efiatan2 - Metamath Proof Explorer

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Theorem efiatan2 26411

Description: Value of the exponential of an artcangent. (Contributed by Mario Carneiro, 3-Apr-2015.)

**Assertion**
Ref	Expression
efiatan2	⊢ (𝐴 ∈ dom arctan → (exp‘(i · (arctan‘𝐴))) = ((1 + (i · 𝐴)) / (√‘(1 + (𝐴↑2)))))

**Proof of Theorem efiatan2**
Step	Hyp	Ref	Expression
1		ax-icn 11165	. . . . 5 ⊢ i ∈ ℂ
2		atancl 26375	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (arctan‘𝐴) ∈ ℂ)
3		mulcl 11190	. . . . 5 ⊢ ((i ∈ ℂ ∧ (arctan‘𝐴) ∈ ℂ) → (i · (arctan‘𝐴)) ∈ ℂ)
4	1, 2, 3	sylancr 587	. . . 4 ⊢ (𝐴 ∈ dom arctan → (i · (arctan‘𝐴)) ∈ ℂ)
5		efcl 16022	. . . 4 ⊢ ((i · (arctan‘𝐴)) ∈ ℂ → (exp‘(i · (arctan‘𝐴))) ∈ ℂ)
6	4, 5	syl 17	. . 3 ⊢ (𝐴 ∈ dom arctan → (exp‘(i · (arctan‘𝐴))) ∈ ℂ)
7		ax-1cn 11164	. . . . 5 ⊢ 1 ∈ ℂ
8		atandm2 26371	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
9	8	simp1bi 1145	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → 𝐴 ∈ ℂ)
10	9	sqcld 14105	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (𝐴↑2) ∈ ℂ)
11		addcl 11188	. . . . 5 ⊢ ((1 ∈ ℂ ∧ (𝐴↑2) ∈ ℂ) → (1 + (𝐴↑2)) ∈ ℂ)
12	7, 10, 11	sylancr 587	. . . 4 ⊢ (𝐴 ∈ dom arctan → (1 + (𝐴↑2)) ∈ ℂ)
13	12	sqrtcld 15380	. . 3 ⊢ (𝐴 ∈ dom arctan → (√‘(1 + (𝐴↑2))) ∈ ℂ)
14	12	sqsqrtd 15382	. . . . 5 ⊢ (𝐴 ∈ dom arctan → ((√‘(1 + (𝐴↑2)))↑2) = (1 + (𝐴↑2)))
15		atandm4 26373	. . . . . 6 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 + (𝐴↑2)) ≠ 0))
16	15	simprbi 497	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (1 + (𝐴↑2)) ≠ 0)
17	14, 16	eqnetrd 3008	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((√‘(1 + (𝐴↑2)))↑2) ≠ 0)
18		sqne0 14084	. . . . 5 ⊢ ((√‘(1 + (𝐴↑2))) ∈ ℂ → (((√‘(1 + (𝐴↑2)))↑2) ≠ 0 ↔ (√‘(1 + (𝐴↑2))) ≠ 0))
19	13, 18	syl 17	. . . 4 ⊢ (𝐴 ∈ dom arctan → (((√‘(1 + (𝐴↑2)))↑2) ≠ 0 ↔ (√‘(1 + (𝐴↑2))) ≠ 0))
20	17, 19	mpbid 231	. . 3 ⊢ (𝐴 ∈ dom arctan → (√‘(1 + (𝐴↑2))) ≠ 0)
21	6, 13, 20	divcan4d 11992	. 2 ⊢ (𝐴 ∈ dom arctan → (((exp‘(i · (arctan‘𝐴))) · (√‘(1 + (𝐴↑2)))) / (√‘(1 + (𝐴↑2)))) = (exp‘(i · (arctan‘𝐴))))
22		halfcn 12423	. . . . . . 7 ⊢ (1 / 2) ∈ ℂ
23	12, 16	logcld 26070	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (log‘(1 + (𝐴↑2))) ∈ ℂ)
24		mulcl 11190	. . . . . . 7 ⊢ (((1 / 2) ∈ ℂ ∧ (log‘(1 + (𝐴↑2))) ∈ ℂ) → ((1 / 2) · (log‘(1 + (𝐴↑2)))) ∈ ℂ)
25	22, 23, 24	sylancr 587	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → ((1 / 2) · (log‘(1 + (𝐴↑2)))) ∈ ℂ)
26		efadd 16033	. . . . . 6 ⊢ (((i · (arctan‘𝐴)) ∈ ℂ ∧ ((1 / 2) · (log‘(1 + (𝐴↑2)))) ∈ ℂ) → (exp‘((i · (arctan‘𝐴)) + ((1 / 2) · (log‘(1 + (𝐴↑2)))))) = ((exp‘(i · (arctan‘𝐴))) · (exp‘((1 / 2) · (log‘(1 + (𝐴↑2)))))))
27	4, 25, 26	syl2anc 584	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (exp‘((i · (arctan‘𝐴)) + ((1 / 2) · (log‘(1 + (𝐴↑2)))))) = ((exp‘(i · (arctan‘𝐴))) · (exp‘((1 / 2) · (log‘(1 + (𝐴↑2)))))))
28		2cn 12283	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
29	28	a1i 11	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom arctan → 2 ∈ ℂ)
30		mulcl 11190	. . . . . . . . . . . . . 14 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
31	1, 9, 30	sylancr 587	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom arctan → (i · 𝐴) ∈ ℂ)
