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Theorem atantayl 26989

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 12871	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12595	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 1 ∈ ℤ)
3		ax-icn 11125	. . . 4 ⊢ i ∈ ℂ
4		halfcl 12440	. . . 4 ⊢ (i ∈ ℂ → (i / 2) ∈ ℂ)
5	3, 4	mp1i 13	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (i / 2) ∈ ℂ)
6		simpl 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ ℂ)
7		mulcl 11150	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
8	3, 6, 7	sylancr 596	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (i · 𝐴) ∈ ℂ)
9	8	negcld 11522	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → -(i · 𝐴) ∈ ℂ)
10	8	absnegd 15469	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘-(i · 𝐴)) = (abs‘(i · 𝐴)))
11		absmul 15311	. . . . . . . . . 10 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (abs‘(i · 𝐴)) = ((abs‘i) · (abs‘𝐴)))
12	3, 6, 11	sylancr 596	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(i · 𝐴)) = ((abs‘i) · (abs‘𝐴)))
13		absi 15303	. . . . . . . . . . 11 ⊢ (abs‘i) = 1
14	13	oveq1i 7400	. . . . . . . . . 10 ⊢ ((abs‘i) · (abs‘𝐴)) = (1 · (abs‘𝐴))
15		abscl 15295	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
16	15	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) ∈ ℝ)
17	16	recnd 11203	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) ∈ ℂ)
18	17	mullidd 11193	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (1 · (abs‘𝐴)) = (abs‘𝐴))
19	14, 18	eqtrid 2808	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((abs‘i) · (abs‘𝐴)) = (abs‘𝐴))
20	10, 12, 19	3eqtrd 2800	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘-(i · 𝐴)) = (abs‘𝐴))
21		simpr 488	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) < 1)
22	20, 21	eqbrtrd 5119	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘-(i · 𝐴)) < 1)
23		logtayl 26712	. . . . . . 7 ⊢ ((-(i · 𝐴) ∈ ℂ ∧ (abs‘-(i · 𝐴)) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))) ⇝ -(log‘(1 − -(i · 𝐴))))
24	9, 22, 23	syl2anc 593	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))) ⇝ -(log‘(1 − -(i · 𝐴))))
25		ax-1cn 11124	. . . . . . . . 9 ⊢ 1 ∈ ℂ
26		subneg 11473	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − -(i · 𝐴)) = (1 + (i · 𝐴)))
27	25, 8, 26	sylancr 596	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (1 − -(i · 𝐴)) = (1 + (i · 𝐴)))
28	27	fveq2d 6865	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (log‘(1 − -(i · 𝐴))) = (log‘(1 + (i · 𝐴))))
29	28	negeqd 11417	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → -(log‘(1 − -(i · 𝐴))) = -(log‘(1 + (i · 𝐴))))
30	24, 29	breqtrd 5123	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))) ⇝ -(log‘(1 + (i · 𝐴))))
31		seqex 14009	. . . . . 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 5121	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(i · 𝐴)) < 1)
34		logtayl 26712	. . . . . 6 ⊢ (((i · 𝐴) ∈ ℂ ∧ (abs‘(i · 𝐴)) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))) ⇝ -(log‘(1 − (i · 𝐴))))
35	8, 33, 34	syl2anc 593	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))) ⇝ -(log‘(1 − (i · 𝐴))))
36		oveq2 7398	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (-(i · 𝐴)↑𝑛) = (-(i · 𝐴)↑𝑚))
37		id 22	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → 𝑛 = 𝑚)
38	36, 37	oveq12d 7408	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → ((-(i · 𝐴)↑𝑛) / 𝑛) = ((-(i · 𝐴)↑𝑚) / 𝑚))
39		eqid 2761	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛)) = (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))
40		ovex 7423	. . . . . . . . . 10 ⊢ ((-(i · 𝐴)↑𝑚) / 𝑚) ∈ V
41	38, 39, 40	fvmpt 6969	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))‘𝑚) = ((-(i · 𝐴)↑𝑚) / 𝑚))
42	41	adantl 485	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))‘𝑚) = ((-(i · 𝐴)↑𝑚) / 𝑚))
43		nnnn0 12481	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ₀)
