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Theorem logf1o2 26614

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 26528	. . . 4 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
2		f1of1 6780	. . . 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 26606	. . 3 ⊢ 𝐷 ⊆ (ℂ ∖ {0})
6		f1ores 6795	. . 3 ⊢ ((log:(ℂ ∖ {0})–1-1→ran log ∧ 𝐷 ⊆ (ℂ ∖ {0})) → (log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷))
7	3, 5, 6	mp2an 693	. 2 ⊢ (log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷)
8		f1ofun 6783	. . . . . . 7 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → Fun log)
9	1, 8	ax-mp 5	. . . . . 6 ⊢ Fun log
10		f1of 6781	. . . . . . . . 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 6680	. . . . . . 7 ⊢ dom log = (ℂ ∖ {0})
13	5, 12	sseqtrri 3972	. . . . . 6 ⊢ 𝐷 ⊆ dom log
14		funimass4 6905	. . . . . 6 ⊢ ((Fun log ∧ 𝐷 ⊆ dom log) → ((log “ 𝐷) ⊆ (^◡ℑ “ (-π(,)π)) ↔ ∀𝑥 ∈ 𝐷 (log‘𝑥) ∈ (^◡ℑ “ (-π(,)π))))
15	9, 13, 14	mp2an 693	. . . . 5 ⊢ ((log “ 𝐷) ⊆ (^◡ℑ “ (-π(,)π)) ↔ ∀𝑥 ∈ 𝐷 (log‘𝑥) ∈ (^◡ℑ “ (-π(,)π)))
16	4	ellogdm 26603	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺)))
17	16	simplbi 496	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ)
18	4	logdmn0 26604	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0)
19	17, 18	logcld 26534	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (log‘𝑥) ∈ ℂ)
20	19	imcld 15157	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ∈ ℝ)
21	17, 18	logimcld 26535	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → (-π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) ≤ π))
22	21	simpld 494	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → -π < (ℑ‘(log‘𝑥)))
23		pire 26421	. . . . . . . . 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 26605	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → ¬ -𝑥 ∈ ℝ⁺)
27		lognegb 26554	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (-𝑥 ∈ ℝ⁺ ↔ (ℑ‘(log‘𝑥)) = π))
28	17, 18, 27	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐷 → (-𝑥 ∈ ℝ⁺ ↔ (ℑ‘(log‘𝑥)) = π))
29	28	necon3bbid 2970	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → (¬ -𝑥 ∈ ℝ⁺ ↔ (ℑ‘(log‘𝑥)) ≠ π))
30	26, 29	mpbid 232	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ≠ π)
31	30	necomd 2988	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → π ≠ (ℑ‘(log‘𝑥)))
32	20, 24, 25, 31	leneltd 11300	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) < π)
33	23	renegcli 11455	. . . . . . . . 9 ⊢ -π ∈ ℝ
34	33	rexri 11203	. . . . . . . 8 ⊢ -π ∈ ℝ^*
35	23	rexri 11203	. . . . . . . 8 ⊢ π ∈ ℝ^*
36		elioo2 13339	. . . . . . . 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 693	. . . . . . 7 ⊢ ((ℑ‘(log‘𝑥)) ∈ (-π(,)π) ↔ ((ℑ‘(log‘𝑥)) ∈ ℝ ∧ -π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) < π))
38	20, 22, 32, 37	syl3anbrc 1345	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ∈ (-π(,)π))
39		imf 15075	. . . . . . 7 ⊢ ℑ:ℂ⟶ℝ
40		ffn 6669	. . . . . . 7 ⊢ (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
41		elpreima 7011	. . . . . . 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 584	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (log‘𝑥) ∈ (^◡ℑ “ (-π(,)π)))
44	15, 43	mprgbir 3059	. . . 4 ⊢ (log “ 𝐷) ⊆ (^◡ℑ “ (-π(,)π))
45		elpreima 7011	. . . . . . 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 13358	. . . . . . . . . . 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 15073	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (ℑ‘𝑥) ∈ ℝ)
53	52	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (ℑ‘𝑥) ∈ ℝ)
54		ltle 11234	. . . . . . . . . . 11 ⊢ (((ℑ‘𝑥) ∈ ℝ ∧ π ∈ ℝ) → ((ℑ‘𝑥) < π → (ℑ‘𝑥) ≤ π))
