atantayl2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > atantayl2		Structured version Visualization version GIF version

Theorem atantayl2 27312

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 11296	. . . . . . . . . . . . . . . 16 ⊢ i ∈ ℂ
3	2	negcli 11663	. . . . . . . . . . . . . . 15 ⊢ -i ∈ ℂ
4	3	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → -i ∈ ℂ)
5		nnnn0 12651	. . . . . . . . . . . . . . 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 14327	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → ( -i↑𝑛) ∈ ℂ)
8		sqneg 14296	. . . . . . . . . . . . . . . . 17 ⊢ (i ∈ ℂ → ( -i↑2) = (i↑2))
9	2, 8	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ( -i↑2) = (i↑2)
10	9	oveq1i 7459	. . . . . . . . . . . . . . 15 ⊢ (( -i↑2)↑(𝑛 / 2)) = ((i↑2)↑(𝑛 / 2))
11		ine0 11788	. . . . . . . . . . . . . . . . . 18 ⊢ i ≠ 0
12	2, 11	negne0i 11670	. . . . . . . . . . . . . . . . 17 ⊢ -i ≠ 0
13	12	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → -i ≠ 0)
14		2z 12766	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℤ
15	14	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 2 ∈ ℤ)
16		2ne0 12487	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ≠ 0
17		nnz 12752	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
18	17	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
19		dvdsval2 16466	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ ℤ ∧ 2 ≠ 0 ∧ 𝑛 ∈ ℤ) → (2 ∥ 𝑛 ↔ (𝑛 / 2) ∈ ℤ))
20	14, 16, 18, 19	mp3an12i 1495	. . . . . . . . . . . . . . . . 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 14289	. . . . . . . . . . . . . . . 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 14289	. . . . . . . . . . . . . . . 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 2824	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → ( -i↑(2 · (𝑛 / 2))) = (i↑(2 · (𝑛 / 2))))
29		nncn 12381	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
30	29	ad2antlr 740	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 𝑛 ∈ ℂ)
31		2cnd 12459	. . . . . . . . . . . . . . . 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 12133	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (2 · (𝑛 / 2)) = 𝑛)
34	33	oveq2d 7465	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → ( -i↑(2 · (𝑛 / 2))) = ( -i↑𝑛))
35	33	oveq2d 7465	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (i↑(2 · (𝑛 / 2))) = (i↑𝑛))
36	28, 34, 35	3eqtr3d 2806	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → ( -i↑𝑛) = (i↑𝑛))
37	7, 36	subeq0bd 11779	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (( -i↑𝑛) − (i↑𝑛)) = 0)
38	37	oveq2d 7465	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (i · (( -i↑𝑛) − (i↑𝑛))) = (i · 0))
39		it0e0 12607	. . . . . . . . . . 11 ⊢ (i · 0) = 0
40	38, 39	eqtrdi 2814	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (i · (( -i↑𝑛) − (i↑𝑛))) = 0)
41	40	oveq1d 7464	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → ((i · (( -i↑𝑛) − (i↑𝑛))) / 2) = (0 / 2))
42		2cn 12456	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
43	42, 16	div0i 12089	. . . . . . . . 9 ⊢ (0 / 2) = 0
44	41, 43	eqtrdi 2814	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → ((i · (( -i↑𝑛) − (i↑𝑛))) / 2) = 0)
45	44	oveq1d 7464	. . . . . . 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 14327	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (𝐴↑𝑛) ∈ ℂ)
48		nnne0 12410	. . . . . . . . . 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 12131	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → ((𝐴↑𝑛) / 𝑛) ∈ ℂ)
51	50	mul02d 11545	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → (0 · ((𝐴↑𝑛) / 𝑛)) = 0)
52	45, 51	eqtr2d 2799	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ 2 ∥ 𝑛) → 0 = (((i · (( -i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛)))
53		2cnd 12459	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → 2 ∈ ℂ)
54		ax-1cn 11295	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
55	54	negcli 11663	. . . . . . . . . 10 ⊢ -1 ∈ ℂ
56	55	a1i 11	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → -1 ∈ ℂ)
57		neg1ne0 12345	. . . . . . . . . 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 11519	. . . . . . . . . . . . . 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 12170	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (((𝑛 + 1) − 2) / 2) = (((𝑛 + 1) / 2) − (2 / 2)))
64		2div2e1 12521	. . . . . . . . . . . . 13 ⊢ (2 / 2) = 1
65	64	oveq2i 7460	. . . . . . . . . . . 12 ⊢ (((𝑛 + 1) / 2) − (2 / 2)) = (((𝑛 + 1) / 2) − 1)
66	63, 65	eqtrdi 2814	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (((𝑛 + 1) − 2) / 2) = (((𝑛 + 1) / 2) − 1))
67		df-2 12443	. . . . . . . . . . . . . 14 ⊢ 2 = (1 + 1)
68	67	oveq2i 7460	. . . . . . . . . . . . 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 11744	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((𝑛 + 1) − (1 + 1)) = (𝑛 − 1))
71	68, 70	eqtrid 2810	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((𝑛 + 1) − 2) = (𝑛 − 1))
72	71	oveq1d 7464	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (((𝑛 + 1) − 2) / 2) = ((𝑛 − 1) / 2))
