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Theorem logf1o2 26681

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 26595	. . . 4 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
2		f1of1 6790	. . . 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 26673	. . 3 ⊢ 𝐷 ⊆ (ℂ ∖ {0})
6		f1ores 6806	. . 3 ⊢ ((log:(ℂ ∖ {0})–1-1→ran log ∧ 𝐷 ⊆ (ℂ ∖ {0})) → (log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷))
7	3, 5, 6	mp2an 700	. 2 ⊢ (log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷)
8		f1ofun 6793	. . . . . . 7 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → Fun log)
9	1, 8	ax-mp 5	. . . . . 6 ⊢ Fun log
10		f1of 6791	. . . . . . . . 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 6688	. . . . . . 7 ⊢ dom log = (ℂ ∖ {0})
13	5, 12	sseqtrri 3976	. . . . . 6 ⊢ 𝐷 ⊆ dom log
14		funimass4 6916	. . . . . 6 ⊢ ((Fun log ∧ 𝐷 ⊆ dom log) → ((log “ 𝐷) ⊆ (^◡ℑ “ (-π(,)π)) ↔ ∀𝑥 ∈ 𝐷 (log‘𝑥) ∈ (^◡ℑ “ (-π(,)π))))
15	9, 13, 14	mp2an 700	. . . . 5 ⊢ ((log “ 𝐷) ⊆ (^◡ℑ “ (-π(,)π)) ↔ ∀𝑥 ∈ 𝐷 (log‘𝑥) ∈ (^◡ℑ “ (-π(,)π)))
16	4	ellogdm 26670	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺)))
17	16	simplbi 499	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ)
18	4	logdmn0 26671	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0)
19	17, 18	logcld 26601	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (log‘𝑥) ∈ ℂ)
20	19	imcld 15194	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ∈ ℝ)
21	17, 18	logimcld 26602	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → (-π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) ≤ π))
22	21	simpld 497	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → -π < (ℑ‘(log‘𝑥)))
23		pire 26485	. . . . . . . . 9 ⊢ π ∈ ℝ
24	23	a1i 11	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → π ∈ ℝ)
25	21	simprd 498	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ≤ π)
26	4	logdmnrp 26672	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → ¬ -𝑥 ∈ ℝ⁺)
27		lognegb 26621	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (-𝑥 ∈ ℝ⁺ ↔ (ℑ‘(log‘𝑥)) = π))
28	17, 18, 27	syl2anc 592	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐷 → (-𝑥 ∈ ℝ⁺ ↔ (ℑ‘(log‘𝑥)) = π))
29	28	necon3bbid 2984	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → (¬ -𝑥 ∈ ℝ⁺ ↔ (ℑ‘(log‘𝑥)) ≠ π))
30	26, 29	mpbid 234	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ≠ π)
31	30	necomd 3002	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → π ≠ (ℑ‘(log‘𝑥)))
32	20, 24, 25, 31	leneltd 11323	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) < π)
33	23	renegcli 11478	. . . . . . . . 9 ⊢ -π ∈ ℝ
34	33	rexri 11226	. . . . . . . 8 ⊢ -π ∈ ℝ^*
35	23	rexri 11226	. . . . . . . 8 ⊢ π ∈ ℝ^*
36		elioo2 13376	. . . . . . . 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 700	. . . . . . 7 ⊢ ((ℑ‘(log‘𝑥)) ∈ (-π(,)π) ↔ ((ℑ‘(log‘𝑥)) ∈ ℝ ∧ -π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) < π))
38	20, 22, 32, 37	syl3anbrc 1353	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ∈ (-π(,)π))
39		imf 15112	. . . . . . 7 ⊢ ℑ:ℂ⟶ℝ
40		ffn 6676	. . . . . . 7 ⊢ (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
41		elpreima 7024	. . . . . . 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 591	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (log‘𝑥) ∈ (^◡ℑ “ (-π(,)π)))
44	15, 43	mprgbir 3073	. . . 4 ⊢ (log “ 𝐷) ⊆ (^◡ℑ “ (-π(,)π))
45		elpreima 7024	. . . . . . 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 485	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → 𝑥 ∈ ℂ)
48		eliooord 13395	. . . . . . . . . . 11 ⊢ ((ℑ‘𝑥) ∈ (-π(,)π) → (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π))
49	48	adantl 484	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π))
50	49	simpld 497	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → -π < (ℑ‘𝑥))
