Theorem argrege0 26969

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 26926	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
2	1	3adant3 1150	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (log‘𝐴) ∈ ℂ)
3	2	imcld 15391	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ ℝ)
4		simp3 1156	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → 0 ≤ (ℜ‘𝐴))
5		simp1 1154	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → 𝐴 ∈ ℂ)
6	5	abscld 15635	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘𝐴) ∈ ℝ)
7	6	recnd 11364	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘𝐴) ∈ ℂ)
8	7	mul01d 11536	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘𝐴) · 0) = 0)
9		absrpcl 15484	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℝ+)
10	9	3adant3 1150	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘𝐴) ∈ ℝ+)
11	10	rpne0d 13196	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘𝐴) ≠ 0)
12	5, 7, 11	divcld 12120	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (𝐴 / (abs‘𝐴)) ∈ ℂ)
13	6, 12	remul2d 15423	. . . . . . . . . 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 12122	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘𝐴) · (𝐴 / (abs‘𝐴))) = 𝐴)
15	14	fveq2d 6912	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℜ‘((abs‘𝐴) · (𝐴 / (abs‘𝐴)))) = (ℜ‘𝐴))
16	13, 15	eqtr3d 2800	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘𝐴) · (ℜ‘(𝐴 / (abs‘𝐴)))) = (ℜ‘𝐴))
17	4, 8, 16	3brtr4d 5140	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘𝐴) · 0) ≤ ((abs‘𝐴) · (ℜ‘(𝐴 / (abs‘𝐴)))))
18		0re 11337	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
19	18	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → 0 ∈ ℝ)
20	12	recld 15390	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℜ‘(𝐴 / (abs‘𝐴))) ∈ ℝ)
21	19, 20, 10	lemul2d 13235	. . . . . . . 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 260	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → 0 ≤ (ℜ‘(𝐴 / (abs‘𝐴))))
23		efiarg 26965	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(i · (ℑ‘(log‘𝐴)))) = (𝐴 / (abs‘𝐴)))
24	23	3adant3 1150	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (exp‘(i · (ℑ‘(log‘𝐴)))) = (𝐴 / (abs‘𝐴)))
25	24	fveq2d 6912	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℜ‘(exp‘(i · (ℑ‘(log‘𝐴))))) = (ℜ‘(𝐴 / (abs‘𝐴))))
26	22, 25	breqtrrd 5136	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → 0 ≤ (ℜ‘(exp‘(i · (ℑ‘(log‘𝐴))))))
27		recosval 16333	. . . . . . 7 ⊢ ((ℑ‘(log‘𝐴)) ∈ ℝ → (cos‘(ℑ‘(log‘𝐴))) = (ℜ‘(exp‘(i · (ℑ‘(log‘𝐴))))))
28	3, 27	syl 18	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (cos‘(ℑ‘(log‘𝐴))) = (ℜ‘(exp‘(i · (ℑ‘(log‘𝐴))))))
29	26, 28	breqtrrd 5136	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → 0 ≤ (cos‘(ℑ‘(log‘𝐴))))
30		halfpire 26823	. . . . . . . . . 10 ⊢ (π / 2) ∈ ℝ
31		pirp 26820	. . . . . . . . . . 11 ⊢ π ∈ ℝ+
32		rphalfcl 13176	. . . . . . . . . . 11 ⊢ (π ∈ ℝ+ → (π / 2) ∈ ℝ+)
33		rpge0 13161	. . . . . . . . . . 11 ⊢ ((π / 2) ∈ ℝ+ → 0 ≤ (π / 2))
34	31, 32, 33	mp2b 10	. . . . . . . . . 10 ⊢ 0 ≤ (π / 2)
35		pire 26813	. . . . . . . . . . 11 ⊢ π ∈ ℝ
36		rphalflt 13178	. . . . . . . . . . . 12 ⊢ (π ∈ ℝ+ → (π / 2) < π)
37	31, 36	ax-mp 5	. . . . . . . . . . 11 ⊢ (π / 2) < π
38	30, 35, 37	ltleii 11460	. . . . . . . . . 10 ⊢ (π / 2) ≤ π
39	18, 35	elicc2i 13570	. . . . . . . . . 10 ⊢ ((π / 2) ∈ (0[,]π) ↔ ((π / 2) ∈ ℝ ∧ 0 ≤ (π / 2) ∧ (π / 2) ≤ π))
40	30, 34, 38, 39	mpbir3an 1360	. . . . . . . . 9 ⊢ (π / 2) ∈ (0[,]π)
41	3	recnd 11364	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ ℂ)
