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Theorem atantayl 26287

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

**Hypothesis**
Ref	Expression
atantayl.1	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((i · ((-i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛)))

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

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

Proof of Theorem atantayl Dummy variables 𝑘 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 12806	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12534	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 1 ∈ ℤ)
3		ax-icn 11110	. . . 4 ⊢ i ∈ ℂ
4		halfcl 12378	. . . 4 ⊢ (i ∈ ℂ → (i / 2) ∈ ℂ)
5	3, 4	mp1i 13	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (i / 2) ∈ ℂ)
6		simpl 483	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ ℂ)
7		mulcl 11135	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
8	3, 6, 7	sylancr 587	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (i · 𝐴) ∈ ℂ)
9	8	negcld 11499	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → -(i · 𝐴) ∈ ℂ)
10	8	absnegd 15334	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘-(i · 𝐴)) = (abs‘(i · 𝐴)))
11		absmul 15179	. . . . . . . . . 10 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (abs‘(i · 𝐴)) = ((abs‘i) · (abs‘𝐴)))
12	3, 6, 11	sylancr 587	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(i · 𝐴)) = ((abs‘i) · (abs‘𝐴)))
13		absi 15171	. . . . . . . . . . 11 ⊢ (abs‘i) = 1
14	13	oveq1i 7367	. . . . . . . . . 10 ⊢ ((abs‘i) · (abs‘𝐴)) = (1 · (abs‘𝐴))
15		abscl 15163	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
16	15	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) ∈ ℝ)
17	16	recnd 11183	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) ∈ ℂ)
18	17	mulid2d 11173	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (1 · (abs‘𝐴)) = (abs‘𝐴))
19	14, 18	eqtrid 2788	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((abs‘i) · (abs‘𝐴)) = (abs‘𝐴))
20	10, 12, 19	3eqtrd 2780	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘-(i · 𝐴)) = (abs‘𝐴))
21		simpr 485	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) < 1)
22	20, 21	eqbrtrd 5127	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘-(i · 𝐴)) < 1)
23		logtayl 26015	. . . . . . 7 ⊢ ((-(i · 𝐴) ∈ ℂ ∧ (abs‘-(i · 𝐴)) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))) ⇝ -(log‘(1 − -(i · 𝐴))))
24	9, 22, 23	syl2anc 584	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))) ⇝ -(log‘(1 − -(i · 𝐴))))
25		ax-1cn 11109	. . . . . . . . 9 ⊢ 1 ∈ ℂ
26		subneg 11450	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − -(i · 𝐴)) = (1 + (i · 𝐴)))
27	25, 8, 26	sylancr 587	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (1 − -(i · 𝐴)) = (1 + (i · 𝐴)))
28	27	fveq2d 6846	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (log‘(1 − -(i · 𝐴))) = (log‘(1 + (i · 𝐴))))
29	28	negeqd 11395	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → -(log‘(1 − -(i · 𝐴))) = -(log‘(1 + (i · 𝐴))))
30	24, 29	breqtrd 5131	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))) ⇝ -(log‘(1 + (i · 𝐴))))
31		seqex 13908	. . . . . 6 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛)))) ∈ V
32	31	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛)))) ∈ V)
33	10, 22	eqbrtrrd 5129	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(i · 𝐴)) < 1)
