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Theorem atantayl 27311

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 13042	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12765	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 1 ∈ ℤ)
3		ax-icn 11296	. . . 4 ⊢ i ∈ ℂ
4		halfcl 12610	. . . 4 ⊢ (i ∈ ℂ → (i / 2) ∈ ℂ)
5	3, 4	mp1i 14	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (i / 2) ∈ ℂ)
6		simpl 488	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ ℂ)
7		mulcl 11321	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
8	3, 6, 7	sylancr 599	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (i · 𝐴) ∈ ℂ)
9	8	negcld 11693	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → -(i · 𝐴) ∈ ℂ)
10	8	absnegd 15660	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘ -(i · 𝐴)) = (abs‘(i · 𝐴)))
11		absmul 15502	. . . . . . . . . 10 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (abs‘(i · 𝐴)) = ((abs‘i) · (abs‘𝐴)))
12	3, 6, 11	sylancr 599	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(i · 𝐴)) = ((abs‘i) · (abs‘𝐴)))
13		absi 15494	. . . . . . . . . . 11 ⊢ (abs‘i) = 1
14	13	oveq1i 7459	. . . . . . . . . 10 ⊢ ((abs‘i) · (abs‘𝐴)) = (1 · (abs‘𝐴))
15		abscl 15486	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
16	15	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) ∈ ℝ)
17	16	recnd 11374	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) ∈ ℂ)
18	17	mullidd 11364	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (1 · (abs‘𝐴)) = (abs‘𝐴))
19	14, 18	eqtrid 2810	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((abs‘i) · (abs‘𝐴)) = (abs‘𝐴))
20	10, 12, 19	3eqtrd 2802	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘ -(i · 𝐴)) = (abs‘𝐴))
21		simpr 490	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) < 1)
22	20, 21	eqbrtrd 5131	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘ -(i · 𝐴)) < 1)
23		logtayl 27034	. . . . . . 7 ⊢ (( -(i · 𝐴) ∈ ℂ ∧ (abs‘ -(i · 𝐴)) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (( -(i · 𝐴)↑𝑛) / 𝑛))) ⇝ -(log‘(1 − -(i · 𝐴))))
24	9, 22, 23	syl2anc 596	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (( -(i · 𝐴)↑𝑛) / 𝑛))) ⇝ -(log‘(1 − -(i · 𝐴))))
25		ax-1cn 11295	. . . . . . . . 9 ⊢ 1 ∈ ℂ
26		subneg 11644	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − -(i · 𝐴)) = (1 + (i · 𝐴)))
27	25, 8, 26	sylancr 599	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (1 − -(i · 𝐴)) = (1 + (i · 𝐴)))
28	27	fveq2d 6918	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (log‘(1 − -(i · 𝐴))) = (log‘(1 + (i · 𝐴))))
29	28	negeqd 11588	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → -(log‘(1 − -(i · 𝐴))) = -(log‘(1 + (i · 𝐴))))
30	24, 29	breqtrd 5135	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (( -(i · 𝐴)↑𝑛) / 𝑛))) ⇝ -(log‘(1 + (i · 𝐴))))
31		seqex 14184	. . . . . 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 5133	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(i · 𝐴)) < 1)
34		logtayl 27034	. . . . . 6 ⊢ (((i · 𝐴) ∈ ℂ ∧ (abs‘(i · 𝐴)) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))) ⇝ -(log‘(1 − (i · 𝐴))))
35	8, 33, 34	syl2anc 596	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))) ⇝ -(log‘(1 − (i · 𝐴))))
36		oveq2 7457	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ( -(i · 𝐴)↑𝑛) = ( -(i · 𝐴)↑𝑚))
37		id 23	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → 𝑛 = 𝑚)
38	36, 37	oveq12d 7467	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (( -(i · 𝐴)↑𝑛) / 𝑛) = (( -(i · 𝐴)↑𝑚) / 𝑚))
39		eqid 2763	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ (( -(i · 𝐴)↑𝑛) / 𝑛)) = (𝑛 ∈ ℕ ↦ (( -(i · 𝐴)↑𝑛) / 𝑛))
40		ovex 7482	. . . . . . . . . 10 ⊢ (( -(i · 𝐴)↑𝑚) / 𝑚) ∈ V
41	38, 39, 40	fvmpt 7022	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (( -(i · 𝐴)↑𝑛) / 𝑛))‘𝑚) = (( -(i · 𝐴)↑𝑚) / 𝑚))
42	41	adantl 487	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (( -(i · 𝐴)↑𝑛) / 𝑛))‘𝑚) = (( -(i · 𝐴)↑𝑚) / 𝑚))
43		nnnn0 12651	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ₀)
44		expcl 14260	. . . . . . . . . 10 ⊢ (( -(i · 𝐴) ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → ( -(i · 𝐴)↑𝑚) ∈ ℂ)
45	9, 43, 44	syl2an 608	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ( -(i · 𝐴)↑𝑚) ∈ ℂ)
46		nncn 12381	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
47	46	adantl 487	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
48		nnne0 12410	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0)
49	48	adantl 487	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0)
50	45, 47, 49	divcld 12131	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (( -(i · 𝐴)↑𝑚) / 𝑚) ∈ ℂ)
51	42, 50	eqeltrd 2864	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (( -(i · 𝐴)↑𝑛) / 𝑛))‘𝑚) ∈ ℂ)