42	41	abscld 15635	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘(ℑ‘(log‘𝐴))) ∈ ℝ)
43	41	absge0d 15643	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → 0 ≤ (abs‘(ℑ‘(log‘𝐴))))
44		logimcl 26927	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ( -π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
45	44	3adant3 1150	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ( -π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
46	45	simpld 500	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → -π < (ℑ‘(log‘𝐴)))
47	35	renegcli 11646	. . . . . . . . . . . . 13 ⊢ -π ∈ ℝ
48		ltle 11425	. . . . . . . . . . . . 13 ⊢ (( -π ∈ ℝ ∧ (ℑ‘(log‘𝐴)) ∈ ℝ) → ( -π < (ℑ‘(log‘𝐴)) → -π ≤ (ℑ‘(log‘𝐴))))
49	47, 3, 48	sylancr 599	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ( -π < (ℑ‘(log‘𝐴)) → -π ≤ (ℑ‘(log‘𝐴))))
50	46, 49	mpd 16	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → -π ≤ (ℑ‘(log‘𝐴)))
51	45	simprd 501	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ≤ π)
52		absle 15512	. . . . . . . . . . . 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 598	. . . . . . . . . . 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 726	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘(ℑ‘(log‘𝐴))) ≤ π)
55	18, 35	elicc2i 13570	. . . . . . . . . 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 1362	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘(ℑ‘(log‘𝐴))) ∈ (0[,]π))
57		cosord 26889	. . . . . . . . 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 599	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((π / 2) < (abs‘(ℑ‘(log‘𝐴))) ↔ (cos‘(abs‘(ℑ‘(log‘𝐴)))) < (cos‘(π / 2))))
59		fveq2 6908	. . . . . . . . . . 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 16344	. . . . . . . . . . . 12 ⊢ ((ℑ‘(log‘𝐴)) ∈ ℂ → (cos‘ -(ℑ‘(log‘𝐴))) = (cos‘(ℑ‘(log‘𝐴))))
62	41, 61	syl 18	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (cos‘ -(ℑ‘(log‘𝐴))) = (cos‘(ℑ‘(log‘𝐴))))
63		fveqeq2 6917	. . . . . . . . . . 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 250	. . . . . . . . . 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 15612	. . . . . . . . . 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 874	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (cos‘(abs‘(ℑ‘(log‘𝐴)))) = (cos‘(ℑ‘(log‘𝐴))))
67		coshalfpi 26828	. . . . . . . . . 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 5119	. . . . . . . 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 282	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((π / 2) < (abs‘(ℑ‘(log‘𝐴))) ↔ (cos‘(ℑ‘(log‘𝐴))) < 0))
71	70	notbid 321	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (¬ (π / 2) < (abs‘(ℑ‘(log‘𝐴))) ↔ ¬ (cos‘(ℑ‘(log‘𝐴))) < 0))
72		lenlt 11415	. . . . . . 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 598	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘(ℑ‘(log‘𝐴))) ≤ (π / 2) ↔ ¬ (π / 2) < (abs‘(ℑ‘(log‘𝐴)))))
74	3	recoscld 16341	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (cos‘(ℑ‘(log‘𝐴))) ∈ ℝ)
75		lenlt 11415	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ (cos‘(ℑ‘(log‘𝐴))) ∈ ℝ) → (0 ≤ (cos‘(ℑ‘(log‘𝐴))) ↔ ¬ (cos‘(ℑ‘(log‘𝐴))) < 0))
76	18, 74, 75	sylancr 599	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (0 ≤ (cos‘(ℑ‘(log‘𝐴))) ↔ ¬ (cos‘(ℑ‘(log‘𝐴))) < 0))
77	71, 73, 76	3bitr4d 314	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((abs‘(ℑ‘(log‘𝐴))) ≤ (π / 2) ↔ 0 ≤ (cos‘(ℑ‘(log‘𝐴)))))
78	29, 77	mpbird 260	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (abs‘(ℑ‘(log‘𝐴))) ≤ (π / 2))