51	49	simprd 498	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (ℑ‘𝑥) < π)
52		imcl 15110	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (ℑ‘𝑥) ∈ ℝ)
53	52	adantr 483	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (ℑ‘𝑥) ∈ ℝ)
54		ltle 11257	. . . . . . . . . . 11 ⊢ (((ℑ‘𝑥) ∈ ℝ ∧ π ∈ ℝ) → ((ℑ‘𝑥) < π → (ℑ‘𝑥) ≤ π))
55	53, 23, 54	sylancl 594	. . . . . . . . . 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 26590	. . . . . . . . 9 ⊢ (𝑥 ∈ ran log ↔ (𝑥 ∈ ℂ ∧ -π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) ≤ π))
58	47, 50, 56, 57	syl3anbrc 1353	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → 𝑥 ∈ ran log)
59		logef 26612	. . . . . . . 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 16084	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) ∈ ℂ)
62	61	adantr 483	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (exp‘𝑥) ∈ ℂ)
63	53	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) ∈ ℝ)
64	63	recnd 11196	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) ∈ ℂ)
65		picn 26487	. . . . . . . . . . . . . 14 ⊢ π ∈ ℂ
66	65	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → π ∈ ℂ)
67		pipos 26489	. . . . . . . . . . . . . . 15 ⊢ 0 < π
68	23, 67	gt0ne0ii 11709	. . . . . . . . . . . . . 14 ⊢ π ≠ 0
69	68	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → π ≠ 0)
70	51	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) < π)
71	65	mulridi 11172	. . . . . . . . . . . . . . . . . 18 ⊢ (π · 1) = π
72	70, 71	breqtrrdi 5132	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) < (π · 1))
73		1re 11167	. . . . . . . . . . . . . . . . . . 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 12053	. . . . . . . . . . . . . . . . . 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 1385	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (((ℑ‘𝑥) / π) < 1 ↔ (ℑ‘𝑥) < (π · 1)))
79	72, 78	mpbird 259	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) < 1)
80		1e0p1 12721	. . . . . . . . . . . . . . . 16 ⊢ 1 = (0 + 1)
81	79, 80	breqtrdi 5131	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) < (0 + 1))
82	63	recoscld 16148	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (cos‘(ℑ‘𝑥)) ∈ ℝ)
83	63	resincld 16147	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (sin‘(ℑ‘𝑥)) ∈ ℝ)
84	82, 83	crimd 15231	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))) = (sin‘(ℑ‘𝑥)))
85		efeul 16166	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) = ((exp‘(ℜ‘𝑥)) · ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))))
86	85	ad2antrr 734	. . . . . . . . . . . . . . . . . . . . . 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 7396	. . . . . . . . . . . . . . . . . . . . 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 11196	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (cos‘(ℑ‘𝑥)) ∈ ℂ)
89		ax-icn 11118	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ i ∈ ℂ
90	83	recnd 11196	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (sin‘(ℑ‘𝑥)) ∈ ℂ)
91		mulcl 11143	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((i ∈ ℂ ∧ (sin‘(ℑ‘𝑥)) ∈ ℂ) → (i · (sin‘(ℑ‘𝑥))) ∈ ℂ)
92	89, 90, 91	sylancr 595	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (i · (sin‘(ℑ‘𝑥))) ∈ ℂ)
93	88, 92	addcld 11187	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥)))) ∈ ℂ)
94		recl 15109	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ ℂ → (ℜ‘𝑥) ∈ ℝ)
95	94	ad2antrr 734	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℜ‘𝑥) ∈ ℝ)
96	95	recnd 11196	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℜ‘𝑥) ∈ ℂ)
97		efcl 16084	. . . . . . . . . . . . . . . . . . . . . . 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 16100	. . . . . . . . . . . . . . . . . . . . . . 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 11958	. . . . . . . . . . . . . . . . . . . . 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 2787	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((exp‘𝑥) / (exp‘(ℜ‘𝑥))) = ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥)))))