34		logtayl 26015	. . . . . 6 ⊢ (((i · 𝐴) ∈ ℂ ∧ (abs‘(i · 𝐴)) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))) ⇝ -(log‘(1 − (i · 𝐴))))
35	8, 33, 34	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))) ⇝ -(log‘(1 − (i · 𝐴))))
36		oveq2 7365	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (-(i · 𝐴)↑𝑛) = (-(i · 𝐴)↑𝑚))
37		id 22	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → 𝑛 = 𝑚)
38	36, 37	oveq12d 7375	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → ((-(i · 𝐴)↑𝑛) / 𝑛) = ((-(i · 𝐴)↑𝑚) / 𝑚))
39		eqid 2736	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛)) = (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))
40		ovex 7390	. . . . . . . . . 10 ⊢ ((-(i · 𝐴)↑𝑚) / 𝑚) ∈ V
41	38, 39, 40	fvmpt 6948	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))‘𝑚) = ((-(i · 𝐴)↑𝑚) / 𝑚))
42	41	adantl 482	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))‘𝑚) = ((-(i · 𝐴)↑𝑚) / 𝑚))
43		nnnn0 12420	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ₀)
44		expcl 13985	. . . . . . . . . 10 ⊢ ((-(i · 𝐴) ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → (-(i · 𝐴)↑𝑚) ∈ ℂ)
45	9, 43, 44	syl2an 596	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (-(i · 𝐴)↑𝑚) ∈ ℂ)
46		nncn 12161	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
47	46	adantl 482	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
48		nnne0 12187	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0)
49	48	adantl 482	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0)
50	45, 47, 49	divcld 11931	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((-(i · 𝐴)↑𝑚) / 𝑚) ∈ ℂ)
51	42, 50	eqeltrd 2838	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))‘𝑚) ∈ ℂ)
52	1, 2, 51	serf 13936	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))):ℕ⟶ℂ)
53	52	ffvelcdmda 7035	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (seq1( + , (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛)))‘𝑘) ∈ ℂ)
54		oveq2 7365	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ((i · 𝐴)↑𝑛) = ((i · 𝐴)↑𝑚))
55	54, 37	oveq12d 7375	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (((i · 𝐴)↑𝑛) / 𝑛) = (((i · 𝐴)↑𝑚) / 𝑚))
56		eqid 2736	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛)) = (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))
57		ovex 7390	. . . . . . . . . 10 ⊢ (((i · 𝐴)↑𝑚) / 𝑚) ∈ V
58	55, 56, 57	fvmpt 6948	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))‘𝑚) = (((i · 𝐴)↑𝑚) / 𝑚))
59	58	adantl 482	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))‘𝑚) = (((i · 𝐴)↑𝑚) / 𝑚))
60		expcl 13985	. . . . . . . . . 10 ⊢ (((i · 𝐴) ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → ((i · 𝐴)↑𝑚) ∈ ℂ)
61	8, 43, 60	syl2an 596	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((i · 𝐴)↑𝑚) ∈ ℂ)
62	61, 47, 49	divcld 11931	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (((i · 𝐴)↑𝑚) / 𝑚) ∈ ℂ)
63	59, 62	eqeltrd 2838	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))‘𝑚) ∈ ℂ)
64	1, 2, 63	serf 13936	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))):ℕ⟶ℂ)
65	64	ffvelcdmda 7035	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (seq1( + , (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛)))‘𝑘) ∈ ℂ)
66		simpr 485	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
67	66, 1	eleqtrdi 2848	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ_≥‘1))
68		simpl 483	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1))
69		elfznn 13470	. . . . . . 7 ⊢ (𝑚 ∈ (1...𝑘) → 𝑚 ∈ ℕ)