55	53, 23, 54	sylancl 587	. . . . . . . . . 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 26523	. . . . . . . . 9 ⊢ (𝑥 ∈ ran log ↔ (𝑥 ∈ ℂ ∧ -π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) ≤ π))
58	47, 50, 56, 57	syl3anbrc 1345	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → 𝑥 ∈ ran log)
59		logef 26545	. . . . . . . 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 16047	. . . . . . . . . 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 11173	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) ∈ ℂ)
65		picn 26422	. . . . . . . . . . . . . 14 ⊢ π ∈ ℂ
66	65	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → π ∈ ℂ)
67		pipos 26423	. . . . . . . . . . . . . . 15 ⊢ 0 < π
68	23, 67	gt0ne0ii 11686	. . . . . . . . . . . . . 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 11149	. . . . . . . . . . . . . . . . . 18 ⊢ (π · 1) = π
72	70, 71	breqtrrdi 5128	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) < (π · 1))
73		1re 11144	. . . . . . . . . . . . . . . . . . 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 12031	. . . . . . . . . . . . . . . . . 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 1377	. . . . . . . . . . . . . . . . 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 12686	. . . . . . . . . . . . . . . 16 ⊢ 1 = (0 + 1)
81	79, 80	breqtrdi 5127	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) < (0 + 1))
82	63	recoscld 16111	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (cos‘(ℑ‘𝑥)) ∈ ℝ)
83	63	resincld 16110	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (sin‘(ℑ‘𝑥)) ∈ ℝ)
84	82, 83	crimd 15194	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))) = (sin‘(ℑ‘𝑥)))
85		efeul 16129	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) = ((exp‘(ℜ‘𝑥)) · ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))))
86	85	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . 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 7382	. . . . . . . . . . . . . . . . . . . . 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 11173	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (cos‘(ℑ‘𝑥)) ∈ ℂ)
89		ax-icn 11097	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ i ∈ ℂ
90	83	recnd 11173	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (sin‘(ℑ‘𝑥)) ∈ ℂ)
91		mulcl 11122	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((i ∈ ℂ ∧ (sin‘(ℑ‘𝑥)) ∈ ℂ) → (i · (sin‘(ℑ‘𝑥))) ∈ ℂ)
92	89, 90, 91	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (i · (sin‘(ℑ‘𝑥))) ∈ ℂ)
93	88, 92	addcld 11164	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥)))) ∈ ℂ)
94		recl 15072	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ ℂ → (ℜ‘𝑥) ∈ ℝ)
95	94	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℜ‘𝑥) ∈ ℝ)
96	95	recnd 11173	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℜ‘𝑥) ∈ ℂ)
97		efcl 16047	. . . . . . . . . . . . . . . . . . . . . . 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 16063	. . . . . . . . . . . . . . . . . . . . . . 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 11936	. . . . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . . . 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 16053	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (exp‘(ℜ‘𝑥)) ∈ ℝ)
105	103, 104, 100	redivcld 11983	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((exp‘𝑥) / (exp‘(ℜ‘𝑥))) ∈ ℝ)
106	102, 105	eqeltrrd 2838	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥)))) ∈ ℝ)
107	106	reim0d 15187	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))) = 0)
108	84, 107	eqtr3d 2774	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (sin‘(ℑ‘𝑥)) = 0)