44		expcl 14085	. . . . . . . . . 10 ⊢ ((-(i · 𝐴) ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → (-(i · 𝐴)↑𝑚) ∈ ℂ)
45	9, 43, 44	syl2an 605	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (-(i · 𝐴)↑𝑚) ∈ ℂ)
46		nncn 12211	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
47	46	adantl 485	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
48		nnne0 12240	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0)
49	48	adantl 485	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0)
50	45, 47, 49	divcld 11960	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((-(i · 𝐴)↑𝑚) / 𝑚) ∈ ℂ)
51	42, 50	eqeltrd 2861	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))‘𝑚) ∈ ℂ)
52	1, 2, 51	serf 14036	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))):ℕ⟶ℂ)
53	52	ffvelcdmda 7059	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (seq1( + , (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛)))‘𝑘) ∈ ℂ)
54		oveq2 7398	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ((i · 𝐴)↑𝑛) = ((i · 𝐴)↑𝑚))
55	54, 37	oveq12d 7408	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (((i · 𝐴)↑𝑛) / 𝑛) = (((i · 𝐴)↑𝑚) / 𝑚))
56		eqid 2761	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛)) = (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))
57		ovex 7423	. . . . . . . . . 10 ⊢ (((i · 𝐴)↑𝑚) / 𝑚) ∈ V
58	55, 56, 57	fvmpt 6969	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))‘𝑚) = (((i · 𝐴)↑𝑚) / 𝑚))
59	58	adantl 485	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))‘𝑚) = (((i · 𝐴)↑𝑚) / 𝑚))
60		expcl 14085	. . . . . . . . . 10 ⊢ (((i · 𝐴) ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → ((i · 𝐴)↑𝑚) ∈ ℂ)
61	8, 43, 60	syl2an 605	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((i · 𝐴)↑𝑚) ∈ ℂ)
62	61, 47, 49	divcld 11960	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (((i · 𝐴)↑𝑚) / 𝑚) ∈ ℂ)
63	59, 62	eqeltrd 2861	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))‘𝑚) ∈ ℂ)
64	1, 2, 63	serf 14036	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))):ℕ⟶ℂ)
65	64	ffvelcdmda 7059	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (seq1( + , (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛)))‘𝑘) ∈ ℂ)
66		simpr 488	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
67	66, 1	eleqtrdi 2871	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ_≥‘1))
68		simpl 486	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1))
69		elfznn 13551	. . . . . . 7 ⊢ (𝑚 ∈ (1...𝑘) → 𝑚 ∈ ℕ)
70	68, 69, 51	syl2an 605	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))‘𝑚) ∈ ℂ)
71	68, 69, 63	syl2an 605	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))‘𝑚) ∈ ℂ)
72	38, 55	oveq12d 7408	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛)) = (((-(i · 𝐴)↑𝑚) / 𝑚) − (((i · 𝐴)↑𝑚) / 𝑚)))
73		eqid 2761	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛))) = (𝑛 ∈ ℕ ↦ (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛)))
74		ovex 7423	. . . . . . . . . 10 ⊢ (((-(i · 𝐴)↑𝑚) / 𝑚) − (((i · 𝐴)↑𝑚) / 𝑚)) ∈ V
75	72, 73, 74	fvmpt 6969	. . . . . . . . 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 485	. . . . . . . 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 7408	. . . . . . . 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 2799	. . . . . . 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 605	. . . . . 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 14051	. . . . 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 15651	. . . 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 11148	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + (i · 𝐴)) ∈ ℂ)
83	25, 8, 82	sylancr 596	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (1 + (i · 𝐴)) ∈ ℂ)
84		bndatandm 26981	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ dom arctan)
85		atandm2 26929	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
