atancj - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > atancj		Structured version Visualization version GIF version

Theorem atancj 26848

Description: The arctangent function distributes under conjugation. (The condition that ℜ(𝐴) ≠ 0 is necessary because the branch cuts are chosen so that the negative imaginary line "agrees with" neighboring values with negative real part, while the positive imaginary line agrees with values with positive real part. This makes atanneg 26845 true unconditionally but messes up conjugation symmetry, and it is impossible to have both in a single-valued function. The claim is true on the imaginary line between -1 and 1, though.) (Contributed by Mario Carneiro, 31-Mar-2015.)

**Assertion**
Ref	Expression
atancj	⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (𝐴 ∈ dom arctan ∧ (∗‘(arctan‘𝐴)) = (arctan‘(∗‘𝐴))))

**Proof of Theorem atancj**
Step	Hyp	Ref	Expression
1		simpl 482	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → 𝐴 ∈ ℂ)
2		simpr 484	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℜ‘𝐴) ≠ 0)
3		fveq2 6822	. . . . . 6 ⊢ (𝐴 = -i → (ℜ‘𝐴) = (ℜ‘-i))
4		ax-icn 11065	. . . . . . . 8 ⊢ i ∈ ℂ
5	4	renegi 15087	. . . . . . 7 ⊢ (ℜ‘-i) = -(ℜ‘i)
6		rei 15063	. . . . . . . 8 ⊢ (ℜ‘i) = 0
7	6	negeqi 11353	. . . . . . 7 ⊢ -(ℜ‘i) = -0
8		neg0 11407	. . . . . . 7 ⊢ -0 = 0
9	5, 7, 8	3eqtri 2758	. . . . . 6 ⊢ (ℜ‘-i) = 0
10	3, 9	eqtrdi 2782	. . . . 5 ⊢ (𝐴 = -i → (ℜ‘𝐴) = 0)
11	10	necon3i 2960	. . . 4 ⊢ ((ℜ‘𝐴) ≠ 0 → 𝐴 ≠ -i)
12	2, 11	syl 17	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → 𝐴 ≠ -i)
13		fveq2 6822	. . . . . 6 ⊢ (𝐴 = i → (ℜ‘𝐴) = (ℜ‘i))
14	13, 6	eqtrdi 2782	. . . . 5 ⊢ (𝐴 = i → (ℜ‘𝐴) = 0)
15	14	necon3i 2960	. . . 4 ⊢ ((ℜ‘𝐴) ≠ 0 → 𝐴 ≠ i)
16	2, 15	syl 17	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → 𝐴 ≠ i)
17		atandm 26814	. . 3 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i))
18	1, 12, 16, 17	syl3anbrc 1344	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → 𝐴 ∈ dom arctan)
19		halfcl 12347	. . . . . 6 ⊢ (i ∈ ℂ → (i / 2) ∈ ℂ)
20	4, 19	ax-mp 5	. . . . 5 ⊢ (i / 2) ∈ ℂ
21		ax-1cn 11064	. . . . . . . 8 ⊢ 1 ∈ ℂ
22		mulcl 11090	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
23	4, 1, 22	sylancr 587	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (i · 𝐴) ∈ ℂ)
24		subcl 11359	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − (i · 𝐴)) ∈ ℂ)
25	21, 23, 24	sylancr 587	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 − (i · 𝐴)) ∈ ℂ)
26		atandm2 26815	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
27	18, 26	sylib 218	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
28	27	simp2d 1143	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 − (i · 𝐴)) ≠ 0)
29	25, 28	logcld 26507	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(1 − (i · 𝐴))) ∈ ℂ)
30		addcl 11088	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + (i · 𝐴)) ∈ ℂ)
31	21, 23, 30	sylancr 587	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 + (i · 𝐴)) ∈ ℂ)
32	27	simp3d 1144	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 + (i · 𝐴)) ≠ 0)
33	31, 32	logcld 26507	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(1 + (i · 𝐴))) ∈ ℂ)
34	29, 33	subcld 11472	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))) ∈ ℂ)
