Theorem argrege0 26576

Description: Closure of the argument of a complex number with nonnegative real part. (Contributed by Mario Carneiro, 2-Apr-2015.)

Assertion

Ref	Expression
argrege0	⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ (-(π / 2)[,](π / 2)))

Proof of Theorem argrege0

Step	Hyp	Ref	Expression
1		logcl 26533	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
2	1	3adant3 1132	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (log‘𝐴) ∈ ℂ)
3	2	imcld 15118	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ ℝ)
4		simp3 1138	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → 0 ≤ (ℜ‘𝐴))
5		simp1 1136	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → 𝐴 ∈ ℂ)
6	5	abscld 15362	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘𝐴) ∈ ℝ)
7	6	recnd 11160	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘𝐴) ∈ ℂ)
8	7	mul01d 11332	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘𝐴) · 0) = 0)
9		absrpcl 15211	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℝ+)
10	9	3adant3 1132	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘𝐴) ∈ ℝ+)
11	10	rpne0d 12954	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘𝐴) ≠ 0)
12	5, 7, 11	divcld 11917	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (𝐴 / (abs‘𝐴)) ∈ ℂ)
13	6, 12	remul2d 15150	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℜ‘((abs‘𝐴) · (𝐴 / (abs‘𝐴)))) = ((abs‘𝐴) · (ℜ‘(𝐴 / (abs‘𝐴)))))
14	5, 7, 11	divcan2d 11919	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘𝐴) · (𝐴 / (abs‘𝐴))) = 𝐴)
15	14	fveq2d 6838	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℜ‘((abs‘𝐴) · (𝐴 / (abs‘𝐴)))) = (ℜ‘𝐴))
16	13, 15	eqtr3d 2773	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘𝐴) · (ℜ‘(𝐴 / (abs‘𝐴)))) = (ℜ‘𝐴))
17	4, 8, 16	3brtr4d 5130	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘𝐴) · 0) ≤ ((abs‘𝐴) · (ℜ‘(𝐴 / (abs‘𝐴)))))
18		0re 11134	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
19	18	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → 0 ∈ ℝ)
20	12	recld 15117	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℜ‘(𝐴 / (abs‘𝐴))) ∈ ℝ)
21	19, 20, 10	lemul2d 12993	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (0 ≤ (ℜ‘(𝐴 / (abs‘𝐴))) ↔ ((abs‘𝐴) · 0) ≤ ((abs‘𝐴) · (ℜ‘(𝐴 / (abs‘𝐴))))))
22	17, 21	mpbird 257	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → 0 ≤ (ℜ‘(𝐴 / (abs‘𝐴))))
23		efiarg 26572	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(i · (ℑ‘(log‘𝐴)))) = (𝐴 / (abs‘𝐴)))
24	23	3adant3 1132	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (exp‘(i · (ℑ‘(log‘𝐴)))) = (𝐴 / (abs‘𝐴)))
25	24	fveq2d 6838	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℜ‘(exp‘(i · (ℑ‘(log‘𝐴))))) = (ℜ‘(𝐴 / (abs‘𝐴))))
26	22, 25	breqtrrd 5126	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → 0 ≤ (ℜ‘(exp‘(i · (ℑ‘(log‘𝐴))))))
27		recosval 16061	. . . . . . 7 ⊢ ((ℑ‘(log‘𝐴)) ∈ ℝ → (cos‘(ℑ‘(log‘𝐴))) = (ℜ‘(exp‘(i · (ℑ‘(log‘𝐴))))))
28	3, 27	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (cos‘(ℑ‘(log‘𝐴))) = (ℜ‘(exp‘(i · (ℑ‘(log‘𝐴))))))
29	26, 28	breqtrrd 5126	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → 0 ≤ (cos‘(ℑ‘(log‘𝐴))))
30		halfpire 26429	. . . . . . . . . 10 ⊢ (π / 2) ∈ ℝ
