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Theorem atantayl 26324

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 12815	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12543	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 1 ∈ ℤ)
3		ax-icn 11119	. . . 4 ⊢ i ∈ ℂ
4		halfcl 12387	. . . 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 11144	. . . . . . . . 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 11508	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → -(i · 𝐴) ∈ ℂ)
10	8	absnegd 15346	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘-(i · 𝐴)) = (abs‘(i · 𝐴)))
11		absmul 15191	. . . . . . . . . 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 15183	. . . . . . . . . . 11 ⊢ (abs‘i) = 1
14	13	oveq1i 7372	. . . . . . . . . 10 ⊢ ((abs‘i) · (abs‘𝐴)) = (1 · (abs‘𝐴))
15		abscl 15175	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ)
16	15	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) ∈ ℝ)
17	16	recnd 11192	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) ∈ ℂ)
18	17	mullidd 11182	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (1 · (abs‘𝐴)) = (abs‘𝐴))
19	14, 18	eqtrid 2783	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((abs‘i) · (abs‘𝐴)) = (abs‘𝐴))
20	10, 12, 19	3eqtrd 2775	. . . . . . . 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 5132	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘-(i · 𝐴)) < 1)
23		logtayl 26052	. . . . . . 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 11118	. . . . . . . . 9 ⊢ 1 ∈ ℂ
26		subneg 11459	. . . . . . . . 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 6851	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (log‘(1 − -(i · 𝐴))) = (log‘(1 + (i · 𝐴))))
29	28	negeqd 11404	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → -(log‘(1 − -(i · 𝐴))) = -(log‘(1 + (i · 𝐴))))
30	24, 29	breqtrd 5136	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))) ⇝ -(log‘(1 + (i · 𝐴))))
31		seqex 13918	. . . . . 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 5134	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(i · 𝐴)) < 1)
34		logtayl 26052	. . . . . 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 7370	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (-(i · 𝐴)↑𝑛) = (-(i · 𝐴)↑𝑚))
37		id 22	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → 𝑛 = 𝑚)
38	36, 37	oveq12d 7380	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → ((-(i · 𝐴)↑𝑛) / 𝑛) = ((-(i · 𝐴)↑𝑚) / 𝑚))
39		eqid 2731	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛)) = (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))
40		ovex 7395	. . . . . . . . . 10 ⊢ ((-(i · 𝐴)↑𝑚) / 𝑚) ∈ V
41	38, 39, 40	fvmpt 6953	. . . . . . . . 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 12429	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ₀)
44		expcl 13995	. . . . . . . . . 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 12170	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
47	46	adantl 482	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
48		nnne0 12196	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0)
49	48	adantl 482	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0)
50	45, 47, 49	divcld 11940	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((-(i · 𝐴)↑𝑚) / 𝑚) ∈ ℂ)
51	42, 50	eqeltrd 2832	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))‘𝑚) ∈ ℂ)
52	1, 2, 51	serf 13946	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛))):ℕ⟶ℂ)
53	52	ffvelcdmda 7040	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑘 ∈ ℕ) → (seq1( + , (𝑛 ∈ ℕ ↦ ((-(i · 𝐴)↑𝑛) / 𝑛)))‘𝑘) ∈ ℂ)
54		oveq2 7370	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ((i · 𝐴)↑𝑛) = ((i · 𝐴)↑𝑚))
55	54, 37	oveq12d 7380	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (((i · 𝐴)↑𝑛) / 𝑛) = (((i · 𝐴)↑𝑚) / 𝑚))
56		eqid 2731	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛)) = (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))
57		ovex 7395	. . . . . . . . . 10 ⊢ (((i · 𝐴)↑𝑚) / 𝑚) ∈ V
58	55, 56, 57	fvmpt 6953	. . . . . . . . 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 13995	. . . . . . . . . 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 11940	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (((i · 𝐴)↑𝑚) / 𝑚) ∈ ℂ)
63	59, 62	eqeltrd 2832	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))‘𝑚) ∈ ℂ)
64	1, 2, 63	serf 13946	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑛 ∈ ℕ ↦ (((i · 𝐴)↑𝑛) / 𝑛))):ℕ⟶ℂ)
65	64	ffvelcdmda 7040	. . . . 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 2842	. . . . . 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 13480	. . . . . . 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 7380	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛)) = (((-(i · 𝐴)↑𝑚) / 𝑚) − (((i · 𝐴)↑𝑚) / 𝑚)))
73		eqid 2731	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛))) = (𝑛 ∈ ℕ ↦ (((-(i · 𝐴)↑𝑛) / 𝑛) − (((i · 𝐴)↑𝑛) / 𝑛)))
74		ovex 7395	. . . . . . . . . 10 ⊢ (((-(i · 𝐴)↑𝑚) / 𝑚) − (((i · 𝐴)↑𝑚) / 𝑚)) ∈ V