35		cjmul 15049	. . . . 5 ⊢ (((i / 2) ∈ ℂ ∧ ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))) ∈ ℂ) → (∗‘((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))) = ((∗‘(i / 2)) · (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
36	20, 34, 35	sylancr 587	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))) = ((∗‘(i / 2)) · (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
37		2ne0 12229	. . . . . . . 8 ⊢ 2 ≠ 0
38		2cn 12200	. . . . . . . . 9 ⊢ 2 ∈ ℂ
39	4, 38	cjdivi 15098	. . . . . . . 8 ⊢ (2 ≠ 0 → (∗‘(i / 2)) = ((∗‘i) / (∗‘2)))
40	37, 39	ax-mp 5	. . . . . . 7 ⊢ (∗‘(i / 2)) = ((∗‘i) / (∗‘2))
41		divneg 11813	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → -(i / 2) = (-i / 2))
42	4, 38, 37, 41	mp3an 1463	. . . . . . . 8 ⊢ -(i / 2) = (-i / 2)
43		cji 15066	. . . . . . . . 9 ⊢ (∗‘i) = -i
44		2re 12199	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
45		cjre 15046	. . . . . . . . . 10 ⊢ (2 ∈ ℝ → (∗‘2) = 2)
46	44, 45	ax-mp 5	. . . . . . . . 9 ⊢ (∗‘2) = 2
47	43, 46	oveq12i 7358	. . . . . . . 8 ⊢ ((∗‘i) / (∗‘2)) = (-i / 2)
48	42, 47	eqtr4i 2757	. . . . . . 7 ⊢ -(i / 2) = ((∗‘i) / (∗‘2))
49	40, 48	eqtr4i 2757	. . . . . 6 ⊢ (∗‘(i / 2)) = -(i / 2)
50	49	oveq1i 7356	. . . . 5 ⊢ ((∗‘(i / 2)) · (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))) = (-(i / 2) · (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
51	34	cjcld 15103	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) ∈ ℂ)
52		mulneg12 11555	. . . . . 6 ⊢ (((i / 2) ∈ ℂ ∧ (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) ∈ ℂ) → (-(i / 2) · (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))) = ((i / 2) · -(∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
53	20, 51, 52	sylancr 587	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (-(i / 2) · (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))) = ((i / 2) · -(∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
54	50, 53	eqtrid 2778	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → ((∗‘(i / 2)) · (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))) = ((i / 2) · -(∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
55		cjsub 15056	. . . . . . . . 9 ⊢ (((log‘(1 − (i · 𝐴))) ∈ ℂ ∧ (log‘(1 + (i · 𝐴))) ∈ ℂ) → (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = ((∗‘(log‘(1 − (i · 𝐴)))) − (∗‘(log‘(1 + (i · 𝐴))))))
56	29, 33, 55	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = ((∗‘(log‘(1 − (i · 𝐴)))) − (∗‘(log‘(1 + (i · 𝐴))))))
57		imsub 15042	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (ℑ‘(1 − (i · 𝐴))) = ((ℑ‘1) − (ℑ‘(i · 𝐴))))
58	21, 23, 57	sylancr 587	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘(1 − (i · 𝐴))) = ((ℑ‘1) − (ℑ‘(i · 𝐴))))
59		reim 15016	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) = (ℑ‘(i · 𝐴)))
60	59	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℜ‘𝐴) = (ℑ‘(i · 𝐴)))
61	60	oveq2d 7362	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → ((ℑ‘1) − (ℜ‘𝐴)) = ((ℑ‘1) − (ℑ‘(i · 𝐴))))
62	58, 61	eqtr4d 2769	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘(1 − (i · 𝐴))) = ((ℑ‘1) − (ℜ‘𝐴)))
63		df-neg 11347	. . . . . . . . . . . . . 14 ⊢ -(ℜ‘𝐴) = (0 − (ℜ‘𝐴))
64		im1 15062	. . . . . . . . . . . . . . 15 ⊢ (ℑ‘1) = 0
65	64	oveq1i 7356	. . . . . . . . . . . . . 14 ⊢ ((ℑ‘1) − (ℜ‘𝐴)) = (0 − (ℜ‘𝐴))