52	1, 2, 51	serf 14211	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (( -(i · 𝐴)↑𝑛) / 𝑛))):ℕ⟶ℂ)
53	52	ffvelcdmda 7114	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (seq1( + , (𝑛 ∈ ℕ ↦ (( -(i · 𝐴)↑𝑛) / 𝑛)))‘𝑘) ∈ ℂ)
54		oveq2 7457	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ((i · 𝐴)↑𝑛) = ((i · 𝐴)↑𝑚))
55	54, 37	oveq12d 7467	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (((i · 𝐴)↑𝑛) / 𝑛) = (((i · 𝐴)↑𝑚) / 𝑚))
56		eqid 2763	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛)) = (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))
57		ovex 7482	. . . . . . . . . 10 ⊢ (((i · 𝐴)↑𝑚) / 𝑚) ∈ V
58	55, 56, 57	fvmpt 7022	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))‘𝑚) = (((i · 𝐴)↑𝑚) / 𝑚))
59	58	adantl 487	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))‘𝑚) = (((i · 𝐴)↑𝑚) / 𝑚))
60		expcl 14260	. . . . . . . . . 10 ⊢ (((i · 𝐴) ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → ((i · 𝐴)↑𝑚) ∈ ℂ)
61	8, 43, 60	syl2an 608	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((i · 𝐴)↑𝑚) ∈ ℂ)
62	61, 47, 49	divcld 12131	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (((i · 𝐴)↑𝑚) / 𝑚) ∈ ℂ)
63	59, 62	eqeltrd 2864	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))‘𝑚) ∈ ℂ)
64	1, 2, 63	serf 14211	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))):ℕ⟶ℂ)
65	64	ffvelcdmda 7114	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (seq1( + , (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛)))‘𝑘) ∈ ℂ)
66		simpr 490	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
67	66, 1	eleqtrdi 2874	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ_≥‘1))
68		simpl 488	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1))
69		elfznn 13725	. . . . . . 7 ⊢ (𝑚 ∈ (1...𝑘) → 𝑚 ∈ ℕ)
70	68, 69, 51	syl2an 608	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((𝑛 ∈ ℕ ↦ (( -(i · 𝐴)↑𝑛) / 𝑛))‘𝑚) ∈ ℂ)
71	68, 69, 63	syl2an 608	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))‘𝑚) ∈ ℂ)
72	38, 55	oveq12d 7467	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → ((( -(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛)) = ((( -(i · 𝐴)↑𝑚) / 𝑚) − (((i · 𝐴)↑𝑚) / 𝑚)))
73		eqid 2763	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ ((( -(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛))) = (𝑛 ∈ ℕ ↦ ((( -(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛)))
74		ovex 7482	. . . . . . . . . 10 ⊢ ((( -(i · 𝐴)↑𝑚) / 𝑚) − (((i · 𝐴)↑𝑚) / 𝑚)) ∈ V
75	72, 73, 74	fvmpt 7022	. . . . . . . . 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 487	. . . . . . . 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 7467	. . . . . . . 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 2801	. . . . . . 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 608	. . . . . 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 14226	. . . . 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 15842	. . . 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 11319	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + (i · 𝐴)) ∈ ℂ)
83	25, 8, 82	sylancr 599	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (1 + (i · 𝐴)) ∈ ℂ)
84		bndatandm 27303	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ dom arctan)
85		atandm2 27251	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
86	84, 85	sylib 221	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
87	86	simp3d 1162	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (1 + (i · 𝐴)) ≠ 0)
88	83, 87	logcld 26944	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (log‘(1 + (i · 𝐴))) ∈ ℂ)
89		subcl 11593	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − (i · 𝐴)) ∈ ℂ)
90	25, 8, 89	sylancr 599	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (1 − (i · 𝐴)) ∈ ℂ)
91	86	simp2d 1161	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (1 − (i · 𝐴)) ≠ 0)
92	90, 91	logcld 26944	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (log‘(1 − (i · 𝐴))) ∈ ℂ)
93	88, 92	neg2subd 11723	. . . 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 5135	. . 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 11706	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((( -(i · 𝐴)↑𝑚) / 𝑚) − (((i · 𝐴)↑𝑚) / 𝑚)) ∈ ℂ)
96	76, 95	eqeltrd 2864	. . 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 11595	. . . . . . . . 9 ⊢ -i ∈ ℂ
99	43	adantl 487	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ₀)
100		expcl 14260	. . . . . . . . 9 ⊢ (( -i ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → ( -i↑𝑚) ∈ ℂ)
101	98, 99, 100	sylancr 599	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ( -i↑𝑚) ∈ ℂ)
102		expcl 14260	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → (i↑𝑚) ∈ ℂ)
103	3, 99, 102	sylancr 599	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (i↑𝑚) ∈ ℂ)