86	84, 85	sylib 220	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
87	86	simp3d 1156	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (1 + (i · 𝐴)) ≠ 0)
88	83, 87	logcld 26622	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (log‘(1 + (i · 𝐴))) ∈ ℂ)
89		subcl 11422	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − (i · 𝐴)) ∈ ℂ)
90	25, 8, 89	sylancr 596	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (1 − (i · 𝐴)) ∈ ℂ)
91	86	simp2d 1155	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (1 − (i · 𝐴)) ≠ 0)
92	90, 91	logcld 26622	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (log‘(1 − (i · 𝐴))) ∈ ℂ)
93	88, 92	neg2subd 11552	. . . 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 5123	. . 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 11535	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (((-(i · 𝐴)↑𝑚) / 𝑚) − (((i · 𝐴)↑𝑚) / 𝑚)) ∈ ℂ)
96	76, 95	eqeltrd 2861	. . 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 11424	. . . . . . . . 9 ⊢ -i ∈ ℂ
99	43	adantl 485	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ₀)
100		expcl 14085	. . . . . . . . 9 ⊢ ((-i ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → (-i↑𝑚) ∈ ℂ)
101	98, 99, 100	sylancr 596	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (-i↑𝑚) ∈ ℂ)
102		expcl 14085	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → (i↑𝑚) ∈ ℂ)
103	3, 99, 102	sylancr 596	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (i↑𝑚) ∈ ℂ)
104	101, 103	subcld 11535	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((-i↑𝑚) − (i↑𝑚)) ∈ ℂ)
105		2cnd 12289	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 2 ∈ ℂ)
106		2ne0 12317	. . . . . . . 8 ⊢ 2 ≠ 0
107	106	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 2 ≠ 0)
108	97, 104, 105, 107	div23d 11997	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((i · ((-i↑𝑚) − (i↑𝑚))) / 2) = ((i / 2) · ((-i↑𝑚) − (i↑𝑚))))
109	108	oveq1d 7405	. . . . 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 484	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (i / 2) ∈ ℂ)
111		expcl 14085	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → (𝐴↑𝑚) ∈ ℂ)
112	6, 43, 111	syl2an 605	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (𝐴↑𝑚) ∈ ℂ)
113	112, 47, 49	divcld 11960	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝐴↑𝑚) / 𝑚) ∈ ℂ)
114	110, 104, 113	mulassd 11198	. . . . 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 11637	. . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 𝐴 ∈ ℂ)
117		mulneg1 11616	. . . . . . . . . . . . 13 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) = -(i · 𝐴))
118	3, 116, 117	sylancr 596	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (-i · 𝐴) = -(i · 𝐴))
119	118	oveq1d 7405	. . . . . . . . . . 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 14167	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((-i · 𝐴)↑𝑚) = ((-i↑𝑚) · (𝐴↑𝑚)))
122	119, 121	eqtr3d 2798	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (-(i · 𝐴)↑𝑚) = ((-i↑𝑚) · (𝐴↑𝑚)))
123	97, 116, 99	mulexpd 14167	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((i · 𝐴)↑𝑚) = ((i↑𝑚) · (𝐴↑𝑚)))
124	122, 123	oveq12d 7408	. . . . . . . . 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 2799	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (((-i↑𝑚) − (i↑𝑚)) · (𝐴↑𝑚)) = ((-(i · 𝐴)↑𝑚) − ((i · 𝐴)↑𝑚)))
126	125	oveq1d 7405	. . . . . . 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 11995	. . . . . . 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 11999	. . . . . . 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 2804	. . . . . 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 7406	. . . . 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 2800	. . . 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 7398	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (-i↑𝑛) = (-i↑𝑚))
