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Theorem atantayl 26980

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 12921	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12648	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 1 ∈ ℤ)
3		ax-icn 11214	. . . 4 ⊢ i ∈ ℂ
4		halfcl 12491	. . . 4 ⊢ (i ∈ ℂ → (i / 2) ∈ ℂ)
5	3, 4	mp1i 13	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (i / 2) ∈ ℂ)
6		simpl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ ℂ)
7		mulcl 11239	. . . . . . . . 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 11607	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → -(i · 𝐴) ∈ ℂ)
10	8	absnegd 15488	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘-(i · 𝐴)) = (abs‘(i · 𝐴)))
11		absmul 15333	. . . . . . . . . 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 15325	. . . . . . . . . . 11 ⊢ (abs‘i) = 1
14	13	oveq1i 7441	. . . . . . . . . 10 ⊢ ((abs‘i) · (abs‘𝐴)) = (1 · (abs‘𝐴))
15		abscl 15317	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
16	15	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) ∈ ℝ)
17	16	recnd 11289	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) ∈ ℂ)
18	17	mullidd 11279	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (1 · (abs‘𝐴)) = (abs‘𝐴))
19	14, 18	eqtrid 2789	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((abs‘i) · (abs‘𝐴)) = (abs‘𝐴))
20	10, 12, 19	3eqtrd 2781	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘-(i · 𝐴)) = (abs‘𝐴))
21		simpr 484	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) < 1)
22	20, 21	eqbrtrd 5165	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘-(i · 𝐴)) < 1)
23		logtayl 26702	. . . . . . 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 11213	. . . . . . . . 9 ⊢ 1 ∈ ℂ
26		subneg 11558	. . . . . . . . 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 6910	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (log‘(1 − -(i · 𝐴))) = (log‘(1 + (i · 𝐴))))
29	28	negeqd 11502	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → -(log‘(1 − -(i · 𝐴))) = -(log‘(1 + (i · 𝐴))))
30	24, 29	breqtrd 5169	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))) ⇝ -(log‘(1 + (i · 𝐴))))
31		seqex 14044	. . . . . 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 5167	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(i · 𝐴)) < 1)
34		logtayl 26702	. . . . . 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 7439	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (-(i · 𝐴)↑𝑛) = (-(i · 𝐴)↑𝑚))
37		id 22	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → 𝑛 = 𝑚)
38	36, 37	oveq12d 7449	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → ((-(i · 𝐴)↑𝑛) / 𝑛) = ((-(i · 𝐴)↑𝑚) / 𝑚))
39		eqid 2737	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛)) = (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))
40		ovex 7464	. . . . . . . . . 10 ⊢ ((-(i · 𝐴)↑𝑚) / 𝑚) ∈ V
41	38, 39, 40	fvmpt 7016	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))‘𝑚) = ((-(i · 𝐴)↑𝑚) / 𝑚))
42	41	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))‘𝑚) = ((-(i · 𝐴)↑𝑚) / 𝑚))
43		nnnn0 12533	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ₀)
44		expcl 14120	. . . . . . . . . 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 12274	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
47	46	adantl 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
48		nnne0 12300	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0)
49	48	adantl 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0)
50	45, 47, 49	divcld 12043	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((-(i · 𝐴)↑𝑚) / 𝑚) ∈ ℂ)
51	42, 50	eqeltrd 2841	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))‘𝑚) ∈ ℂ)
52	1, 2, 51	serf 14071	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))):ℕ⟶ℂ)
53	52	ffvelcdmda 7104	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (seq1( + , (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛)))‘𝑘) ∈ ℂ)
54		oveq2 7439	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ((i · 𝐴)↑𝑛) = ((i · 𝐴)↑𝑚))
55	54, 37	oveq12d 7449	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (((i · 𝐴)↑𝑛) / 𝑛) = (((i · 𝐴)↑𝑚) / 𝑚))
56		eqid 2737	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛)) = (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))
57		ovex 7464	. . . . . . . . . 10 ⊢ (((i · 𝐴)↑𝑚) / 𝑚) ∈ V
58	55, 56, 57	fvmpt 7016	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))‘𝑚) = (((i · 𝐴)↑𝑚) / 𝑚))
59	58	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))‘𝑚) = (((i · 𝐴)↑𝑚) / 𝑚))
60		expcl 14120	. . . . . . . . . 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 12043	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (((i · 𝐴)↑𝑚) / 𝑚) ∈ ℂ)
63	59, 62	eqeltrd 2841	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))‘𝑚) ∈ ℂ)
64	1, 2, 63	serf 14071	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))):ℕ⟶ℂ)
65	64	ffvelcdmda 7104	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (seq1( + , (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛)))‘𝑘) ∈ ℂ)
66		simpr 484	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
67	66, 1	eleqtrdi 2851	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ_≥‘1))
68		simpl 482	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1))
69		elfznn 13593	. . . . . . 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 7449	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛)) = (((-(i · 𝐴)↑𝑚) / 𝑚) − (((i · 𝐴)↑𝑚) / 𝑚)))
73		eqid 2737	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛))) = (𝑛 ∈ ℕ ↦ (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛)))
74		ovex 7464	. . . . . . . . . 10 ⊢ (((-(i · 𝐴)↑𝑚) / 𝑚) − (((i · 𝐴)↑𝑚) / 𝑚)) ∈ V