75	72, 73, 74	fvmpt 6953	. . . . . . . . 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 7380	. . . . . . . 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 2774	. . . . . . 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 13961	. . . . 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 15528	. . . 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 11142	. . . . . . 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 26316	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ dom arctan)
85		atandm2 26264	. . . . . . . 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 25963	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (log‘(1 + (i · 𝐴))) ∈ ℂ)
89		subcl 11409	. . . . . . 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 25963	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (log‘(1 − (i · 𝐴))) ∈ ℂ)
93	88, 92	neg2subd 11538	. . . 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 5136	. . 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 11521	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (((-(i · 𝐴)↑𝑚) / 𝑚) − (((i · 𝐴)↑𝑚) / 𝑚)) ∈ ℂ)
96	76, 95	eqeltrd 2832	. . 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 11411	. . . . . . . . 9 ⊢ -i ∈ ℂ
99	43	adantl 482	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ₀)
100		expcl 13995	. . . . . . . . 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 13995	. . . . . . . . 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 11521	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((-i↑𝑚) − (i↑𝑚)) ∈ ℂ)
105		2cnd 12240	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 2 ∈ ℂ)
106		2ne0 12266	. . . . . . . 8 ⊢ 2 ≠ 0
107	106	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → 2 ≠ 0)
108	97, 104, 105, 107	div23d 11977	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((i · ((-i↑𝑚) − (i↑𝑚))) / 2) = ((i / 2) · ((-i↑𝑚) − (i↑𝑚))))
109	108	oveq1d 7377	. . . . 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 13995	. . . . . . . 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 11940	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((𝐴↑𝑚) / 𝑚) ∈ ℂ)
114	110, 104, 113	mulassd 11187	. . . . 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 11621	. . . . . . . . 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 11600	. . . . . . . . . . . . 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 7377	. . . . . . . . . . 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 14076	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((-i · 𝐴)↑𝑚) = ((-i↑𝑚) · (𝐴↑𝑚)))
122	119, 121	eqtr3d 2773	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (-(i · 𝐴)↑𝑚) = ((-i↑𝑚) · (𝐴↑𝑚)))
123	97, 116, 99	mulexpd 14076	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → ((i · 𝐴)↑𝑚) = ((i↑𝑚) · (𝐴↑𝑚)))
124	122, 123	oveq12d 7380	. . . . . . . . 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 2774	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) ∧ 𝑚 ∈ ℕ) → (((-i↑𝑚) − (i↑𝑚)) · (𝐴↑𝑚)) = ((-(i · 𝐴)↑𝑚) − ((i · 𝐴)↑𝑚)))
126	125	oveq1d 7377	. . . . . . 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 11975	. . . . . . 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 11979	. . . . . . 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 2779	. . . . . 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 7378	. . . . 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 2775	. . . 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 7370	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (-i↑𝑛) = (-i↑𝑚))
133		oveq2 7370	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (i↑𝑛) = (i↑𝑚))
134	132, 133	oveq12d 7380	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → ((-i↑𝑛) − (i↑𝑛)) = ((-i↑𝑚) − (i↑𝑚)))
135	134	oveq2d 7378	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (i · ((-i↑𝑛) − (i↑𝑛))) = (i · ((-i↑𝑚) − (i↑𝑚))))
136	135	oveq1d 7377	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((i · ((-i↑𝑛) − (i↑𝑛))) / 2) = ((i · ((-i↑𝑚) − (i↑𝑚))) / 2))
137		oveq2 7370	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝐴↑𝑛) = (𝐴↑𝑚))
138	137, 37	oveq12d 7380	. . . . . . 7 ⊢ (𝑛 = 𝑚 → ((𝐴↑𝑛) / 𝑛) = ((𝐴↑𝑚) / 𝑚))
139	136, 138	oveq12d 7380	. . . . . 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 7395	. . . . . 6 ⊢ (((i · ((-i↑𝑚) − (i↑𝑚))) / 2) · ((𝐴↑𝑚) / 𝑚)) ∈ V
142	139, 140, 141	fvmpt 6953	. . . . 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 7378	. . . 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 2781	. . 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 15554	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , 𝐹) ⇝ ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
147		atanval 26271	. . 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 5138	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 2939 Vcvv 3446 class class class wbr 5110 ↦ cmpt 5193 dom cdm 5638 ‘cfv 6501 (class class class)co 7362 ℂcc 11058 ℝcr 11059 0cc0 11060 1c1 11061 ici 11062 + caddc 11063 · cmul 11065 < clt 11198 − cmin 11394 -cneg 11395 / cdiv 11821 ℕcn 12162 2c2 12217 ℕ₀cn0 12422 ℤ_≥cuz 12772 ...cfz 13434 seqcseq 13916 ↑cexp 13977 abscabs 15131 ⇝ cli 15378 logclog 25947 arctancatan 26251