103		simpr 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (exp‘𝑥) ∈ ℝ)
104	95	reefcld 16090	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (exp‘(ℜ‘𝑥)) ∈ ℝ)
105	103, 104, 100	redivcld 12005	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((exp‘𝑥) / (exp‘(ℜ‘𝑥))) ∈ ℝ)
106	102, 105	eqeltrrd 2853	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥)))) ∈ ℝ)
107	106	reim0d 15224	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))) = 0)
108	84, 107	eqtr3d 2789	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (sin‘(ℑ‘𝑥)) = 0)
109		sineq0 26555	. . . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) ∈ ℤ)
112		0z 12565	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
113		zleltp1 12608	. . . . . . . . . . . . . . . 16 ⊢ ((((ℑ‘𝑥) / π) ∈ ℤ ∧ 0 ∈ ℤ) → (((ℑ‘𝑥) / π) ≤ 0 ↔ ((ℑ‘𝑥) / π) < (0 + 1)))
114	111, 112, 113	sylancl 594	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (((ℑ‘𝑥) / π) ≤ 0 ↔ ((ℑ‘𝑥) / π) < (0 + 1)))
115	81, 114	mpbird 259	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) ≤ 0)
116		df-neg 11403	. . . . . . . . . . . . . . . 16 ⊢ -1 = (0 − 1)
117	65	mulm1i 11618	. . . . . . . . . . . . . . . . . 18 ⊢ (-1 · π) = -π
118	50	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → -π < (ℑ‘𝑥))
119	117, 118	eqbrtrid 5125	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (-1 · π) < (ℑ‘𝑥))
120	73	renegcli 11478	. . . . . . . . . . . . . . . . . . 19 ⊢ -1 ∈ ℝ
121	120	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → -1 ∈ ℝ)
122		ltmuldiv 12051	. . . . . . . . . . . . . . . . . 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 1385	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((-1 · π) < (ℑ‘𝑥) ↔ -1 < ((ℑ‘𝑥) / π)))
124	119, 123	mpbid 234	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → -1 < ((ℑ‘𝑥) / π))
125	116, 124	eqbrtrrid 5126	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (0 − 1) < ((ℑ‘𝑥) / π))
126		zlem1lt 12609	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℤ ∧ ((ℑ‘𝑥) / π) ∈ ℤ) → (0 ≤ ((ℑ‘𝑥) / π) ↔ (0 − 1) < ((ℑ‘𝑥) / π)))
127	112, 111, 126	sylancr 595	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (0 ≤ ((ℑ‘𝑥) / π) ↔ (0 − 1) < ((ℑ‘𝑥) / π)))
128	125, 127	mpbird 259	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → 0 ≤ ((ℑ‘𝑥) / π))
129	63, 75, 69	redivcld 12005	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) ∈ ℝ)
130		0re 11169	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
131		letri3 11254	. . . . . . . . . . . . . . 15 ⊢ ((((ℑ‘𝑥) / π) ∈ ℝ ∧ 0 ∈ ℝ) → (((ℑ‘𝑥) / π) = 0 ↔ (((ℑ‘𝑥) / π) ≤ 0 ∧ 0 ≤ ((ℑ‘𝑥) / π))))
132	129, 130, 131	sylancl 594	. . . . . . . . . . . . . 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 721	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) = 0)
134	64, 66, 69, 133	diveq0d 11960	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) = 0)
135		reim0b 15118	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (𝑥 ∈ ℝ ↔ (ℑ‘𝑥) = 0))
136	135	ad2antrr 734	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (𝑥 ∈ ℝ ↔ (ℑ‘𝑥) = 0))
137	134, 136	mpbird 259	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → 𝑥 ∈ ℝ)
138	137	rpefcld 16109	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (exp‘𝑥) ∈ ℝ⁺)
139	138	ex 415	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → ((exp‘𝑥) ∈ ℝ → (exp‘𝑥) ∈ ℝ⁺))
140	4	ellogdm 26670	. . . . . . . . 9 ⊢ ((exp‘𝑥) ∈ 𝐷 ↔ ((exp‘𝑥) ∈ ℂ ∧ ((exp‘𝑥) ∈ ℝ → (exp‘𝑥) ∈ ℝ⁺)))
141	62, 139, 140	sylanbrc 591	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (exp‘𝑥) ∈ 𝐷)
142		funfvima2 7200	. . . . . . . . 9 ⊢ ((Fun log ∧ 𝐷 ⊆ dom log) → ((exp‘𝑥) ∈ 𝐷 → (log‘(exp‘𝑥)) ∈ (log “ 𝐷)))