75	72, 73, 74	fvmpt 7016	. . . . . . . . 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 481	. . . . . . . 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 7449	. . . . . . . 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 2780	. . . . . . 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 14086	. . . . 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 15670	. . . 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 11237	. . . . . . 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 26972	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ dom arctan)
85		atandm2 26920	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
86	84, 85	sylib 218	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
87	86	simp3d 1145	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (1 + (i · 𝐴)) ≠ 0)
88	83, 87	logcld 26612	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (log‘(1 + (i · 𝐴))) ∈ ℂ)
89		subcl 11507	. . . . . . 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 1144	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (1 − (i · 𝐴)) ≠ 0)
92	90, 91	logcld 26612	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (log‘(1 − (i · 𝐴))) ∈ ℂ)
93	88, 92	neg2subd 11637	. . . 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 5169	. . 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 11620	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (((-(i · 𝐴)↑𝑚) / 𝑚) − (((i · 𝐴)↑𝑚) / 𝑚)) ∈ ℂ)
96	76, 95	eqeltrd 2841	. . 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 11509	. . . . . . . . 9 ⊢ -i ∈ ℂ
99	43	adantl 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ₀)
100		expcl 14120	. . . . . . . . 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 14120	. . . . . . . . 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 11620	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((-i↑𝑚) − (i↑𝑚)) ∈ ℂ)
105		2cnd 12344	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 2 ∈ ℂ)
106		2ne0 12370	. . . . . . . 8 ⊢ 2 ≠ 0
107	106	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 2 ≠ 0)
108	97, 104, 105, 107	div23d 12080	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((i · ((-i↑𝑚) − (i↑𝑚))) / 2) = ((i / 2) · ((-i↑𝑚) − (i↑𝑚))))
109	108	oveq1d 7446	. . . . 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 480	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (i / 2) ∈ ℂ)
111		expcl 14120	. . . . . . . 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 12043	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝐴↑𝑚) / 𝑚) ∈ ℂ)
114	110, 104, 113	mulassd 11284	. . . . 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 11720	. . . . . . . . 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 480	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 𝐴 ∈ ℂ)
117		mulneg1 11699	. . . . . . . . . . . . 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 7446	. . . . . . . . . . 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 14201	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((-i · 𝐴)↑𝑚) = ((-i↑𝑚) · (𝐴↑𝑚)))
122	119, 121	eqtr3d 2779	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (-(i · 𝐴)↑𝑚) = ((-i↑𝑚) · (𝐴↑𝑚)))
123	97, 116, 99	mulexpd 14201	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((i · 𝐴)↑𝑚) = ((i↑𝑚) · (𝐴↑𝑚)))
124	122, 123	oveq12d 7449	. . . . . . . . 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 2780	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (((-i↑𝑚) − (i↑𝑚)) · (𝐴↑𝑚)) = ((-(i · 𝐴)↑𝑚) − ((i · 𝐴)↑𝑚)))
126	125	oveq1d 7446	. . . . . . 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 12078	. . . . . . 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 12082	. . . . . . 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 2785	. . . . . 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 7447	. . . . 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 2781	. . . 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 7439	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (-i↑𝑛) = (-i↑𝑚))
133		oveq2 7439	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (i↑𝑛) = (i↑𝑚))
134	132, 133	oveq12d 7449	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → ((-i↑𝑛) − (i↑𝑛)) = ((-i↑𝑚) − (i↑𝑚)))
135	134	oveq2d 7447	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (i · ((-i↑𝑛) − (i↑𝑛))) = (i · ((-i↑𝑚) − (i↑𝑚))))
136	135	oveq1d 7446	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((i · ((-i↑𝑛) − (i↑𝑛))) / 2) = ((i · ((-i↑𝑚) − (i↑𝑚))) / 2))
137		oveq2 7439	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝐴↑𝑛) = (𝐴↑𝑚))
138	137, 37	oveq12d 7449	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((𝐴↑𝑛) / 𝑛) = ((𝐴↑𝑚) / 𝑚))
139	136, 138	oveq12d 7449	. . . . . 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 7464	. . . . . 6 ⊢ (((i · ((-i↑𝑚) − (i↑𝑚))) / 2) · ((𝐴↑𝑚) / 𝑚)) ∈ V
142	139, 140, 141	fvmpt 7016	. . . . 5 ⊢ (𝑚 ∈ ℕ → (𝐹‘𝑚) = (((i · ((-i↑𝑚) − (i↑𝑚))) / 2) · ((𝐴↑𝑚) / 𝑚)))
143	142	adantl 481	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) = (((i · ((-i↑𝑚) − (i↑𝑚))) / 2) · ((𝐴↑𝑚) / 𝑚)))
144	76	oveq2d 7447	. . . 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 2787	. . 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 15694	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , 𝐹) ⇝ ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
147		atanval 26927	. . 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 5171	1 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , 𝐹) ⇝ (arctan‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 ici 11157 + caddc 11158 · cmul 11160 < clt 11295 − cmin 11492 -cneg 11493 / cdiv 11920 ℕcn 12266 2c2 12321 ℕ₀cn0 12526 ℤ_≥cuz 12878 ...cfz 13547 seqcseq 14042 ↑cexp 14102 abscabs 15273 ⇝ cli 15520 logclog 26596 arctancatan 26907

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-tan 16107 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-ulm 26420 df-log 26598 df-atan 26910

This theorem is referenced by: atantayl2 26981

W3C validator