109		sineq0 26488	. . . . . . . . . . . . . . . . . 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 12535	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
113		zleltp1 12578	. . . . . . . . . . . . . . . 16 ⊢ ((((ℑ‘𝑥) / π) ∈ ℤ ∧ 0 ∈ ℤ) → (((ℑ‘𝑥) / π) ≤ 0 ↔ ((ℑ‘𝑥) / π) < (0 + 1)))
114	111, 112, 113	sylancl 587	. . . . . . . . . . . . . . 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 11380	. . . . . . . . . . . . . . . 16 ⊢ -1 = (0 − 1)
117	65	mulm1i 11595	. . . . . . . . . . . . . . . . . 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 5121	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (-1 · π) < (ℑ‘𝑥))
120	73	renegcli 11455	. . . . . . . . . . . . . . . . . . 19 ⊢ -1 ∈ ℝ
121	120	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → -1 ∈ ℝ)
122		ltmuldiv 12029	. . . . . . . . . . . . . . . . . 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 1377	. . . . . . . . . . . . . . . . 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 5122	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (0 − 1) < ((ℑ‘𝑥) / π))
126		zlem1lt 12579	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℤ ∧ ((ℑ‘𝑥) / π) ∈ ℤ) → (0 ≤ ((ℑ‘𝑥) / π) ↔ (0 − 1) < ((ℑ‘𝑥) / π)))
127	112, 111, 126	sylancr 588	. . . . . . . . . . . . . . 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 11983	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) ∈ ℝ)
130		0re 11146	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
131		letri3 11231	. . . . . . . . . . . . . . 15 ⊢ ((((ℑ‘𝑥) / π) ∈ ℝ ∧ 0 ∈ ℝ) → (((ℑ‘𝑥) / π) = 0 ↔ (((ℑ‘𝑥) / π) ≤ 0 ∧ 0 ≤ ((ℑ‘𝑥) / π))))
132	129, 130, 131	sylancl 587	. . . . . . . . . . . . . 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 714	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) = 0)
134	64, 66, 69, 133	diveq0d 11938	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) = 0)
135		reim0b 15081	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (𝑥 ∈ ℝ ↔ (ℑ‘𝑥) = 0))
136	135	ad2antrr 727	. . . . . . . . . . . 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 16072	. . . . . . . . . 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 26603	. . . . . . . . 9 ⊢ ((exp‘𝑥) ∈ 𝐷 ↔ ((exp‘𝑥) ∈ ℂ ∧ ((exp‘𝑥) ∈ ℝ → (exp‘𝑥) ∈ ℝ⁺)))
141	62, 139, 140	sylanbrc 584	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (exp‘𝑥) ∈ 𝐷)
142		funfvima2 7186	. . . . . . . . 9 ⊢ ((Fun log ∧ 𝐷 ⊆ dom log) → ((exp‘𝑥) ∈ 𝐷 → (log‘(exp‘𝑥)) ∈ (log “ 𝐷)))
143	9, 13, 142	mp2an 693	. . . . . . . 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 2838	. . . . . 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 3926	. . . 4 ⊢ (^◡ℑ “ (-π(,)π)) ⊆ (log “ 𝐷)
148	44, 147	eqssi 3939	. . 3 ⊢ (log “ 𝐷) = (^◡ℑ “ (-π(,)π))
149		f1oeq3 6771	. . 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))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∖ cdif 3887 ⊆ wss 3890 {csn 4568 class class class wbr 5086 ^◡ccnv 5630 dom cdm 5631 ran crn 5632 ↾ cres 5633 “ cima 5634 Fun wfun 6493 Fn wfn 6494 ⟶wf 6495 –1-1→wf1 6496 –1-1-onto→wf1o 6498 ‘cfv 6499 (class class class)co 7367 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 ici 11040 + caddc 11041 · cmul 11043 -∞cmnf 11177 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 − cmin 11377 -cneg 11378 / cdiv 11807 ℤcz 12524 ℝ⁺crp 12942 (,)cioo 13298 (,]cioc 13299 ℜcre 15059 ℑcim 15060 expce 16026 sincsin 16028 cosccos 16029 πcpi 16031 logclog 26518

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6266 df-ord 6327 df-on 6328 df-lim 6329 df-suc 6330 df-iota 6455 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-ef 16032 df-sin 16034 df-cos 16035 df-pi 16037 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834 df-log 26520

This theorem is referenced by: efopnlem2 26621

W3C validator