70	68, 69, 51	syl2an 596	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))‘𝑚) ∈ ℂ)
71	68, 69, 63	syl2an 596	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))‘𝑚) ∈ ℂ)
72	38, 55	oveq12d 7375	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛)) = (((-(i · 𝐴)↑𝑚) / 𝑚) − (((i · 𝐴)↑𝑚) / 𝑚)))
73		eqid 2736	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛))) = (𝑛 ∈ ℕ ↦ (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛)))
74		ovex 7390	. . . . . . . . . 10 ⊢ (((-(i · 𝐴)↑𝑚) / 𝑚) − (((i · 𝐴)↑𝑚) / 𝑚)) ∈ V
75	72, 73, 74	fvmpt 6948	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛)))‘𝑚) = (((-(i · 𝐴)↑𝑚) / 𝑚) − (((i · 𝐴)↑𝑚) / 𝑚)))
76	75	adantl 482	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛)))‘𝑚) = (((-(i · 𝐴)↑𝑚) / 𝑚) − (((i · 𝐴)↑𝑚) / 𝑚)))
77	42, 59	oveq12d 7375	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (((𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))‘𝑚) − ((𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))‘𝑚)) = (((-(i · 𝐴)↑𝑚) / 𝑚) − (((i · 𝐴)↑𝑚) / 𝑚)))
78	76, 77	eqtr4d 2779	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛)))‘𝑚) = (((𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))‘𝑚) − ((𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))‘𝑚)))
79	68, 69, 78	syl2an 596	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((𝑛 ∈ ℕ ↦ (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛)))‘𝑚) = (((𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))‘𝑚) − ((𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))‘𝑚)))
80	67, 70, 71, 79	sersub 13951	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (seq1( + , (𝑛 ∈ ℕ ↦ (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛))))‘𝑘) = ((seq1( + , (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛)))‘𝑘) − (seq1( + , (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛)))‘𝑘)))
81	1, 2, 30, 32, 35, 53, 65, 80	climsub 15516	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛)))) ⇝ (-(log‘(1 + (i · 𝐴))) − -(log‘(1 − (i · 𝐴)))))
82		addcl 11133	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + (i · 𝐴)) ∈ ℂ)
83	25, 8, 82	sylancr 587	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (1 + (i · 𝐴)) ∈ ℂ)
84		bndatandm 26279	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ dom arctan)
85		atandm2 26227	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
86	84, 85	sylib 217	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
87	86	simp3d 1144	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (1 + (i · 𝐴)) ≠ 0)
88	83, 87	logcld 25926	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (log‘(1 + (i · 𝐴))) ∈ ℂ)
89		subcl 11400	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − (i · 𝐴)) ∈ ℂ)
90	25, 8, 89	sylancr 587	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (1 − (i · 𝐴)) ∈ ℂ)
91	86	simp2d 1143	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (1 − (i · 𝐴)) ≠ 0)
92	90, 91	logcld 25926	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (log‘(1 − (i · 𝐴))) ∈ ℂ)
93	88, 92	neg2subd 11529	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (-(log‘(1 + (i · 𝐴))) − -(log‘(1 − (i · 𝐴)))) = ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
94	81, 93	breqtrd 5131	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛)))) ⇝ ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
95	50, 62	subcld 11512	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (((-(i · 𝐴)↑𝑚) / 𝑚) − (((i · 𝐴)↑𝑚) / 𝑚)) ∈ ℂ)
