Theorem logf1o2 26707

Description: The logarithm maps its continuous domain bijectively onto the set of numbers with imaginary part -π < ℑ(𝑧) < π. The negative reals are mapped to the numbers with imaginary part equal to π. (Contributed by Mario Carneiro, 2-May-2015.)

Hypothesis

Ref	Expression
logcn.d	⊢ 𝐷 = (ℂ ∖ (-∞(,]0))

Assertion

Ref	Expression
logf1o2	⊢ (log ↾ 𝐷):𝐷–1-1-onto→(◡ℑ “ (-π(,)π))

Proof of Theorem logf1o2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		logf1o 26621	. . . 4 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
2		f1of1 6848	. . . 4 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})–1-1→ran log)
3	1, 2	ax-mp 5	. . 3 ⊢ log:(ℂ ∖ {0})–1-1→ran log
4		logcn.d	. . . 4 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0))
5	4	logdmss 26699	. . 3 ⊢ 𝐷 ⊆ (ℂ ∖ {0})
6		f1ores 6863	. . 3 ⊢ ((log:(ℂ ∖ {0})–1-1→ran log ∧ 𝐷 ⊆ (ℂ ∖ {0})) → (log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷))
7	3, 5, 6	mp2an 692	. 2 ⊢ (log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷)
8		f1ofun 6851	. . . . . . 7 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → Fun log)
9	1, 8	ax-mp 5	. . . . . 6 ⊢ Fun log
10		f1of 6849	. . . . . . . . 9 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})⟶ran log)
11	1, 10	ax-mp 5	. . . . . . . 8 ⊢ log:(ℂ ∖ {0})⟶ran log
12	11	fdmi 6748	. . . . . . 7 ⊢ dom log = (ℂ ∖ {0})
13	5, 12	sseqtrri 4033	. . . . . 6 ⊢ 𝐷 ⊆ dom log
14		funimass4 6973	. . . . . 6 ⊢ ((Fun log ∧ 𝐷 ⊆ dom log) → ((log “ 𝐷) ⊆ (◡ℑ “ (-π(,)π)) ↔ ∀𝑥 ∈ 𝐷 (log‘𝑥) ∈ (◡ℑ “ (-π(,)π))))
15	9, 13, 14	mp2an 692	. . . . 5 ⊢ ((log “ 𝐷) ⊆ (◡ℑ “ (-π(,)π)) ↔ ∀𝑥 ∈ 𝐷 (log‘𝑥) ∈ (◡ℑ “ (-π(,)π)))
16	4	ellogdm 26696	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ+)))
17	16	simplbi 497	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ)
18	4	logdmn0 26697	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0)
19	17, 18	logcld 26627	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (log‘𝑥) ∈ ℂ)
20	19	imcld 15231	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ∈ ℝ)
21	17, 18	logimcld 26628	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → (-π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) ≤ π))
22	21	simpld 494	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → -π < (ℑ‘(log‘𝑥)))
23		pire 26515	. . . . . . . . 9 ⊢ π ∈ ℝ
24	23	a1i 11	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → π ∈ ℝ)
25	21	simprd 495	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ≤ π)
26	4	logdmnrp 26698	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → ¬ -𝑥 ∈ ℝ+)
27		lognegb 26647	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (-𝑥 ∈ ℝ+ ↔ (ℑ‘(log‘𝑥)) = π))
28	17, 18, 27	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐷 → (-𝑥 ∈ ℝ+ ↔ (ℑ‘(log‘𝑥)) = π))
29	28	necon3bbid 2976	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → (¬ -𝑥 ∈ ℝ+ ↔ (ℑ‘(log‘𝑥)) ≠ π))
30	26, 29	mpbid 232	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ≠ π)
31	30	necomd 2994	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → π ≠ (ℑ‘(log‘𝑥)))
32	20, 24, 25, 31	leneltd 11413	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) < π)
33	23	renegcli 11568	. . . . . . . . 9 ⊢ -π ∈ ℝ
34	33	rexri 11317	. . . . . . . 8 ⊢ -π ∈ ℝ*
