eqtrd 2795	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ( -1↑((𝑛 − 1) / 2)) = -(i · (i↑𝑛)))
130	112, 129	oveq12d 7439	. . . . . . . . 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 11288	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ ( -i↑𝑛) ∈ ℂ) → (i · ( -i↑𝑛)) ∈ ℂ)
132	2, 99, 131	sylancr 599	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (i · ( -i↑𝑛)) ∈ ℂ)
133		mulcl 11288	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ (i↑𝑛) ∈ ℂ) → (i · (i↑𝑛)) ∈ ℂ)
134	2, 121, 133	sylancr 599	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (i · (i↑𝑛)) ∈ ℂ)
135	132, 134	negsubd 11679	. . . . . . . . . 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 11776	. . . . . . . . . 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 2798	. . . . . . . . 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 2799	. . . . . . . 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 12153	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ( -1↑((𝑛 − 1) / 2)) = ((i · (( -i↑𝑛) − (i↑𝑛))) / 2))
140	139	oveq1d 7436	. . . . . 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 4518	. . . . 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 5197	. . . 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 2807	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐹 = (𝑛 ∈ ℕ ↦ (((i · (( -i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛))))
144	143	seqeq3d 14156	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , 𝐹) = seq1( + , (𝑛 ∈ ℕ ↦ (((i · (( -i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛)))))
145		eqid 2760	. . 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 27271	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (((i · (( -i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛)))) ⇝ (arctan‘𝐴))
147	144, 146	eqbrtrd 5126	1 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , 𝐹) ⇝ (arctan‘𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ifcif 4481 class class class wbr 5102 ↦ cmpt 5185 ‘cfv 6539 (class class class)co 7421 ℂcc 11202 0cc0 11204 1c1 11205 ici 11206 + caddc 11207 · cmul 11209 < clt 11347 − cmin 11545 -cneg 11546 / cdiv 11977 ℕcn 12339 2c2 12401 ℕ₀cn0 12610 ℤcz 12697 seqcseq 14148 ↑cexp 14208 abscabs 15407 ⇝ cli 15657 ∥ cdvds 16428 arctancatan 27198

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-inf2 9646 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281 ax-pre-sup 11282 ax-addf 11283

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-of 7694 df-om 7879 df-1st 8002 df-2nd 8003 df-supp 8181 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-2o 8481 df-er 8721 df-map 8853 df-pm 8854 df-ixp 8930 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-fsupp 9358 df-fi 9407 df-sup 9438 df-inf 9439 df-oi 9508 df-card 10024 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-div 11978 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-5 12412 df-6 12413 df-7 12414 df-8 12415 df-9 12416 df-n0 12611 df-z 12698 df-dec 12819 df-uz 12970 df-q 13080 df-rp 13125 df-xneg 13245 df-xadd 13246 df-xmul 13247 df-ioo 13484 df-ioc 13485 df-ico 13486 df-icc 13487 df-fz 13644 df-fzo 13793 df-fl 13936 df-mod 14014 df-seq 14149 df-exp 14209 df-fac 14422 df-bc 14451 df-hash 14479 df-shft 15226 df-cj 15272 df-re 15273 df-im 15274 df-sqrt 15408 df-abs 15409 df-limsup 15644 df-clim 15661 df-rlim 15662 df-sum 15860 df-ef 16239 df-sin 16241 df-cos 16242 df-tan 16243 df-pi 16244 df-dvds 16429 df-struct 17331 df-sets 17348 df-slot 17366 df-ndx 17378 df-base 17394 df-ress 17415 df-plusg 17447 df-mulr 17448 df-starv 17449 df-sca 17450 df-vsca 17451 df-ip 17452 df-tset 17453 df-ple 17454 df-ds 17456 df-unif 17457 df-hom 17458 df-cco 17459 df-rest 17599 df-topn 17600 df-0g 17618 df-gsum 17619 df-topgen 17620 df-pt 17621 df-prds 17624 df-xrs 17680 df-qtop 17685 df-imas 17686 df-xps 17688 df-mre 17762 df-mrc 17763 df-acs 17765 df-mgm 18822 df-sgrp 18914 df-mnd 18930 df-submnd 18985 df-mulg 19284 df-cntz 19537 df-cmn 20002 df-psmet 21676 df-xmet 21677 df-met 21678 df-bl 21679 df-mopn 21680 df-fbas 21681 df-fg 21682 df-cnfld 21685 df-top 23218 df-topon 23235 df-topsp 23257 df-bases 23270 df-cld 23343 df-ntr 23344 df-cls 23345 df-nei 23422 df-lp 23460 df-perf 23461 df-cn 23551 df-cnp 23552 df-haus 23639 df-cmp 23711 df-tx 23887 df-hmeo 24080 df-fil 24171 df-fm 24263 df-flim 24264 df-flf 24265 df-xms 24645 df-ms 24646 df-tms 24647 df-cncf 25205 df-limc 26192 df-dv 26193 df-ulm 26710 df-log 26890 df-atan 27201

This theorem is used by: atantayl3 27273 leibpi 27276

W3C validator