31		pirp 26426	. . . . . . . . . . 11 ⊢ π ∈ ℝ+
32		rphalfcl 12934	. . . . . . . . . . 11 ⊢ (π ∈ ℝ+ → (π / 2) ∈ ℝ+)
33		rpge0 12919	. . . . . . . . . . 11 ⊢ ((π / 2) ∈ ℝ+ → 0 ≤ (π / 2))
34	31, 32, 33	mp2b 10	. . . . . . . . . 10 ⊢ 0 ≤ (π / 2)
35		pire 26422	. . . . . . . . . . 11 ⊢ π ∈ ℝ
36		rphalflt 12936	. . . . . . . . . . . 12 ⊢ (π ∈ ℝ+ → (π / 2) < π)
37	31, 36	ax-mp 5	. . . . . . . . . . 11 ⊢ (π / 2) < π
38	30, 35, 37	ltleii 11256	. . . . . . . . . 10 ⊢ (π / 2) ≤ π
39	18, 35	elicc2i 13328	. . . . . . . . . 10 ⊢ ((π / 2) ∈ (0[,]π) ↔ ((π / 2) ∈ ℝ ∧ 0 ≤ (π / 2) ∧ (π / 2) ≤ π))
40	30, 34, 38, 39	mpbir3an 1342	. . . . . . . . 9 ⊢ (π / 2) ∈ (0[,]π)
41	3	recnd 11160	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ ℂ)
42	41	abscld 15362	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘(ℑ‘(log‘𝐴))) ∈ ℝ)
43	41	absge0d 15370	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → 0 ≤ (abs‘(ℑ‘(log‘𝐴))))
44		logimcl 26534	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
45	44	3adant3 1132	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
46	45	simpld 494	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → -π < (ℑ‘(log‘𝐴)))
47	35	renegcli 11442	. . . . . . . . . . . . 13 ⊢ -π ∈ ℝ
48		ltle 11221	. . . . . . . . . . . . 13 ⊢ ((-π ∈ ℝ ∧ (ℑ‘(log‘𝐴)) ∈ ℝ) → (-π < (ℑ‘(log‘𝐴)) → -π ≤ (ℑ‘(log‘𝐴))))
49	47, 3, 48	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (-π < (ℑ‘(log‘𝐴)) → -π ≤ (ℑ‘(log‘𝐴))))
50	46, 49	mpd 15	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → -π ≤ (ℑ‘(log‘𝐴)))
51	45	simprd 495	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ≤ π)
52		absle 15239	. . . . . . . . . . . 12 ⊢ (((ℑ‘(log‘𝐴)) ∈ ℝ ∧ π ∈ ℝ) → ((abs‘(ℑ‘(log‘𝐴))) ≤ π ↔ (-π ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π)))
53	3, 35, 52	sylancl 586	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘(ℑ‘(log‘𝐴))) ≤ π ↔ (-π ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π)))
54	50, 51, 53	mpbir2and 713	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘(ℑ‘(log‘𝐴))) ≤ π)
55	18, 35	elicc2i 13328	. . . . . . . . . 10 ⊢ ((abs‘(ℑ‘(log‘𝐴))) ∈ (0[,]π) ↔ ((abs‘(ℑ‘(log‘𝐴))) ∈ ℝ ∧ 0 ≤ (abs‘(ℑ‘(log‘𝐴))) ∧ (abs‘(ℑ‘(log‘𝐴))) ≤ π))
56	42, 43, 54, 55	syl3anbrc 1344	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘(ℑ‘(log‘𝐴))) ∈ (0[,]π))
57		cosord 26496	. . . . . . . . 9 ⊢ (((π / 2) ∈ (0[,]π) ∧ (abs‘(ℑ‘(log‘𝐴))) ∈ (0[,]π)) → ((π / 2) < (abs‘(ℑ‘(log‘𝐴))) ↔ (cos‘(abs‘(ℑ‘(log‘𝐴)))) < (cos‘(π / 2))))
58	40, 56, 57	sylancr 587	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((π / 2) < (abs‘(ℑ‘(log‘𝐴))) ↔ (cos‘(abs‘(ℑ‘(log‘𝐴)))) < (cos‘(π / 2))))
59		fveq2 6834	. . . . . . . . . . 11 ⊢ ((abs‘(ℑ‘(log‘𝐴))) = (ℑ‘(log‘𝐴)) → (cos‘(abs‘(ℑ‘(log‘𝐴)))) = (cos‘(ℑ‘(log‘𝐴))))
60	59	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘(ℑ‘(log‘𝐴))) = (ℑ‘(log‘𝐴)) → (cos‘(abs‘(ℑ‘(log‘𝐴)))) = (cos‘(ℑ‘(log‘𝐴)))))
61		cosneg 16072	. . . . . . . . . . . 12 ⊢ ((ℑ‘(log‘𝐴)) ∈ ℂ → (cos‘-(ℑ‘(log‘𝐴))) = (cos‘(ℑ‘(log‘𝐴))))
62	41, 61	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (cos‘-(ℑ‘(log‘𝐴))) = (cos‘(ℑ‘(log‘𝐴))))
63		fveqeq2 6843	. . . . . . . . . . 11 ⊢ ((abs‘(ℑ‘(log‘𝐴))) = -(ℑ‘(log‘𝐴)) → ((cos‘(abs‘(ℑ‘(log‘𝐴)))) = (cos‘(ℑ‘(log‘𝐴))) ↔ (cos‘-(ℑ‘(log‘𝐴))) = (cos‘(ℑ‘(log‘𝐴)))))