104	101, 103	subcld 11706	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (( -i↑𝑚) − (i↑𝑚)) ∈ ℂ)
105		2cnd 12459	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 2 ∈ ℂ)
106		2ne0 12487	. . . . . . . 8 ⊢ 2 ≠ 0
107	106	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 2 ≠ 0)
108	97, 104, 105, 107	div23d 12168	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((i · (( -i↑𝑚) − (i↑𝑚))) / 2) = ((i / 2) · (( -i↑𝑚) − (i↑𝑚))))
109	108	oveq1d 7464	. . . . 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 486	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (i / 2) ∈ ℂ)
111		expcl 14260	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑚 ∈ ℕ₀) → (𝐴↑𝑚) ∈ ℂ)
112	6, 43, 111	syl2an 608	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (𝐴↑𝑚) ∈ ℂ)
113	112, 47, 49	divcld 12131	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝐴↑𝑚) / 𝑚) ∈ ℂ)
114	110, 104, 113	mulassd 11369	. . . . 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 11810	. . . . . . . . 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 486	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 𝐴 ∈ ℂ)
117		mulneg1 11789	. . . . . . . . . . . . 13 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → ( -i · 𝐴) = -(i · 𝐴))
118	3, 116, 117	sylancr 599	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ( -i · 𝐴) = -(i · 𝐴))
119	118	oveq1d 7464	. . . . . . . . . . 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 14342	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (( -i · 𝐴)↑𝑚) = (( -i↑𝑚) · (𝐴↑𝑚)))
122	119, 121	eqtr3d 2800	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ( -(i · 𝐴)↑𝑚) = (( -i↑𝑚) · (𝐴↑𝑚)))
123	97, 116, 99	mulexpd 14342	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((i · 𝐴)↑𝑚) = ((i↑𝑚) · (𝐴↑𝑚)))
124	122, 123	oveq12d 7467	. . . . . . . . 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 2801	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((( -i↑𝑚) − (i↑𝑚)) · (𝐴↑𝑚)) = (( -(i · 𝐴)↑𝑚) − ((i · 𝐴)↑𝑚)))
126	125	oveq1d 7464	. . . . . . 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 12166	. . . . . . 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 12170	. . . . . . 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 2806	. . . . . 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 7465	. . . . 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 2802	. . . 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 7457	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → ( -i↑𝑛) = ( -i↑𝑚))
133		oveq2 7457	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (i↑𝑛) = (i↑𝑚))
134	132, 133	oveq12d 7467	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (( -i↑𝑛) − (i↑𝑛)) = (( -i↑𝑚) − (i↑𝑚)))
135	134	oveq2d 7465	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (i · (( -i↑𝑛) − (i↑𝑛))) = (i · (( -i↑𝑚) − (i↑𝑚))))
136	135	oveq1d 7464	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((i · (( -i↑𝑛) − (i↑𝑛))) / 2) = ((i · (( -i↑𝑚) − (i↑𝑚))) / 2))
137		oveq2 7457	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝐴↑𝑛) = (𝐴↑𝑚))
138	137, 37	oveq12d 7467	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((𝐴↑𝑛) / 𝑛) = ((𝐴↑𝑚) / 𝑚))
139	136, 138	oveq12d 7467	. . . . . 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 7482	. . . . . 6 ⊢ (((i · (( -i↑𝑚) − (i↑𝑚))) / 2) · ((𝐴↑𝑚) / 𝑚)) ∈ V
142	139, 140, 141	fvmpt 7022	. . . . 5 ⊢ (𝑚 ∈ ℕ → (𝐹‘𝑚) = (((i · (( -i↑𝑚) − (i↑𝑚))) / 2) · ((𝐴↑𝑚) / 𝑚)))
143	142	adantl 487	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) = (((i · (( -i↑𝑚) − (i↑𝑚))) / 2) · ((𝐴↑𝑚) / 𝑚)))
144	76	oveq2d 7465	. . . 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 2808	. . 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 15866	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , 𝐹) ⇝ ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
147		atanval 27258	. . 3 ⊢ (𝐴 ∈ dom arctan → (arctan‘𝐴) = ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
148	84, 147	syl 18	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (arctan‘𝐴) = ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
149	146, 148	breqtrrd 5137	1 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , 𝐹) ⇝ (arctan‘𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 Vcvv 3454 class class class wbr 5106 ↦ cmpt 5190 dom cdm 5656 ‘cfv 6564 (class class class)co 7449 ℂcc 11235 ℝcr 11236 0cc0 11237 1c1 11238 ici 11239 + caddc 11240 · cmul 11242 < clt 11380 − cmin 11578 -cneg 11579 / cdiv 12011 ℕcn 12373 2c2 12435 ℕ₀cn0 12644 ℤ_≥cuz 13003 ...cfz 13677 seqcseq 14182 ↑cexp 14242 abscabs 15442 ⇝ cli 15692 logclog 26928 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-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: atantayl2 27312

W3C validator