73	66, 72	eqtr3d 2800	. . . . . . . . . 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 12823	. . . . . . . . . . . . . . 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 12777	. . . . . . . . . . 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 2865	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((𝑛 − 1) / 2) ∈ ℤ)
83	56, 58, 82	expclzd 14332	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ( -1↑((𝑛 − 1) / 2)) ∈ ℂ)
84	83	2timesd 12627	. . . . . . . . 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 11593	. . . . . . . . . . . . . . . 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 12133	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (2 · ((𝑛 − 1) / 2)) = (𝑛 − 1))
88	87	oveq2d 7465	. . . . . . . . . . . . 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 14337	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ( -i↑(𝑛 − 1)) = (( -i↑𝑛) / -i))
93	88, 92	eqtrd 2798	. . . . . . . . . . . 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 14289	. . . . . . . . . . . . 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 14260	. . . . . . . . . . . . . 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 12135	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (( -i↑𝑛) / -i) = ((1 / -i) · ( -i↑𝑛)))
101	93, 96, 100	3eqtr3d 2806	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (( -i↑2)↑((𝑛 − 1) / 2)) = ((1 / -i) · ( -i↑𝑛)))
102		i2 14384	. . . . . . . . . . . . 13 ⊢ (i↑2) = -1
103	9, 102	eqtri 2786	. . . . . . . . . . . 12 ⊢ ( -i↑2) = -1
104	103	oveq1i 7459	. . . . . . . . . . 11 ⊢ (( -i↑2)↑((𝑛 − 1) / 2)) = ( -1↑((𝑛 − 1) / 2))
105		irec 14383	. . . . . . . . . . . . . 14 ⊢ (1 / i) = -i
106	105	negeqi 11587	. . . . . . . . . . . . 13 ⊢ -(1 / i) = - -i
107		divneg2 12079	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℂ ∧ i ∈ ℂ ∧ i ≠ 0) → -(1 / i) = (1 / -i))
108	54, 2, 11, 107	mp3an 1491	. . . . . . . . . . . . 13 ⊢ -(1 / i) = (1 / -i)
109	2	negnegi 11665	. . . . . . . . . . . . 13 ⊢ - -i = i
110	106, 108, 109	3eqtr3i 2794	. . . . . . . . . . . 12 ⊢ (1 / -i) = i
111	110	oveq1i 7459	. . . . . . . . . . 11 ⊢ ((1 / -i) · ( -i↑𝑛)) = (i · ( -i↑𝑛))
112	101, 104, 111	3eqtr3g 2821	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ( -1↑((𝑛 − 1) / 2)) = (i · ( -i↑𝑛)))
113	87	oveq2d 7465	. . . . . . . . . . . . . 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 14337	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (i↑(𝑛 − 1)) = ((i↑𝑛) / i))
117	113, 116	eqtrd 2798	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → (i↑(2 · ((𝑛 − 1) / 2))) = ((i↑𝑛) / i))
118		expmulz 14289	. . . . . . . . . . . . . 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 14260	. . . . . . . . . . . . . . 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 12135	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((i↑𝑛) / i) = ((1 / i) · (i↑𝑛)))
123	117, 119, 122	3eqtr3d 2806	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ((i↑2)↑((𝑛 − 1) / 2)) = ((1 / i) · (i↑𝑛)))
124	102	oveq1i 7459	. . . . . . . . . . . 12 ⊢ ((i↑2)↑((𝑛 − 1) / 2)) = ( -1↑((𝑛 − 1) / 2))
125	105	oveq1i 7459	. . . . . . . . . . . 12 ⊢ ((1 / i) · (i↑𝑛)) = ( -i · (i↑𝑛))
126	123, 124, 125	3eqtr3g 2821	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ( -1↑((𝑛 − 1) / 2)) = ( -i · (i↑𝑛)))
127		mulneg1 11789	. . . . . . . . . . . 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 2798	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ( -1↑((𝑛 − 1) / 2)) = -(i · (i↑𝑛)))
130	112, 129	oveq12d 7467	. . . . . . . . 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 11321	. . . . . . . . . . . 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 11321	. . . . . . . . . . . 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 11712	. . . . . . . . . 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 11809	. . . . . . . . . 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 2801	. . . . . . . . 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 2802	. . . . . . . 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 12187	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑛 ∈ ℕ) ∧ ¬ 2 ∥ 𝑛) → ( -1↑((𝑛 − 1) / 2)) = ((i · (( -i↑𝑛) − (i↑𝑛))) / 2))
140	139	oveq1d 7464	. . . . . 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 5202	. . . 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 2810	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐹 = (𝑛 ∈ ℕ ↦ (((i · (( -i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛))))
144	143	seqeq3d 14190	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , 𝐹) = seq1( + , (𝑛 ∈ ℕ ↦ (((i · (( -i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛)))))
145		eqid 2763	. . 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 27311	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (((i · (( -i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛)))) ⇝ (arctan‘𝐴))
147	144, 146	eqbrtrd 5131	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 1570 ∈ wcel 2147 ≠ wne 2959 ifcif 4485 class class class wbr 5106 ↦ cmpt 5190 ‘cfv 6564 (class class class)co 7449 ℂcc 11235 0cc0 11237 1c1 11238 ici 11239 + caddc 11240 · cmul 11242 < clt 11380 − cmin 11578 -cneg 11579 / cdiv 12011 ℕcn 12373 2c2 12435 ℕ₀cn0 12644 ℤcz 12731 seqcseq 14182 ↑cexp 14242 abscabs 15442 ⇝ cli 15692 ∥ cdvds 16463 arctancatan 27238