64	62, 63	syl5ibrcom 247	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘(ℑ‘(log‘𝐴))) = -(ℑ‘(log‘𝐴)) → (cos‘(abs‘(ℑ‘(log‘𝐴)))) = (cos‘(ℑ‘(log‘𝐴)))))
65	3	absord 15339	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘(ℑ‘(log‘𝐴))) = (ℑ‘(log‘𝐴)) ∨ (abs‘(ℑ‘(log‘𝐴))) = -(ℑ‘(log‘𝐴))))
66	60, 64, 65	mpjaod 860	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (cos‘(abs‘(ℑ‘(log‘𝐴)))) = (cos‘(ℑ‘(log‘𝐴))))
67		coshalfpi 26434	. . . . . . . . . 10 ⊢ (cos‘(π / 2)) = 0
68	67	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (cos‘(π / 2)) = 0)
69	66, 68	breq12d 5111	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((cos‘(abs‘(ℑ‘(log‘𝐴)))) < (cos‘(π / 2)) ↔ (cos‘(ℑ‘(log‘𝐴))) < 0))
70	58, 69	bitrd 279	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((π / 2) < (abs‘(ℑ‘(log‘𝐴))) ↔ (cos‘(ℑ‘(log‘𝐴))) < 0))
71	70	notbid 318	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (¬ (π / 2) < (abs‘(ℑ‘(log‘𝐴))) ↔ ¬ (cos‘(ℑ‘(log‘𝐴))) < 0))
72		lenlt 11211	. . . . . . 7 ⊢ (((abs‘(ℑ‘(log‘𝐴))) ∈ ℝ ∧ (π / 2) ∈ ℝ) → ((abs‘(ℑ‘(log‘𝐴))) ≤ (π / 2) ↔ ¬ (π / 2) < (abs‘(ℑ‘(log‘𝐴)))))
73	42, 30, 72	sylancl 586	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘(ℑ‘(log‘𝐴))) ≤ (π / 2) ↔ ¬ (π / 2) < (abs‘(ℑ‘(log‘𝐴)))))
74	3	recoscld 16069	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (cos‘(ℑ‘(log‘𝐴))) ∈ ℝ)
75		lenlt 11211	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ (cos‘(ℑ‘(log‘𝐴))) ∈ ℝ) → (0 ≤ (cos‘(ℑ‘(log‘𝐴))) ↔ ¬ (cos‘(ℑ‘(log‘𝐴))) < 0))
76	18, 74, 75	sylancr 587	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (0 ≤ (cos‘(ℑ‘(log‘𝐴))) ↔ ¬ (cos‘(ℑ‘(log‘𝐴))) < 0))
77	71, 73, 76	3bitr4d 311	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘(ℑ‘(log‘𝐴))) ≤ (π / 2) ↔ 0 ≤ (cos‘(ℑ‘(log‘𝐴)))))
78	29, 77	mpbird 257	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘(ℑ‘(log‘𝐴))) ≤ (π / 2))
79		absle 15239	. . . . 5 ⊢ (((ℑ‘(log‘𝐴)) ∈ ℝ ∧ (π / 2) ∈ ℝ) → ((abs‘(ℑ‘(log‘𝐴))) ≤ (π / 2) ↔ (-(π / 2) ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ (π / 2))))
80	3, 30, 79	sylancl 586	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘(ℑ‘(log‘𝐴))) ≤ (π / 2) ↔ (-(π / 2) ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ (π / 2))))
81	78, 80	mpbid 232	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (-(π / 2) ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ (π / 2)))
82	81	simpld 494	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → -(π / 2) ≤ (ℑ‘(log‘𝐴)))
83	81	simprd 495	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ≤ (π / 2))
84	30	renegcli 11442	. . 3 ⊢ -(π / 2) ∈ ℝ
85	84, 30	elicc2i 13328	. 2 ⊢ ((ℑ‘(log‘𝐴)) ∈ (-(π / 2)[,](π / 2)) ↔ ((ℑ‘(log‘𝐴)) ∈ ℝ ∧ -(π / 2) ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ (π / 2)))
86	3, 82, 83, 85	syl3anbrc 1344	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ (-(π / 2)[,](π / 2)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358  ℂcc 11024  ℝcr 11025  0cc0 11026  ici 11028   · cmul 11031   < clt 11166   ≤ cle 11167  -cneg 11365   / cdiv 11794  2c2 12200  ℝ⁺crp 12905  [,]cicc 13264  ℜcre 15020  ℑcim 15021  abscabs 15157  expce 15984  cosccos 15987  πcpi 15989  logclog 26519