66	63, 65	eqtr4i 2757	. . . . . . . . . . . . 13 ⊢ -(ℜ‘𝐴) = ((ℑ‘1) − (ℜ‘𝐴))
67	62, 66	eqtr4di 2784	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘(1 − (i · 𝐴))) = -(ℜ‘𝐴))
68		recl 15017	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ)
69	68	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℜ‘𝐴) ∈ ℝ)
70	69	recnd 11140	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℜ‘𝐴) ∈ ℂ)
71	70, 2	negne0d 11470	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → -(ℜ‘𝐴) ≠ 0)
72	67, 71	eqnetrd 2995	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘(1 − (i · 𝐴))) ≠ 0)
73		logcj 26543	. . . . . . . . . . 11 ⊢ (((1 − (i · 𝐴)) ∈ ℂ ∧ (ℑ‘(1 − (i · 𝐴))) ≠ 0) → (log‘(∗‘(1 − (i · 𝐴)))) = (∗‘(log‘(1 − (i · 𝐴)))))
74	25, 72, 73	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(∗‘(1 − (i · 𝐴)))) = (∗‘(log‘(1 − (i · 𝐴)))))
75		cjsub 15056	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (∗‘(1 − (i · 𝐴))) = ((∗‘1) − (∗‘(i · 𝐴))))
76	21, 23, 75	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(1 − (i · 𝐴))) = ((∗‘1) − (∗‘(i · 𝐴))))
77		1re 11112	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
78		cjre 15046	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℝ → (∗‘1) = 1)
79	77, 78	mp1i 13	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘1) = 1)
80		cjmul 15049	. . . . . . . . . . . . . . 15 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (∗‘(i · 𝐴)) = ((∗‘i) · (∗‘𝐴)))
81	4, 1, 80	sylancr 587	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(i · 𝐴)) = ((∗‘i) · (∗‘𝐴)))
82	43	oveq1i 7356	. . . . . . . . . . . . . . 15 ⊢ ((∗‘i) · (∗‘𝐴)) = (-i · (∗‘𝐴))
83		cjcl 15012	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ)
84	83	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘𝐴) ∈ ℂ)
85		mulneg1 11553	. . . . . . . . . . . . . . . 16 ⊢ ((i ∈ ℂ ∧ (∗‘𝐴) ∈ ℂ) → (-i · (∗‘𝐴)) = -(i · (∗‘𝐴)))
86	4, 84, 85	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (-i · (∗‘𝐴)) = -(i · (∗‘𝐴)))
87	82, 86	eqtrid 2778	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → ((∗‘i) · (∗‘𝐴)) = -(i · (∗‘𝐴)))
88	81, 87	eqtrd 2766	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(i · 𝐴)) = -(i · (∗‘𝐴)))
89	79, 88	oveq12d 7364	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → ((∗‘1) − (∗‘(i · 𝐴))) = (1 − -(i · (∗‘𝐴))))
90		mulcl 11090	. . . . . . . . . . . . . 14 ⊢ ((i ∈ ℂ ∧ (∗‘𝐴) ∈ ℂ) → (i · (∗‘𝐴)) ∈ ℂ)
91	4, 84, 90	sylancr 587	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (i · (∗‘𝐴)) ∈ ℂ)
92		subneg 11410	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ (i · (∗‘𝐴)) ∈ ℂ) → (1 − -(i · (∗‘𝐴))) = (1 + (i · (∗‘𝐴))))
93	21, 91, 92	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 − -(i · (∗‘𝐴))) = (1 + (i · (∗‘𝐴))))
94	76, 89, 93	3eqtrd 2770	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(1 − (i · 𝐴))) = (1 + (i · (∗‘𝐴))))
95	94	fveq2d 6826	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(∗‘(1 − (i · 𝐴)))) = (log‘(1 + (i · (∗‘𝐴)))))
96	74, 95	eqtr3d 2768	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(log‘(1 − (i · 𝐴)))) = (log‘(1 + (i · (∗‘𝐴)))))
97		imadd 15041	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (ℑ‘(1 + (i · 𝐴))) = ((ℑ‘1) + (ℑ‘(i · 𝐴))))
98	21, 23, 97	sylancr 587	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘(1 + (i · 𝐴))) = ((ℑ‘1) + (ℑ‘(i · 𝐴))))