32		addcl 11188	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + (i · 𝐴)) ∈ ℂ)
33	7, 31, 32	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan → (1 + (i · 𝐴)) ∈ ℂ)
34	8	simp3bi 1147	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan → (1 + (i · 𝐴)) ≠ 0)
35	33, 34	logcld 26070	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom arctan → (log‘(1 + (i · 𝐴))) ∈ ℂ)
36	29, 35, 4	subdid 11666	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom arctan → (2 · ((log‘(1 + (i · 𝐴))) − (i · (arctan‘𝐴)))) = ((2 · (log‘(1 + (i · 𝐴)))) − (2 · (i · (arctan‘𝐴)))))
37		atanval 26378	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom arctan → (arctan‘𝐴) = ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
38	37	oveq2d 7421	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan → ((2 · i) · (arctan‘𝐴)) = ((2 · i) · ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
39	1	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom arctan → i ∈ ℂ)
40	29, 39, 2	mulassd 11233	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan → ((2 · i) · (arctan‘𝐴)) = (2 · (i · (arctan‘𝐴))))
41		halfcl 12433	. . . . . . . . . . . . . . . . . 18 ⊢ (i ∈ ℂ → (i / 2) ∈ ℂ)
42	1, 41	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (i / 2) ∈ ℂ
43	28, 1, 42	mulassi 11221	. . . . . . . . . . . . . . . 16 ⊢ ((2 · i) · (i / 2)) = (2 · (i · (i / 2)))
44	28, 1, 42	mul12i 11405	. . . . . . . . . . . . . . . 16 ⊢ (2 · (i · (i / 2))) = (i · (2 · (i / 2)))
45		2ne0 12312	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ≠ 0
46	1, 28, 45	divcan2i 11953	. . . . . . . . . . . . . . . . . 18 ⊢ (2 · (i / 2)) = i
47	46	oveq2i 7416	. . . . . . . . . . . . . . . . 17 ⊢ (i · (2 · (i / 2))) = (i · i)
48		ixi 11839	. . . . . . . . . . . . . . . . 17 ⊢ (i · i) = -1
49	47, 48	eqtri 2760	. . . . . . . . . . . . . . . 16 ⊢ (i · (2 · (i / 2))) = -1
50	43, 44, 49	3eqtri 2764	. . . . . . . . . . . . . . 15 ⊢ ((2 · i) · (i / 2)) = -1
51	50	oveq1i 7415	. . . . . . . . . . . . . 14 ⊢ (((2 · i) · (i / 2)) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = (-1 · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
52		subcl 11455	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − (i · 𝐴)) ∈ ℂ)
53	7, 31, 52	sylancr 587	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ dom arctan → (1 − (i · 𝐴)) ∈ ℂ)
54	8	simp2bi 1146	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ dom arctan → (1 − (i · 𝐴)) ≠ 0)
55	53, 54	logcld 26070	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ dom arctan → (log‘(1 − (i · 𝐴))) ∈ ℂ)
56	55, 35	subcld 11567	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom arctan → ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))) ∈ ℂ)
57	56	mulm1d 11662	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom arctan → (-1 · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = -((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
58	51, 57	eqtrid 2784	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom arctan → (((2 · i) · (i / 2)) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = -((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
59		2mulicn 12431	. . . . . . . . . . . . . . 15 ⊢ (2 · i) ∈ ℂ