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104  ax-addf 11105

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-map 8765  df-pm 8766  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-fi 9314  df-sup 9345  df-inf 9346  df-oi 9415  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-q 12862  df-rp 12906  df-xneg 13026  df-xadd 13027  df-xmul 13028  df-ioo 13265  df-ioc 13266  df-ico 13267  df-icc 13268  df-fz 13424  df-fzo 13571  df-fl 13712  df-mod 13790  df-seq 13925  df-exp 13985  df-fac 14197  df-bc 14226  df-hash 14254  df-shft 14990  df-cj 15022  df-re 15023  df-im 15024  df-sqrt 15158  df-abs 15159  df-limsup 15394  df-clim 15411  df-rlim 15412  df-sum 15610  df-ef 15990  df-sin 15992  df-cos 15993  df-pi 15995  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-mulr 17191  df-starv 17192  df-sca 17193  df-vsca 17194  df-ip 17195  df-tset 17196  df-ple 17197  df-ds 17199  df-unif 17200  df-hom 17201  df-cco 17202  df-rest 17342  df-topn 17343  df-0g 17361  df-gsum 17362  df-topgen 17363  df-pt 17364  df-prds 17367  df-xrs 17423  df-qtop 17428  df-imas 17429  df-xps 17431  df-mre 17505  df-mrc 17506  df-acs 17508  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18709  df-mulg 18998  df-cntz 19246  df-cmn 19711  df-psmet 21301  df-xmet 21302  df-met 21303  df-bl 21304  df-mopn 21305  df-fbas 21306  df-fg 21307  df-cnfld 21310  df-top 22838  df-topon 22855  df-topsp 22877  df-bases 22890  df-cld 22963  df-ntr 22964  df-cls 22965  df-nei 23042  df-lp 23080  df-perf 23081  df-cn 23171  df-cnp 23172  df-haus 23259  df-tx 23506  df-hmeo 23699  df-fil 23790  df-fm 23882  df-flim 23883  df-flf 23884  df-xms 24264  df-ms 24265  df-tms 24266  df-cncf 24827  df-limc 25823  df-dv 25824  df-log 26521

This theorem is referenced by:  logimul  26579  isosctrlem1  26784  asinbnd  26865  isosctrlem1ALT  45170