99	60	oveq2d 7362	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (0 + (ℜ‘𝐴)) = (0 + (ℑ‘(i · 𝐴))))
100	64	oveq1i 7356	. . . . . . . . . . . . . 14 ⊢ ((ℑ‘1) + (ℑ‘(i · 𝐴))) = (0 + (ℑ‘(i · 𝐴)))
101	99, 100	eqtr4di 2784	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (0 + (ℜ‘𝐴)) = ((ℑ‘1) + (ℑ‘(i · 𝐴))))
102	70	addlidd 11314	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (0 + (ℜ‘𝐴)) = (ℜ‘𝐴))
103	98, 101, 102	3eqtr2d 2772	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘(1 + (i · 𝐴))) = (ℜ‘𝐴))
104	103, 2	eqnetrd 2995	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘(1 + (i · 𝐴))) ≠ 0)
105		logcj 26543	. . . . . . . . . . 11 ⊢ (((1 + (i · 𝐴)) ∈ ℂ ∧ (ℑ‘(1 + (i · 𝐴))) ≠ 0) → (log‘(∗‘(1 + (i · 𝐴)))) = (∗‘(log‘(1 + (i · 𝐴)))))
106	31, 104, 105	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(∗‘(1 + (i · 𝐴)))) = (∗‘(log‘(1 + (i · 𝐴)))))
107		cjadd 15048	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (∗‘(1 + (i · 𝐴))) = ((∗‘1) + (∗‘(i · 𝐴))))
108	21, 23, 107	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(1 + (i · 𝐴))) = ((∗‘1) + (∗‘(i · 𝐴))))
109	79, 88	oveq12d 7364	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → ((∗‘1) + (∗‘(i · 𝐴))) = (1 + -(i · (∗‘𝐴))))
110		negsub 11409	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ (i · (∗‘𝐴)) ∈ ℂ) → (1 + -(i · (∗‘𝐴))) = (1 − (i · (∗‘𝐴))))
111	21, 91, 110	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 + -(i · (∗‘𝐴))) = (1 − (i · (∗‘𝐴))))
112	108, 109, 111	3eqtrd 2770	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(1 + (i · 𝐴))) = (1 − (i · (∗‘𝐴))))
113	112	fveq2d 6826	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(∗‘(1 + (i · 𝐴)))) = (log‘(1 − (i · (∗‘𝐴)))))
114	106, 113	eqtr3d 2768	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(log‘(1 + (i · 𝐴)))) = (log‘(1 − (i · (∗‘𝐴)))))
115	96, 114	oveq12d 7364	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → ((∗‘(log‘(1 − (i · 𝐴)))) − (∗‘(log‘(1 + (i · 𝐴))))) = ((log‘(1 + (i · (∗‘𝐴)))) − (log‘(1 − (i · (∗‘𝐴))))))
116	56, 115	eqtrd 2766	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = ((log‘(1 + (i · (∗‘𝐴)))) − (log‘(1 − (i · (∗‘𝐴))))))
117	116	negeqd 11354	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → -(∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = -((log‘(1 + (i · (∗‘𝐴)))) − (log‘(1 − (i · (∗‘𝐴))))))
118		addcl 11088	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ (i · (∗‘𝐴)) ∈ ℂ) → (1 + (i · (∗‘𝐴))) ∈ ℂ)
119	21, 91, 118	sylancr 587	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 + (i · (∗‘𝐴))) ∈ ℂ)
120		atandmcj 26847	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom arctan → (∗‘𝐴) ∈ dom arctan)
121	18, 120	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘𝐴) ∈ dom arctan)
122		atandm2 26815	. . . . . . . . . 10 ⊢ ((∗‘𝐴) ∈ dom arctan ↔ ((∗‘𝐴) ∈ ℂ ∧ (1 − (i · (∗‘𝐴))) ≠ 0 ∧ (1 + (i · (∗‘𝐴))) ≠ 0))
123	122	simp3bi 1147	. . . . . . . . 9 ⊢ ((∗‘𝐴) ∈ dom arctan → (1 + (i · (∗‘𝐴))) ≠ 0)
124	121, 123	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 + (i · (∗‘𝐴))) ≠ 0)
125	119, 124	logcld 26507	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(1 + (i · (∗‘𝐴)))) ∈ ℂ)
126		subcl 11359	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ (i · (∗‘𝐴)) ∈ ℂ) → (1 − (i · (∗‘𝐴))) ∈ ℂ)