79		absle 15512	. . . . 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 598	. . . 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 235	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ( -(π / 2) ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ (π / 2)))
82	81	simpld 500	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → -(π / 2) ≤ (ℑ‘(log‘𝐴)))
83	81	simprd 501	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ≤ (π / 2))
84	30	renegcli 11646	. . 3 ⊢ -(π / 2) ∈ ℝ
85	84, 30	elicc2i 13570	. 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 1362	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ ( -(π / 2)[,](π / 2)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   class class class wbr 5106  ‘cfv 6558  (class class class)co 7443  ℂcc 11225  ℝcr 11226  0cc0 11227  ici 11229   · cmul 11232   < clt 11370   ≤ cle 11371   -cneg 11569   / cdiv 12000  2c2 12424  ℝ⁺crp 13147  [,]cicc 13506  ℜcre 15293  ℑcim 15294  abscabs 15430  expce 16256  cosccos 16259  πcpi 16261  logclog 26912

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 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-inf2 9669  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304  ax-pre-sup 11305  ax-addf 11306

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 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-of 7717  df-om 7902  df-1st 8025  df-2nd 8026  df-supp 8205  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-er 8745  df-map 8877  df-pm 8878  df-ixp 8954  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-fsupp 9382  df-fi 9430  df-sup 9461  df-inf 9462  df-oi 9531  df-card 10047  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-div 12001  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-5 12435  df-6 12436  df-7 12437  df-8 12438  df-9 12439  df-n0 12634  df-z 12721  df-dec 12842  df-uz 12993  df-q 13103  df-rp 13148  df-xneg 13268  df-xadd 13269  df-xmul 13270  df-ioo 13507  df-ioc 13508  df-ico 13509  df-icc 13510  df-fz 13667  df-fzo 13816  df-fl 13959  df-mod 14037  df-seq 14172  df-exp 14232  df-fac 14445  df-bc 14474  df-hash 14502  df-shft 15249  df-cj 15295  df-re 15296  df-im 15297  df-sqrt 15431  df-abs 15432  df-limsup 15667  df-clim 15684  df-rlim 15685  df-sum 15883  df-ef 16262  df-sin 16264  df-cos 16265  df-pi 16267  df-struct 17354  df-sets 17371  df-slot 17389  df-ndx 17401  df-base 17417  df-ress 17438  df-plusg 17470  df-mulr 17471  df-starv 17472  df-sca 17473  df-vsca 17474  df-ip 17475  df-tset 17476  df-ple 17477  df-ds 17479  df-unif 17480  df-hom 17481  df-cco 17482  df-rest 17622  df-topn 17623  df-0g 17641  df-gsum 17642  df-topgen 17643  df-pt 17644  df-prds 17647  df-xrs 17703  df-qtop 17708  df-imas 17709  df-xps 17711  df-mre 17785  df-mrc 17786  df-acs 17788  df-mgm 18845  df-sgrp 18937  df-mnd 18953  df-submnd 19008  df-mulg 19307  df-cntz 19561  df-cmn 20026  df-psmet 21700  df-xmet 21701  df-met 21702  df-bl 21703  df-mopn 21704  df-fbas 21705  df-fg 21706  df-cnfld 21709  df-top 23242  df-topon 23259  df-topsp 23281  df-bases 23294  df-cld 23367  df-ntr 23368  df-cls 23369  df-nei 23446  df-lp 23484  df-perf 23485  df-cn 23575  df-cnp 23576  df-haus 23663  df-tx 23911  df-hmeo 24104  df-fil 24195  df-fm 24287  df-flim 24288  df-flf 24289  df-xms 24669  df-ms 24670  df-tms 24671  df-cncf 25229  df-limc 26216  df-dv 26217  df-log 26914

This theorem is used by:  logimul  26972  isosctrlem1  27176  asinbnd  27257  isosctrlem1ALT  46107