60	59	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom arctan → (2 · i) ∈ ℂ)
61	42	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom arctan → (i / 2) ∈ ℂ)
62	60, 61, 56	mulassd 11233	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom arctan → (((2 · i) · (i / 2)) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = ((2 · i) · ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
63	55, 35	negsubdi2d 11583	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom arctan → -((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))) = ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))))
64	58, 62, 63	3eqtr3d 2780	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan → ((2 · i) · ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))) = ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))))
65	38, 40, 64	3eqtr3d 2780	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom arctan → (2 · (i · (arctan‘𝐴))) = ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))))
66	65	oveq2d 7421	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom arctan → ((2 · (log‘(1 + (i · 𝐴)))) − (2 · (i · (arctan‘𝐴)))) = ((2 · (log‘(1 + (i · 𝐴)))) − ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
67		mulcl 11190	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ (log‘(1 + (i · 𝐴))) ∈ ℂ) → (2 · (log‘(1 + (i · 𝐴)))) ∈ ℂ)
68	28, 35, 67	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan → (2 · (log‘(1 + (i · 𝐴)))) ∈ ℂ)
69	68, 35, 55	subsubd 11595	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom arctan → ((2 · (log‘(1 + (i · 𝐴)))) − ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) = (((2 · (log‘(1 + (i · 𝐴)))) − (log‘(1 + (i · 𝐴)))) + (log‘(1 − (i · 𝐴)))))
70	35	2timesd 12451	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom arctan → (2 · (log‘(1 + (i · 𝐴)))) = ((log‘(1 + (i · 𝐴))) + (log‘(1 + (i · 𝐴)))))
71	35, 35, 70	mvrladdd 11623	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan → ((2 · (log‘(1 + (i · 𝐴)))) − (log‘(1 + (i · 𝐴)))) = (log‘(1 + (i · 𝐴))))
72	71	oveq1d 7420	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom arctan → (((2 · (log‘(1 + (i · 𝐴)))) − (log‘(1 + (i · 𝐴)))) + (log‘(1 − (i · 𝐴)))) = ((log‘(1 + (i · 𝐴))) + (log‘(1 − (i · 𝐴)))))
73		atanlogadd 26408	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom arctan → ((log‘(1 + (i · 𝐴))) + (log‘(1 − (i · 𝐴)))) ∈ ran log)
74		logef 26081	. . . . . . . . . . . . 13 ⊢ (((log‘(1 + (i · 𝐴))) + (log‘(1 − (i · 𝐴)))) ∈ ran log → (log‘(exp‘((log‘(1 + (i · 𝐴))) + (log‘(1 − (i · 𝐴)))))) = ((log‘(1 + (i · 𝐴))) + (log‘(1 − (i · 𝐴)))))
75	73, 74	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan → (log‘(exp‘((log‘(1 + (i · 𝐴))) + (log‘(1 − (i · 𝐴)))))) = ((log‘(1 + (i · 𝐴))) + (log‘(1 − (i · 𝐴)))))
76		efadd 16033	. . . . . . . . . . . . . . 15 ⊢ (((log‘(1 + (i · 𝐴))) ∈ ℂ ∧ (log‘(1 − (i · 𝐴))) ∈ ℂ) → (exp‘((log‘(1 + (i · 𝐴))) + (log‘(1 − (i · 𝐴))))) = ((exp‘(log‘(1 + (i · 𝐴)))) · (exp‘(log‘(1 − (i · 𝐴))))))
77	35, 55, 76	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom arctan → (exp‘((log‘(1 + (i · 𝐴))) + (log‘(1 − (i · 𝐴))))) = ((exp‘(log‘(1 + (i · 𝐴)))) · (exp‘(log‘(1 − (i · 𝐴))))))