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 2702 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9586 ax-cnex 11116 ax-resscn 11117 ax-1cn 11118 ax-icn 11119 ax-addcl 11120 ax-addrcl 11121 ax-mulcl 11122 ax-mulrcl 11123 ax-mulcom 11124 ax-addass 11125 ax-mulass 11126 ax-distr 11127 ax-i2m1 11128 ax-1ne0 11129 ax-1rid 11130 ax-rnegex 11131 ax-rrecex 11132 ax-cnre 11133 ax-pre-lttri 11134 ax-pre-lttrn 11135 ax-pre-ltadd 11136 ax-pre-mulgt0 11137 ax-pre-sup 11138 ax-addf 11139 ax-mulf 11140

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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3448 df-sbc 3743 df-csb 3859 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-er 8655 df-map 8774 df-pm 8775 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-fi 9356 df-sup 9387 df-inf 9388 df-oi 9455 df-card 9884 df-pnf 11200 df-mnf 11201 df-xr 11202 df-ltxr 11203 df-le 11204 df-sub 11396 df-neg 11397 df-div 11822 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12423 df-z 12509 df-dec 12628 df-uz 12773 df-q 12883 df-rp 12925 df-xneg 13042 df-xadd 13043 df-xmul 13044 df-ioo 13278 df-ioc 13279 df-ico 13280 df-icc 13281 df-fz 13435 df-fzo 13578 df-fl 13707 df-mod 13785 df-seq 13917 df-exp 13978 df-fac 14184 df-bc 14213 df-hash 14241 df-shft 14964 df-cj 14996 df-re 14997 df-im 14998 df-sqrt 15132 df-abs 15133 df-limsup 15365 df-clim 15382 df-rlim 15383 df-sum 15583 df-ef 15961 df-sin 15963 df-cos 15964 df-tan 15965 df-pi 15966 df-struct 17030 df-sets 17047 df-slot 17065 df-ndx 17077 df-base 17095 df-ress 17124 df-plusg 17160 df-mulr 17161 df-starv 17162 df-sca 17163 df-vsca 17164 df-ip 17165 df-tset 17166 df-ple 17167 df-ds 17169 df-unif 17170 df-hom 17171 df-cco 17172 df-rest 17318 df-topn 17319 df-0g 17337 df-gsum 17338 df-topgen 17339 df-pt 17340 df-prds 17343 df-xrs 17398 df-qtop 17403 df-imas 17404 df-xps 17406 df-mre 17480 df-mrc 17481 df-acs 17483 df-mgm 18511 df-sgrp 18560 df-mnd 18571 df-submnd 18616 df-mulg 18887 df-cntz 19111 df-cmn 19578 df-psmet 20825 df-xmet 20826 df-met 20827 df-bl 20828 df-mopn 20829 df-fbas 20830 df-fg 20831 df-cnfld 20834 df-top 22280 df-topon 22297 df-topsp 22319 df-bases 22333 df-cld 22407 df-ntr 22408 df-cls 22409 df-nei 22486 df-lp 22524 df-perf 22525 df-cn 22615 df-cnp 22616 df-haus 22703 df-cmp 22775 df-tx 22950 df-hmeo 23143 df-fil 23234 df-fm 23326 df-flim 23327 df-flf 23328 df-xms 23710 df-ms 23711 df-tms 23712 df-cncf 24278 df-limc 25267 df-dv 25268 df-ulm 25773 df-log 25949 df-atan 26254

This theorem is referenced by: atantayl2 26325

W3C validator