35	23	rexri 11317	. . . . . . . 8 ⊢ π ∈ ℝ*
36		elioo2 13425	. . . . . . . 8 ⊢ ((-π ∈ ℝ* ∧ π ∈ ℝ*) → ((ℑ‘(log‘𝑥)) ∈ (-π(,)π) ↔ ((ℑ‘(log‘𝑥)) ∈ ℝ ∧ -π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) < π)))
37	34, 35, 36	mp2an 692	. . . . . . 7 ⊢ ((ℑ‘(log‘𝑥)) ∈ (-π(,)π) ↔ ((ℑ‘(log‘𝑥)) ∈ ℝ ∧ -π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) < π))
38	20, 22, 32, 37	syl3anbrc 1342	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ∈ (-π(,)π))
39		imf 15149	. . . . . . 7 ⊢ ℑ:ℂ⟶ℝ
40		ffn 6737	. . . . . . 7 ⊢ (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
41		elpreima 7078	. . . . . . 7 ⊢ (ℑ Fn ℂ → ((log‘𝑥) ∈ (◡ℑ “ (-π(,)π)) ↔ ((log‘𝑥) ∈ ℂ ∧ (ℑ‘(log‘𝑥)) ∈ (-π(,)π))))
42	39, 40, 41	mp2b 10	. . . . . 6 ⊢ ((log‘𝑥) ∈ (◡ℑ “ (-π(,)π)) ↔ ((log‘𝑥) ∈ ℂ ∧ (ℑ‘(log‘𝑥)) ∈ (-π(,)π)))
43	19, 38, 42	sylanbrc 583	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (log‘𝑥) ∈ (◡ℑ “ (-π(,)π)))
44	15, 43	mprgbir 3066	. . . 4 ⊢ (log “ 𝐷) ⊆ (◡ℑ “ (-π(,)π))
45		elpreima 7078	. . . . . . 7 ⊢ (ℑ Fn ℂ → (𝑥 ∈ (◡ℑ “ (-π(,)π)) ↔ (𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π))))
46	39, 40, 45	mp2b 10	. . . . . 6 ⊢ (𝑥 ∈ (◡ℑ “ (-π(,)π)) ↔ (𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)))
47		simpl 482	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → 𝑥 ∈ ℂ)
48		eliooord 13443	. . . . . . . . . . 11 ⊢ ((ℑ‘𝑥) ∈ (-π(,)π) → (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π))
49	48	adantl 481	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π))
50	49	simpld 494	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → -π < (ℑ‘𝑥))
51	49	simprd 495	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (ℑ‘𝑥) < π)
52		imcl 15147	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (ℑ‘𝑥) ∈ ℝ)
53	52	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (ℑ‘𝑥) ∈ ℝ)
54		ltle 11347	. . . . . . . . . . 11 ⊢ (((ℑ‘𝑥) ∈ ℝ ∧ π ∈ ℝ) → ((ℑ‘𝑥) < π → (ℑ‘𝑥) ≤ π))
55	53, 23, 54	sylancl 586	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → ((ℑ‘𝑥) < π → (ℑ‘𝑥) ≤ π))
56	51, 55	mpd 15	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (ℑ‘𝑥) ≤ π)
57		ellogrn 26616	. . . . . . . . 9 ⊢ (𝑥 ∈ ran log ↔ (𝑥 ∈ ℂ ∧ -π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) ≤ π))
58	47, 50, 56, 57	syl3anbrc 1342	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → 𝑥 ∈ ran log)
59		logef 26638	. . . . . . . 8 ⊢ (𝑥 ∈ ran log → (log‘(exp‘𝑥)) = 𝑥)
60	58, 59	syl 17	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (log‘(exp‘𝑥)) = 𝑥)
61		efcl 16115	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) ∈ ℂ)
62	61	adantr 480	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (exp‘𝑥) ∈ ℂ)
63	53	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) ∈ ℝ)
64	63	recnd 11287	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) ∈ ℂ)
65		picn 26516	. . . . . . . . . . . . . 14 ⊢ π ∈ ℂ
66	65	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → π ∈ ℂ)
67		pipos 26517	. . . . . . . . . . . . . . 15 ⊢ 0 < π
68	23, 67	gt0ne0ii 11797	. . . . . . . . . . . . . 14 ⊢ π ≠ 0
69	68	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → π ≠ 0)
70	51	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) < π)
71	65	mulridi 11263	. . . . . . . . . . . . . . . . . 18 ⊢ (π · 1) = π