143	9, 13, 142	mp2an 700	. . . . . . . 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 2853	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → 𝑥 ∈ (log “ 𝐷))
146	46, 145	sylbi 219	. . . . 5 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → 𝑥 ∈ (log “ 𝐷))
147	146	ssriv 3931	. . . 4 ⊢ (^◡ℑ “ (-π(,)π)) ⊆ (log “ 𝐷)
148	44, 147	eqssi 3943	. . 3 ⊢ (log “ 𝐷) = (^◡ℑ “ (-π(,)π))
149		f1oeq3 6781	. . 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 232	1 ⊢ (log ↾ 𝐷):𝐷–1-1-onto→(^◡ℑ “ (-π(,)π))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1095 = wceq 1550 ∈ wcel 2132 ≠ wne 2947 ∀wral 3066 ∖ cdif 3892 ⊆ wss 3895 {csn 4572 class class class wbr 5090 ^◡ccnv 5635 dom cdm 5636 ran crn 5637 ↾ cres 5638 “ cima 5639 Fun wfun 6500 Fn wfn 6501 ⟶wf 6502 –1-1→wf1 6503 –1-1-onto→wf1o 6505 ‘cfv 6506 (class class class)co 7381 ℂcc 11057 ℝcr 11058 0cc0 11059 1c1 11060 ici 11061 + caddc 11062 · cmul 11064 -∞cmnf 11200 ℝ^*cxr 11201 < clt 11202 ≤ cle 11203 − cmin 11400 -cneg 11401 / cdiv 11830 ℤcz 12554 ℝ⁺crp 12979 (,)cioo 13335 (,]cioc 13336 ℜcre 15096 ℑcim 15097 expce 16063 sincsin 16065 cosccos 16066 πcpi 16068 logclog 26585

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-rep 5217 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703 ax-inf2 9582 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-nel 3052 df-ral 3067 df-rex 3077 df-rmo 3357 df-reu 3358 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-pss 3915 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4856 df-int 4896 df-iun 4941 df-iin 4942 df-br 5091 df-opab 5153 df-mpt 5172 df-tr 5198 df-id 5531 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5589 df-se 5590 df-we 5591 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-pred 6273 df-ord 6334 df-on 6335 df-lim 6336 df-suc 6337 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-isom 6515 df-riota 7338 df-ov 7384 df-oprab 7385 df-mpo 7386 df-of 7645 df-om 7832 df-1st 7955 df-2nd 7956 df-supp 8125 df-frecs 8246 df-wrecs 8277 df-recs 8326 df-rdg 8365 df-1o 8421 df-2o 8422 df-er 8662 df-map 8794 df-pm 8795 df-ixp 8865 df-en 8913 df-dom 8914 df-sdom 8915 df-fin 8916 df-fsupp 9294 df-fi 9343 df-sup 9374 df-inf 9375 df-oi 9444 df-card 9883 df-pnf 11204 df-mnf 11205 df-xr 11206 df-ltxr 11207 df-le 11208 df-sub 11402 df-neg 11403 df-div 11831 df-nn 12197 df-2 12266 df-3 12267 df-4 12268 df-5 12269 df-6 12270 df-7 12271 df-8 12272 df-9 12273 df-n0 12468 df-z 12555 df-dec 12675 df-uz 12826 df-q 12936 df-rp 12980 df-xneg 13100 df-xadd 13101 df-xmul 13102 df-ioo 13339 df-ioc 13340 df-ico 13341 df-icc 13342 df-fz 13499 df-fzo 13646 df-fl 13788 df-mod 13866 df-seq 14001 df-exp 14061 df-fac 14273 df-bc 14302 df-hash 14330 df-shft 15066 df-cj 15098 df-re 15099 df-im 15100 df-sqrt 15234 df-abs 15235 df-limsup 15470 df-clim 15487 df-rlim 15488 df-sum 15686 df-ef 16069 df-sin 16071 df-cos 16072 df-pi 16074 df-struct 17155 df-sets 17172 df-slot 17190 df-ndx 17202 df-base 17218 df-ress 17239 df-plusg 17271 df-mulr 17272 df-starv 17273 df-sca 17274 df-vsca 17275 df-ip 17276 df-tset 17277 df-ple 17278 df-ds 17280 df-unif 17281 df-hom 17282 df-cco 17283 df-rest 17423 df-topn 17424 df-0g 17442 df-gsum 17443 df-topgen 17444 df-pt 17445 df-prds 17448 df-xrs 17504 df-qtop 17509 df-imas 17510 df-xps 17512 df-mre 17586 df-mrc 17587 df-acs 17589 df-mgm 18646 df-sgrp 18725 df-mnd 18741 df-submnd 18790 df-mulg 19082 df-cntz 19329 df-cmn 19794 df-psmet 21385 df-xmet 21386 df-met 21387 df-bl 21388 df-mopn 21389 df-fbas 21390 df-fg 21391 df-cnfld 21394 df-top 22923 df-topon 22940 df-topsp 22962 df-bases 22975 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24349 df-ms 24350 df-tms 24351 df-cncf 24909 df-limc 25897 df-dv 25898 df-log 26587

This theorem is referenced by: efopnlem2 26688

W3C validator