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7781 ax-inf2 9677 ax-cnex 11293 ax-resscn 11294 ax-1cn 11295 ax-icn 11296 ax-addcl 11297 ax-addrcl 11298 ax-mulcl 11299 ax-mulrcl 11300 ax-mulcom 11301 ax-addass 11302 ax-mulass 11303 ax-distr 11304 ax-i2m1 11305 ax-1ne0 11306 ax-1rid 11307 ax-rnegex 11308 ax-rrecex 11309 ax-cnre 11310 ax-pre-lttri 11311 ax-pre-lttrn 11312 ax-pre-ltadd 11313 ax-pre-mulgt0 11314 ax-pre-sup 11315 ax-addf 11316

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-se 5610 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6328 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6518 df-fun 6566 df-fn 6567 df-f 6568 df-f1 6569 df-fo 6570 df-f1o 6571 df-fv 6572 df-isom 6573 df-riota 7406 df-ov 7452 df-oprab 7453 df-mpo 7454 df-of 7723 df-om 7910 df-1st 8033 df-2nd 8034 df-supp 8213 df-frecs 8335 df-wrecs 8366 df-recs 8415 df-rdg 8454 df-1o 8512 df-2o 8513 df-er 8753 df-map 8885 df-pm 8886 df-ixp 8962 df-en 9010 df-dom 9011 df-sdom 9012 df-fin 9013 df-fsupp 9390 df-fi 9438 df-sup 9469 df-inf 9470 df-oi 9539 df-card 10055 df-pnf 11382 df-mnf 11383 df-xr 11384 df-ltxr 11385 df-le 11386 df-sub 11580 df-neg 11581 df-div 12012 df-nn 12374 df-2 12443 df-3 12444 df-4 12445 df-5 12446 df-6 12447 df-7 12448 df-8 12449 df-9 12450 df-n0 12645 df-z 12732 df-dec 12853 df-uz 13004 df-q 13114 df-rp 13159 df-xneg 13279 df-xadd 13280 df-xmul 13281 df-ioo 13518 df-ioc 13519 df-ico 13520 df-icc 13521 df-fz 13678 df-fzo 13827 df-fl 13970 df-mod 14048 df-seq 14183 df-exp 14243 df-fac 14457 df-bc 14486 df-hash 14514 df-shft 15261 df-cj 15307 df-re 15308 df-im 15309 df-sqrt 15443 df-abs 15444 df-limsup 15679 df-clim 15696 df-rlim 15697 df-sum 15895 df-ef 16274 df-sin 16276 df-cos 16277 df-tan 16278 df-pi 16279 df-dvds 16464 df-struct 17366 df-sets 17383 df-slot 17401 df-ndx 17413 df-base 17429 df-ress 17450 df-plusg 17482 df-mulr 17483 df-starv 17484 df-sca 17485 df-vsca 17486 df-ip 17487 df-tset 17488 df-ple 17489 df-ds 17491 df-unif 17492 df-hom 17493 df-cco 17494 df-rest 17634 df-topn 17635 df-0g 17653 df-gsum 17654 df-topgen 17655 df-pt 17656 df-prds 17659 df-xrs 17715 df-qtop 17720 df-imas 17721 df-xps 17723 df-mre 17797 df-mrc 17798 df-acs 17800 df-mgm 18857 df-sgrp 18949 df-mnd 18965 df-submnd 19020 df-mulg 19319 df-cntz 19573 df-cmn 20038 df-psmet 21712 df-xmet 21713 df-met 21714 df-bl 21715 df-mopn 21716 df-fbas 21717 df-fg 21718 df-cnfld 21721 df-top 23256 df-topon 23273 df-topsp 23295 df-bases 23308 df-cld 23381 df-ntr 23382 df-cls 23383 df-nei 23460 df-lp 23498 df-perf 23499 df-cn 23589 df-cnp 23590 df-haus 23677 df-cmp 23749 df-tx 23925 df-hmeo 24118 df-fil 24209 df-fm 24301 df-flim 24302 df-flf 24303 df-xms 24683 df-ms 24684 df-tms 24685 df-cncf 25243 df-limc 26230 df-dv 26231 df-ulm 26750 df-log 26930 df-atan 27241

This theorem is used by: atantayl3 27313 leibpi 27316

W3C validator