78		eflog 26076	. . . . . . . . . . . . . . . 16 ⊢ (((1 + (i · 𝐴)) ∈ ℂ ∧ (1 + (i · 𝐴)) ≠ 0) → (exp‘(log‘(1 + (i · 𝐴)))) = (1 + (i · 𝐴)))
79	33, 34, 78	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom arctan → (exp‘(log‘(1 + (i · 𝐴)))) = (1 + (i · 𝐴)))
80		eflog 26076	. . . . . . . . . . . . . . . 16 ⊢ (((1 − (i · 𝐴)) ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0) → (exp‘(log‘(1 − (i · 𝐴)))) = (1 − (i · 𝐴)))
81	53, 54, 80	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom arctan → (exp‘(log‘(1 − (i · 𝐴)))) = (1 − (i · 𝐴)))
82	79, 81	oveq12d 7423	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom arctan → ((exp‘(log‘(1 + (i · 𝐴)))) · (exp‘(log‘(1 − (i · 𝐴))))) = ((1 + (i · 𝐴)) · (1 − (i · 𝐴))))
83		sq1 14155	. . . . . . . . . . . . . . . . 17 ⊢ (1↑2) = 1
84	83	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ dom arctan → (1↑2) = 1)
85		sqmul 14080	. . . . . . . . . . . . . . . . . 18 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((i · 𝐴)↑2) = ((i↑2) · (𝐴↑2)))
86	1, 9, 85	sylancr 587	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ dom arctan → ((i · 𝐴)↑2) = ((i↑2) · (𝐴↑2)))
87		i2 14162	. . . . . . . . . . . . . . . . . . 19 ⊢ (i↑2) = -1
88	87	oveq1i 7415	. . . . . . . . . . . . . . . . . 18 ⊢ ((i↑2) · (𝐴↑2)) = (-1 · (𝐴↑2))
89	10	mulm1d 11662	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ dom arctan → (-1 · (𝐴↑2)) = -(𝐴↑2))
90	88, 89	eqtrid 2784	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ dom arctan → ((i↑2) · (𝐴↑2)) = -(𝐴↑2))
91	86, 90	eqtrd 2772	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ dom arctan → ((i · 𝐴)↑2) = -(𝐴↑2))
92	84, 91	oveq12d 7423	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom arctan → ((1↑2) − ((i · 𝐴)↑2)) = (1 − -(𝐴↑2)))
93		subsq 14170	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → ((1↑2) − ((i · 𝐴)↑2)) = ((1 + (i · 𝐴)) · (1 − (i · 𝐴))))
94	7, 31, 93	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom arctan → ((1↑2) − ((i · 𝐴)↑2)) = ((1 + (i · 𝐴)) · (1 − (i · 𝐴))))
95		subneg 11505	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℂ ∧ (𝐴↑2) ∈ ℂ) → (1 − -(𝐴↑2)) = (1 + (𝐴↑2)))
96	7, 10, 95	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom arctan → (1 − -(𝐴↑2)) = (1 + (𝐴↑2)))
97	92, 94, 96	3eqtr3d 2780	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom arctan → ((1 + (i · 𝐴)) · (1 − (i · 𝐴))) = (1 + (𝐴↑2)))
98	77, 82, 97	3eqtrd 2776	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom arctan → (exp‘((log‘(1 + (i · 𝐴))) + (log‘(1 − (i · 𝐴))))) = (1 + (𝐴↑2)))
99	98	fveq2d 6892	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan → (log‘(exp‘((log‘(1 + (i · 𝐴))) + (log‘(1 − (i · 𝐴)))))) = (log‘(1 + (𝐴↑2))))
100	75, 99	eqtr3d 2774	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom arctan → ((log‘(1 + (i · 𝐴))) + (log‘(1 − (i · 𝐴)))) = (log‘(1 + (𝐴↑2))))
101	69, 72, 100	3eqtrd 2776	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom arctan → ((2 · (log‘(1 + (i · 𝐴)))) − ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) = (log‘(1 + (𝐴↑2))))
102	36, 66, 101	3eqtrd 2776	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (2 · ((log‘(1 + (i · 𝐴))) − (i · (arctan‘𝐴)))) = (log‘(1 + (𝐴↑2))))