96	76, 95	eqeltrd 2838	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛)))‘𝑚) ∈ ℂ)
97	3	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → i ∈ ℂ)
98		negicn 11402	. . . . . . . . 9 ⊢ -i ∈ ℂ
99	43	adantl 482	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ₀)
100		expcl 13985	. . . . . . . . 9 ⊢ ((-i ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → (-i↑𝑚) ∈ ℂ)
101	98, 99, 100	sylancr 587	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (-i↑𝑚) ∈ ℂ)
102		expcl 13985	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → (i↑𝑚) ∈ ℂ)
103	3, 99, 102	sylancr 587	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (i↑𝑚) ∈ ℂ)
104	101, 103	subcld 11512	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((-i↑𝑚) − (i↑𝑚)) ∈ ℂ)
105		2cnd 12231	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 2 ∈ ℂ)
106		2ne0 12257	. . . . . . . 8 ⊢ 2 ≠ 0
107	106	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 2 ≠ 0)
108	97, 104, 105, 107	div23d 11968	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((i · ((-i↑𝑚) − (i↑𝑚))) / 2) = ((i / 2) · ((-i↑𝑚) − (i↑𝑚))))
109	108	oveq1d 7372	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (((i · ((-i↑𝑚) − (i↑𝑚))) / 2) · ((𝐴↑𝑚) / 𝑚)) = (((i / 2) · ((-i↑𝑚) − (i↑𝑚))) · ((𝐴↑𝑚) / 𝑚)))
110	5	adantr 481	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (i / 2) ∈ ℂ)
111		expcl 13985	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → (𝐴↑𝑚) ∈ ℂ)
112	6, 43, 111	syl2an 596	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (𝐴↑𝑚) ∈ ℂ)
113	112, 47, 49	divcld 11931	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝐴↑𝑚) / 𝑚) ∈ ℂ)
114	110, 104, 113	mulassd 11178	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (((i / 2) · ((-i↑𝑚) − (i↑𝑚))) · ((𝐴↑𝑚) / 𝑚)) = ((i / 2) · (((-i↑𝑚) − (i↑𝑚)) · ((𝐴↑𝑚) / 𝑚))))
115	101, 103, 112	subdird 11612	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (((-i↑𝑚) − (i↑𝑚)) · (𝐴↑𝑚)) = (((-i↑𝑚) · (𝐴↑𝑚)) − ((i↑𝑚) · (𝐴↑𝑚))))
116	6	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 𝐴 ∈ ℂ)
117		mulneg1 11591	. . . . . . . . . . . . 13 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) = -(i · 𝐴))
118	3, 116, 117	sylancr 587	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (-i · 𝐴) = -(i · 𝐴))
119	118	oveq1d 7372	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((-i · 𝐴)↑𝑚) = (-(i · 𝐴)↑𝑚))
120	98	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → -i ∈ ℂ)
121	120, 116, 99	mulexpd 14066	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((-i · 𝐴)↑𝑚) = ((-i↑𝑚) · (𝐴↑𝑚)))
122	119, 121	eqtr3d 2778	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (-(i · 𝐴)↑𝑚) = ((-i↑𝑚) · (𝐴↑𝑚)))
123	97, 116, 99	mulexpd 14066	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((i · 𝐴)↑𝑚) = ((i↑𝑚) · (𝐴↑𝑚)))
124	122, 123	oveq12d 7375	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((-(i · 𝐴)↑𝑚) − ((i · 𝐴)↑𝑚)) = (((-i↑𝑚) · (𝐴↑𝑚)) − ((i↑𝑚) · (𝐴↑𝑚))))
125	115, 124	eqtr4d 2779	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (((-i↑𝑚) − (i↑𝑚)) · (𝐴↑𝑚)) = ((-(i · 𝐴)↑𝑚) − ((i · 𝐴)↑𝑚)))
126	125	oveq1d 7372	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((((-i↑𝑚) − (i↑𝑚)) · (𝐴↑𝑚)) / 𝑚) = (((-(i · 𝐴)↑𝑚) − ((i · 𝐴)↑𝑚)) / 𝑚))
127	104, 112, 47, 49	divassd 11966	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((((-i↑𝑚) − (i↑𝑚)) · (𝐴↑𝑚)) / 𝑚) = (((-i↑𝑚) − (i↑𝑚)) · ((𝐴↑𝑚) / 𝑚)))