127	21, 91, 126	sylancr 587	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 − (i · (∗‘𝐴))) ∈ ℂ)
128	122	simp2bi 1146	. . . . . . . . 9 ⊢ ((∗‘𝐴) ∈ dom arctan → (1 − (i · (∗‘𝐴))) ≠ 0)
129	121, 128	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (1 − (i · (∗‘𝐴))) ≠ 0)
130	127, 129	logcld 26507	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (log‘(1 − (i · (∗‘𝐴)))) ∈ ℂ)
131	125, 130	negsubdi2d 11488	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → -((log‘(1 + (i · (∗‘𝐴)))) − (log‘(1 − (i · (∗‘𝐴))))) = ((log‘(1 − (i · (∗‘𝐴)))) − (log‘(1 + (i · (∗‘𝐴))))))
132	117, 131	eqtrd 2766	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → -(∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = ((log‘(1 − (i · (∗‘𝐴)))) − (log‘(1 + (i · (∗‘𝐴))))))
133	132	oveq2d 7362	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → ((i / 2) · -(∗‘((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))) = ((i / 2) · ((log‘(1 − (i · (∗‘𝐴)))) − (log‘(1 + (i · (∗‘𝐴)))))))
134	36, 54, 133	3eqtrd 2770	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))) = ((i / 2) · ((log‘(1 − (i · (∗‘𝐴)))) − (log‘(1 + (i · (∗‘𝐴)))))))
135		atanval 26822	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (arctan‘𝐴) = ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
136	18, 135	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (arctan‘𝐴) = ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
137	136	fveq2d 6826	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(arctan‘𝐴)) = (∗‘((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
138		atanval 26822	. . . 4 ⊢ ((∗‘𝐴) ∈ dom arctan → (arctan‘(∗‘𝐴)) = ((i / 2) · ((log‘(1 − (i · (∗‘𝐴)))) − (log‘(1 + (i · (∗‘𝐴)))))))
139	121, 138	syl 17	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (arctan‘(∗‘𝐴)) = ((i / 2) · ((log‘(1 − (i · (∗‘𝐴)))) − (log‘(1 + (i · (∗‘𝐴)))))))
140	134, 137, 139	3eqtr4d 2776	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (∗‘(arctan‘𝐴)) = (arctan‘(∗‘𝐴)))
141	18, 140	jca 511	1 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (𝐴 ∈ dom arctan ∧ (∗‘(arctan‘𝐴)) = (arctan‘(∗‘𝐴))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 dom cdm 5616 ‘cfv 6481 (class class class)co 7346 ℂcc 11004 ℝcr 11005 0cc0 11006 1c1 11007 ici 11008 + caddc 11009 · cmul 11011 − cmin 11344 -cneg 11345 / cdiv 11774 2c2 12180 ∗ccj 15003 ℜcre 15004 ℑcim 15005 logclog 26491 arctancatan 26802

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-se 5570 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ioc 13250 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-sin 15976 df-cos 15977 df-pi 15979 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19230 df-cmn 19695 df-psmet 21284 df-xmet 21285 df-met 21286 df-bl 21287 df-mopn 21288 df-fbas 21289 df-fg 21290 df-cnfld 21293 df-top 22810 df-topon 22827 df-topsp 22849 df-bases 22862 df-cld 22935 df-ntr 22936 df-cls 22937 df-nei 23014 df-lp 23052 df-perf 23053 df-cn 23143 df-cnp 23144 df-haus 23231 df-tx 23478 df-hmeo 23671 df-fil 23762 df-fm 23854 df-flim 23855 df-flf 23856 df-xms 24236 df-ms 24237 df-tms 24238 df-cncf 24799 df-limc 25795 df-dv 25796 df-log 26493 df-atan 26805

This theorem is referenced by: atanrecl 26849

W3C validator