133		oveq2 7398	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (i↑𝑛) = (i↑𝑚))
134	132, 133	oveq12d 7408	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → ((-i↑𝑛) − (i↑𝑛)) = ((-i↑𝑚) − (i↑𝑚)))
135	134	oveq2d 7406	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (i · ((-i↑𝑛) − (i↑𝑛))) = (i · ((-i↑𝑚) − (i↑𝑚))))
136	135	oveq1d 7405	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((i · ((-i↑𝑛) − (i↑𝑛))) / 2) = ((i · ((-i↑𝑚) − (i↑𝑚))) / 2))
137		oveq2 7398	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝐴↑𝑛) = (𝐴↑𝑚))
138	137, 37	oveq12d 7408	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((𝐴↑𝑛) / 𝑛) = ((𝐴↑𝑚) / 𝑚))
139	136, 138	oveq12d 7408	. . . . . 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 7423	. . . . . 6 ⊢ (((i · ((-i↑𝑚) − (i↑𝑚))) / 2) · ((𝐴↑𝑚) / 𝑚)) ∈ V
142	139, 140, 141	fvmpt 6969	. . . . 5 ⊢ (𝑚 ∈ ℕ → (𝐹‘𝑚) = (((i · ((-i↑𝑚) − (i↑𝑚))) / 2) · ((𝐴↑𝑚) / 𝑚)))
143	142	adantl 485	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) = (((i · ((-i↑𝑚) − (i↑𝑚))) / 2) · ((𝐴↑𝑚) / 𝑚)))
144	76	oveq2d 7406	. . . 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 2806	. . 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 15675	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , 𝐹) ⇝ ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
147		atanval 26936	. . 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 5125	1 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , 𝐹) ⇝ (arctan‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 Vcvv 3453 class class class wbr 5097 ↦ cmpt 5178 dom cdm 5643 ‘cfv 6515 (class class class)co 7390 ℂcc 11064 ℝcr 11065 0cc0 11066 1c1 11067 ici 11068 + caddc 11069 · cmul 11071 < clt 11209 − cmin 11407 -cneg 11408 / cdiv 11837 ℕcn 12203 2c2 12265 ℕ₀cn0 12474 ℤ_≥cuz 12832 ...cfz 13505 seqcseq 14007 ↑cexp 14067 abscabs 15251 ⇝ cli 15501 logclog 26606 arctancatan 26916

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-inf2 9589 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143 ax-pre-sup 11144 ax-addf 11145

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-isom 6524 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7654 df-om 7841 df-1st 7964 df-2nd 7965 df-supp 8134 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-2o 8431 df-er 8671 df-map 8803 df-pm 8804 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9301 df-fi 9350 df-sup 9381 df-inf 9382 df-oi 9451 df-card 9890 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-div 11838 df-nn 12204 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12475 df-z 12562 df-dec 12682 df-uz 12833 df-q 12943 df-rp 12987 df-xneg 13107 df-xadd 13108 df-xmul 13109 df-ioo 13346 df-ioc 13347 df-ico 13348 df-icc 13349 df-fz 13506 df-fzo 13653 df-fl 13795 df-mod 13873 df-seq 14008 df-exp 14068 df-fac 14280 df-bc 14309 df-hash 14337 df-shft 15073 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-limsup 15488 df-clim 15505 df-rlim 15506 df-sum 15704 df-ef 16087 df-sin 16089 df-cos 16090 df-tan 16091 df-pi 16092 df-struct 17173 df-sets 17190 df-slot 17208 df-ndx 17220 df-base 17236 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-xrs 17522 df-qtop 17527 df-imas 17528 df-xps 17530 df-mre 17604 df-mrc 17605 df-acs 17607 df-mgm 18664 df-sgrp 18743 df-mnd 18759 df-submnd 18808 df-mulg 19100 df-cntz 19347 df-cmn 19812 df-psmet 21403 df-xmet 21404 df-met 21405 df-bl 21406 df-mopn 21407 df-fbas 21408 df-fg 21409 df-cnfld 21412 df-top 22941 df-topon 22958 df-topsp 22980 df-bases 22993 df-cld 23066 df-ntr 23067 df-cls 23068 df-nei 23145 df-lp 23183 df-perf 23184 df-cn 23274 df-cnp 23275 df-haus 23362 df-cmp 23434 df-tx 23609 df-hmeo 23802 df-fil 23893 df-fm 23985 df-flim 23986 df-flf 23987 df-xms 24367 df-ms 24368 df-tms 24369 df-cncf 24927 df-limc 25915 df-dv 25916 df-ulm 26427 df-log 26608 df-atan 26919

This theorem is referenced by: atantayl2 26990

W3C validator