128	45, 61, 47, 49	divsubdird 11970	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (((-(i · 𝐴)↑𝑚) − ((i · 𝐴)↑𝑚)) / 𝑚) = (((-(i · 𝐴)↑𝑚) / 𝑚) − (((i · 𝐴)↑𝑚) / 𝑚)))
129	126, 127, 128	3eqtr3d 2784	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (((-i↑𝑚) − (i↑𝑚)) · ((𝐴↑𝑚) / 𝑚)) = (((-(i · 𝐴)↑𝑚) / 𝑚) − (((i · 𝐴)↑𝑚) / 𝑚)))
130	129	oveq2d 7373	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((i / 2) · (((-i↑𝑚) − (i↑𝑚)) · ((𝐴↑𝑚) / 𝑚))) = ((i / 2) · (((-(i · 𝐴)↑𝑚) / 𝑚) − (((i · 𝐴)↑𝑚) / 𝑚))))
131	109, 114, 130	3eqtrd 2780	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (((i · ((-i↑𝑚) − (i↑𝑚))) / 2) · ((𝐴↑𝑚) / 𝑚)) = ((i / 2) · (((-(i · 𝐴)↑𝑚) / 𝑚) − (((i · 𝐴)↑𝑚) / 𝑚))))
132		oveq2 7365	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (-i↑𝑛) = (-i↑𝑚))
133		oveq2 7365	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (i↑𝑛) = (i↑𝑚))
134	132, 133	oveq12d 7375	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → ((-i↑𝑛) − (i↑𝑛)) = ((-i↑𝑚) − (i↑𝑚)))
135	134	oveq2d 7373	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (i · ((-i↑𝑛) − (i↑𝑛))) = (i · ((-i↑𝑚) − (i↑𝑚))))
136	135	oveq1d 7372	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((i · ((-i↑𝑛) − (i↑𝑛))) / 2) = ((i · ((-i↑𝑚) − (i↑𝑚))) / 2))
137		oveq2 7365	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝐴↑𝑛) = (𝐴↑𝑚))
138	137, 37	oveq12d 7375	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((𝐴↑𝑛) / 𝑛) = ((𝐴↑𝑚) / 𝑚))
139	136, 138	oveq12d 7375	. . . . . 6 ⊢ (𝑛 = 𝑚 → (((i · ((-i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛)) = (((i · ((-i↑𝑚) − (i↑𝑚))) / 2) · ((𝐴↑𝑚) / 𝑚)))
140		atantayl.1	. . . . . 6 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((i · ((-i↑𝑛) − (i↑𝑛))) / 2) · ((𝐴↑𝑛) / 𝑛)))
141		ovex 7390	. . . . . 6 ⊢ (((i · ((-i↑𝑚) − (i↑𝑚))) / 2) · ((𝐴↑𝑚) / 𝑚)) ∈ V
142	139, 140, 141	fvmpt 6948	. . . . 5 ⊢ (𝑚 ∈ ℕ → (𝐹‘𝑚) = (((i · ((-i↑𝑚) − (i↑𝑚))) / 2) · ((𝐴↑𝑚) / 𝑚)))
143	142	adantl 482	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) = (((i · ((-i↑𝑚) − (i↑𝑚))) / 2) · ((𝐴↑𝑚) / 𝑚)))
144	76	oveq2d 7373	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((i / 2) · ((𝑛 ∈ ℕ ↦ (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛)))‘𝑚)) = ((i / 2) · (((-(i · 𝐴)↑𝑚) / 𝑚) − (((i · 𝐴)↑𝑚) / 𝑚))))
145	131, 143, 144	3eqtr4d 2786	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) = ((i / 2) · ((𝑛 ∈ ℕ ↦ (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛)))‘𝑚)))
146	1, 2, 5, 94, 96, 145	isermulc2 15542	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , 𝐹) ⇝ ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
147		atanval 26234	. . 3 ⊢ (𝐴 ∈ dom arctan → (arctan‘𝐴) = ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
148	84, 147	syl 17	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (arctan‘𝐴) = ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
149	146, 148	breqtrrd 5133	1 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , 𝐹) ⇝ (arctan‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 Vcvv 3445 class class class wbr 5105 ↦ cmpt 5188 dom cdm 5633 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 ici 11053 + caddc 11054 · cmul 11056 < clt 11189 − cmin 11385 -cneg 11386 / cdiv 11812 ℕcn 12153 2c2 12208 ℕ₀cn0 12413 ℤ_≥cuz 12763 ...cfz 13424 seqcseq 13906 ↑cexp 13967 abscabs 15119 ⇝ cli 15366 logclog 25910 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-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: atantayl2 26288

W3C validator