72	70, 71	breqtrrdi 5190	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) < (π · 1))
73		1re 11259	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ
74	73	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → 1 ∈ ℝ)
75	23	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → π ∈ ℝ)
76	67	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → 0 < π)
77		ltdivmul 12141	. . . . . . . . . . . . . . . . . 18 ⊢ (((ℑ‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧ (π ∈ ℝ ∧ 0 < π)) → (((ℑ‘𝑥) / π) < 1 ↔ (ℑ‘𝑥) < (π · 1)))
78	63, 74, 75, 76, 77	syl112anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (((ℑ‘𝑥) / π) < 1 ↔ (ℑ‘𝑥) < (π · 1)))
79	72, 78	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) < 1)
80		1e0p1 12773	. . . . . . . . . . . . . . . 16 ⊢ 1 = (0 + 1)
81	79, 80	breqtrdi 5189	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) < (0 + 1))
82	63	recoscld 16177	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (cos‘(ℑ‘𝑥)) ∈ ℝ)
83	63	resincld 16176	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (sin‘(ℑ‘𝑥)) ∈ ℝ)
84	82, 83	crimd 15268	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))) = (sin‘(ℑ‘𝑥)))
85		efeul 16195	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) = ((exp‘(ℜ‘𝑥)) · ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))))
86	85	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (exp‘𝑥) = ((exp‘(ℜ‘𝑥)) · ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))))
87	86	oveq1d 7446	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((exp‘𝑥) / (exp‘(ℜ‘𝑥))) = (((exp‘(ℜ‘𝑥)) · ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))) / (exp‘(ℜ‘𝑥))))
88	82	recnd 11287	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (cos‘(ℑ‘𝑥)) ∈ ℂ)
89		ax-icn 11212	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ i ∈ ℂ
90	83	recnd 11287	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (sin‘(ℑ‘𝑥)) ∈ ℂ)
91		mulcl 11237	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((i ∈ ℂ ∧ (sin‘(ℑ‘𝑥)) ∈ ℂ) → (i · (sin‘(ℑ‘𝑥))) ∈ ℂ)
92	89, 90, 91	sylancr 587	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (i · (sin‘(ℑ‘𝑥))) ∈ ℂ)
93	88, 92	addcld 11278	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥)))) ∈ ℂ)
94		recl 15146	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ ℂ → (ℜ‘𝑥) ∈ ℝ)
95	94	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℜ‘𝑥) ∈ ℝ)
96	95	recnd 11287	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℜ‘𝑥) ∈ ℂ)
97		efcl 16115	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℜ‘𝑥) ∈ ℂ → (exp‘(ℜ‘𝑥)) ∈ ℂ)
98	96, 97	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (exp‘(ℜ‘𝑥)) ∈ ℂ)
99		efne0 16130	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℜ‘𝑥) ∈ ℂ → (exp‘(ℜ‘𝑥)) ≠ 0)
100	96, 99	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (exp‘(ℜ‘𝑥)) ≠ 0)
101	93, 98, 100	divcan3d 12046	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (((exp‘(ℜ‘𝑥)) · ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))) / (exp‘(ℜ‘𝑥))) = ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥)))))
102	87, 101	eqtrd 2775	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((exp‘𝑥) / (exp‘(ℜ‘𝑥))) = ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥)))))
103		simpr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (exp‘𝑥) ∈ ℝ)