103	102	oveq1d 7420	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → ((2 · ((log‘(1 + (i · 𝐴))) − (i · (arctan‘𝐴)))) / 2) = ((log‘(1 + (𝐴↑2))) / 2))
104	35, 4	subcld 11567	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → ((log‘(1 + (i · 𝐴))) − (i · (arctan‘𝐴))) ∈ ℂ)
105	45	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → 2 ≠ 0)
106	104, 29, 105	divcan3d 11991	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → ((2 · ((log‘(1 + (i · 𝐴))) − (i · (arctan‘𝐴)))) / 2) = ((log‘(1 + (i · 𝐴))) − (i · (arctan‘𝐴))))
107	23, 29, 105	divrec2d 11990	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → ((log‘(1 + (𝐴↑2))) / 2) = ((1 / 2) · (log‘(1 + (𝐴↑2)))))
108	103, 106, 107	3eqtr3d 2780	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((log‘(1 + (i · 𝐴))) − (i · (arctan‘𝐴))) = ((1 / 2) · (log‘(1 + (𝐴↑2)))))
109	35, 4, 25	subaddd 11585	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (((log‘(1 + (i · 𝐴))) − (i · (arctan‘𝐴))) = ((1 / 2) · (log‘(1 + (𝐴↑2)))) ↔ ((i · (arctan‘𝐴)) + ((1 / 2) · (log‘(1 + (𝐴↑2))))) = (log‘(1 + (i · 𝐴)))))
110	108, 109	mpbid 231	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → ((i · (arctan‘𝐴)) + ((1 / 2) · (log‘(1 + (𝐴↑2))))) = (log‘(1 + (i · 𝐴))))
111	110	fveq2d 6892	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (exp‘((i · (arctan‘𝐴)) + ((1 / 2) · (log‘(1 + (𝐴↑2)))))) = (exp‘(log‘(1 + (i · 𝐴)))))
112	27, 111	eqtr3d 2774	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((exp‘(i · (arctan‘𝐴))) · (exp‘((1 / 2) · (log‘(1 + (𝐴↑2)))))) = (exp‘(log‘(1 + (i · 𝐴)))))
113	22	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (1 / 2) ∈ ℂ)
114	12, 16, 113	cxpefd 26211	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → ((1 + (𝐴↑2))↑_𝑐(1 / 2)) = (exp‘((1 / 2) · (log‘(1 + (𝐴↑2))))))
115		cxpsqrt 26202	. . . . . . 7 ⊢ ((1 + (𝐴↑2)) ∈ ℂ → ((1 + (𝐴↑2))↑_𝑐(1 / 2)) = (√‘(1 + (𝐴↑2))))
116	12, 115	syl 17	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → ((1 + (𝐴↑2))↑_𝑐(1 / 2)) = (√‘(1 + (𝐴↑2))))
117	114, 116	eqtr3d 2774	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (exp‘((1 / 2) · (log‘(1 + (𝐴↑2))))) = (√‘(1 + (𝐴↑2))))
118	117	oveq2d 7421	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((exp‘(i · (arctan‘𝐴))) · (exp‘((1 / 2) · (log‘(1 + (𝐴↑2)))))) = ((exp‘(i · (arctan‘𝐴))) · (√‘(1 + (𝐴↑2)))))
119	112, 118, 79	3eqtr3d 2780	. . 3 ⊢ (𝐴 ∈ dom arctan → ((exp‘(i · (arctan‘𝐴))) · (√‘(1 + (𝐴↑2)))) = (1 + (i · 𝐴)))
120	119	oveq1d 7420	. 2 ⊢ (𝐴 ∈ dom arctan → (((exp‘(i · (arctan‘𝐴))) · (√‘(1 + (𝐴↑2)))) / (√‘(1 + (𝐴↑2)))) = ((1 + (i · 𝐴)) / (√‘(1 + (𝐴↑2)))))
121	21, 120	eqtr3d 2774	1 ⊢ (𝐴 ∈ dom arctan → (exp‘(i · (arctan‘𝐴))) = ((1 + (i · 𝐴)) / (√‘(1 + (𝐴↑2)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 dom cdm 5675 ran crn 5676 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 0cc0 11106 1c1 11107 ici 11108 + caddc 11109 · cmul 11111 − cmin 11440 -cneg 11441 / cdiv 11867 2c2 12263 ↑cexp 14023 √csqrt 15176 expce 16001 logclog 26054 ↑_𝑐ccxp 26055 arctancatan 26358

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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-limc 25374 df-dv 25375 df-log 26056 df-cxp 26057 df-atan 26361

This theorem is referenced by: cosatan 26415

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