104	95	reefcld 16121	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (exp‘(ℜ‘𝑥)) ∈ ℝ)
105	103, 104, 100	redivcld 12093	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((exp‘𝑥) / (exp‘(ℜ‘𝑥))) ∈ ℝ)
106	102, 105	eqeltrrd 2840	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥)))) ∈ ℝ)
107	106	reim0d 15261	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))) = 0)
108	84, 107	eqtr3d 2777	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (sin‘(ℑ‘𝑥)) = 0)
109		sineq0 26581	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℑ‘𝑥) ∈ ℂ → ((sin‘(ℑ‘𝑥)) = 0 ↔ ((ℑ‘𝑥) / π) ∈ ℤ))
110	64, 109	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((sin‘(ℑ‘𝑥)) = 0 ↔ ((ℑ‘𝑥) / π) ∈ ℤ))
111	108, 110	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) ∈ ℤ)
112		0z 12622	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
113		zleltp1 12666	. . . . . . . . . . . . . . . 16 ⊢ ((((ℑ‘𝑥) / π) ∈ ℤ ∧ 0 ∈ ℤ) → (((ℑ‘𝑥) / π) ≤ 0 ↔ ((ℑ‘𝑥) / π) < (0 + 1)))
114	111, 112, 113	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (((ℑ‘𝑥) / π) ≤ 0 ↔ ((ℑ‘𝑥) / π) < (0 + 1)))
115	81, 114	mpbird 257	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) ≤ 0)
116		df-neg 11493	. . . . . . . . . . . . . . . 16 ⊢ -1 = (0 − 1)
117	65	mulm1i 11706	. . . . . . . . . . . . . . . . . 18 ⊢ (-1 · π) = -π
118	50	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → -π < (ℑ‘𝑥))
119	117, 118	eqbrtrid 5183	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (-1 · π) < (ℑ‘𝑥))
120	73	renegcli 11568	. . . . . . . . . . . . . . . . . . 19 ⊢ -1 ∈ ℝ
121	120	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → -1 ∈ ℝ)
122		ltmuldiv 12139	. . . . . . . . . . . . . . . . . 18 ⊢ ((-1 ∈ ℝ ∧ (ℑ‘𝑥) ∈ ℝ ∧ (π ∈ ℝ ∧ 0 < π)) → ((-1 · π) < (ℑ‘𝑥) ↔ -1 < ((ℑ‘𝑥) / π)))
123	121, 63, 75, 76, 122	syl112anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((-1 · π) < (ℑ‘𝑥) ↔ -1 < ((ℑ‘𝑥) / π)))
124	119, 123	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → -1 < ((ℑ‘𝑥) / π))
125	116, 124	eqbrtrrid 5184	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (0 − 1) < ((ℑ‘𝑥) / π))
126		zlem1lt 12667	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℤ ∧ ((ℑ‘𝑥) / π) ∈ ℤ) → (0 ≤ ((ℑ‘𝑥) / π) ↔ (0 − 1) < ((ℑ‘𝑥) / π)))
127	112, 111, 126	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (0 ≤ ((ℑ‘𝑥) / π) ↔ (0 − 1) < ((ℑ‘𝑥) / π)))
128	125, 127	mpbird 257	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → 0 ≤ ((ℑ‘𝑥) / π))
129	63, 75, 69	redivcld 12093	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) ∈ ℝ)
130		0re 11261	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
131		letri3 11344	. . . . . . . . . . . . . . 15 ⊢ ((((ℑ‘𝑥) / π) ∈ ℝ ∧ 0 ∈ ℝ) → (((ℑ‘𝑥) / π) = 0 ↔ (((ℑ‘𝑥) / π) ≤ 0 ∧ 0 ≤ ((ℑ‘𝑥) / π))))
132	129, 130, 131	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (((ℑ‘𝑥) / π) = 0 ↔ (((ℑ‘𝑥) / π) ≤ 0 ∧ 0 ≤ ((ℑ‘𝑥) / π))))
133	115, 128, 132	mpbir2and 713	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) = 0)
134	64, 66, 69, 133	diveq0d 12048	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) = 0)
135		reim0b 15155	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (𝑥 ∈ ℝ ↔ (ℑ‘𝑥) = 0))
136	135	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (𝑥 ∈ ℝ ↔ (ℑ‘𝑥) = 0))
137	134, 136	mpbird 257	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → 𝑥 ∈ ℝ)
138	137	rpefcld 16138	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (exp‘𝑥) ∈ ℝ+)
139	138	ex 412	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → ((exp‘𝑥) ∈ ℝ → (exp‘𝑥) ∈ ℝ+))
140	4	ellogdm 26696	. . . . . . . . 9 ⊢ ((exp‘𝑥) ∈ 𝐷 ↔ ((exp‘𝑥) ∈ ℂ ∧ ((exp‘𝑥) ∈ ℝ → (exp‘𝑥) ∈ ℝ+)))
141	62, 139, 140	sylanbrc 583	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (exp‘𝑥) ∈ 𝐷)
142		funfvima2 7251	. . . . . . . . 9 ⊢ ((Fun log ∧ 𝐷 ⊆ dom log) → ((exp‘𝑥) ∈ 𝐷 → (log‘(exp‘𝑥)) ∈ (log “ 𝐷)))
143	9, 13, 142	mp2an 692	. . . . . . . 8 ⊢ ((exp‘𝑥) ∈ 𝐷 → (log‘(exp‘𝑥)) ∈ (log “ 𝐷))
144	141, 143	syl 17	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (log‘(exp‘𝑥)) ∈ (log “ 𝐷))
145	60, 144	eqeltrrd 2840	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → 𝑥 ∈ (log “ 𝐷))
146	46, 145	sylbi 217	. . . . 5 ⊢ (𝑥 ∈ (◡ℑ “ (-π(,)π)) → 𝑥 ∈ (log “ 𝐷))
147	146	ssriv 3999	. . . 4 ⊢ (◡ℑ “ (-π(,)π)) ⊆ (log “ 𝐷)
148	44, 147	eqssi 4012	. . 3 ⊢ (log “ 𝐷) = (◡ℑ “ (-π(,)π))
149		f1oeq3 6839	. . 3 ⊢ ((log “ 𝐷) = (◡ℑ “ (-π(,)π)) → ((log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷) ↔ (log ↾ 𝐷):𝐷–1-1-onto→(◡ℑ “ (-π(,)π))))
150	148, 149	ax-mp 5	. 2 ⊢ ((log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷) ↔ (log ↾ 𝐷):𝐷–1-1-onto→(◡ℑ “ (-π(,)π)))
151	7, 150	mpbi 230	1 ⊢ (log ↾ 𝐷):𝐷–1-1-onto→(◡ℑ “ (-π(,)π))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059   ∖ cdif 3960   ⊆ wss 3963  {csn 4631   class class class wbr 5148  ^◡ccnv 5688  dom cdm 5689  ran crn 5690   ↾ cres 5691   “ cima 5692  Fun wfun 6557   Fn wfn 6558  ⟶wf 6559  –1-1→wf1 6560  –1-1-onto→wf1o 6562  ‘cfv 6563  (class class class)co 7431  ℂcc 11151  ℝcr 11152  0cc0 11153  1c1 11154  ici 11155   + caddc 11156   · cmul 11158  -∞cmnf 11291  ℝ^*cxr 11292   < clt 11293   ≤ cle 11294   − cmin 11490  -cneg 11491   / cdiv 11918  ℤcz 12611  ℝ⁺crp 13032  (,)cioo 13384  (,]cioc 13385  ℜcre 15133  ℑcim 15134  expce 16094  sincsin 16096  cosccos 16097  πcpi 16099  logclog 26611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ioo 13388  df-ioc 13389  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-fl 13829  df-mod 13907  df-seq 14040  df-exp 14100  df-fac 14310  df-bc 14339  df-hash 14367  df-shft 15103  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-limsup 15504  df-clim 15521  df-rlim 15522  df-sum 15720  df-ef 16100  df-sin 16102  df-cos 16103  df-pi 16105  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-rest 17469  df-topn 17470  df-0g 17488  df-gsum 17489  df-topgen 17490  df-pt 17491  df-prds 17494  df-xrs 17549  df-qtop 17554  df-imas 17555  df-xps 17557  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-mulg 19099  df-cntz 19348  df-cmn 19815  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-fbas 21379  df-fg 21380  df-cnfld 21383  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-lp 23160  df-perf 23161  df-cn 23251  df-cnp 23252  df-haus 23339  df-tx 23586  df-hmeo 23779  df-fil 23870  df-fm 23962  df-flim 23963  df-flf 23964  df-xms 24346  df-ms 24347  df-tms 24348  df-cncf 24918  df-limc 25916  df-dv 25917  df-log 26613

This theorem is referenced by:  efopnlem2  26714