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Theorem logf1o2 25233

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 25148	. . . 4 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
2		f1of1 6614	. . . 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 25225	. . 3 ⊢ 𝐷 ⊆ (ℂ ∖ {0})
6		f1ores 6629	. . 3 ⊢ ((log:(ℂ ∖ {0})–1-1→ran log ∧ 𝐷 ⊆ (ℂ ∖ {0})) → (log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷))
7	3, 5, 6	mp2an 690	. 2 ⊢ (log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷)
8		f1ofun 6617	. . . . . . 7 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → Fun log)
9	1, 8	ax-mp 5	. . . . . 6 ⊢ Fun log
10		f1of 6615	. . . . . . . . 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 6524	. . . . . . 7 ⊢ dom log = (ℂ ∖ {0})
13	5, 12	sseqtrri 4004	. . . . . 6 ⊢ 𝐷 ⊆ dom log
14		funimass4 6730	. . . . . 6 ⊢ ((Fun log ∧ 𝐷 ⊆ dom log) → ((log “ 𝐷) ⊆ (^◡ℑ “ (-π(,)π)) ↔ ∀𝑥 ∈ 𝐷 (log‘𝑥) ∈ (^◡ℑ “ (-π(,)π))))
15	9, 13, 14	mp2an 690	. . . . 5 ⊢ ((log “ 𝐷) ⊆ (^◡ℑ “ (-π(,)π)) ↔ ∀𝑥 ∈ 𝐷 (log‘𝑥) ∈ (^◡ℑ “ (-π(,)π)))
16	4	ellogdm 25222	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺)))
17	16	simplbi 500	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ)
18	4	logdmn0 25223	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0)
19	17, 18	logcld 25154	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (log‘𝑥) ∈ ℂ)
20	19	imcld 14554	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ∈ ℝ)
21	17, 18	logimcld 25155	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → (-π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) ≤ π))
22	21	simpld 497	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → -π < (ℑ‘(log‘𝑥)))
23		pire 25044	. . . . . . . . 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 25224	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → ¬ -𝑥 ∈ ℝ⁺)
27		lognegb 25173	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (-𝑥 ∈ ℝ⁺ ↔ (ℑ‘(log‘𝑥)) = π))
28	17, 18, 27	syl2anc 586	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐷 → (-𝑥 ∈ ℝ⁺ ↔ (ℑ‘(log‘𝑥)) = π))
29	28	necon3bbid 3053	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → (¬ -𝑥 ∈ ℝ⁺ ↔ (ℑ‘(log‘𝑥)) ≠ π))
30	26, 29	mpbid 234	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ≠ π)
31	30	necomd 3071	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → π ≠ (ℑ‘(log‘𝑥)))
32	20, 24, 25, 31	leneltd 10794	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) < π)
33	23	renegcli 10947	. . . . . . . . 9 ⊢ -π ∈ ℝ
34	33	rexri 10699	. . . . . . . 8 ⊢ -π ∈ ℝ^*
35	23	rexri 10699	. . . . . . . 8 ⊢ π ∈ ℝ^*
36		elioo2 12780	. . . . . . . 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 690	. . . . . . 7 ⊢ ((ℑ‘(log‘𝑥)) ∈ (-π(,)π) ↔ ((ℑ‘(log‘𝑥)) ∈ ℝ ∧ -π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) < π))
38	20, 22, 32, 37	syl3anbrc 1339	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ∈ (-π(,)π))
39		imf 14472	. . . . . . 7 ⊢ ℑ:ℂ⟶ℝ
40		ffn 6514	. . . . . . 7 ⊢ (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
41		elpreima 6828	. . . . . . 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 585	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (log‘𝑥) ∈ (^◡ℑ “ (-π(,)π)))
44	15, 43	mprgbir 3153	. . . 4 ⊢ (log “ 𝐷) ⊆ (^◡ℑ “ (-π(,)π))
45		elpreima 6828	. . . . . . 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 12797	. . . . . . . . . . 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 14470	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (ℑ‘𝑥) ∈ ℝ)
53	52	adantr 483	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (ℑ‘𝑥) ∈ ℝ)
54		ltle 10729	. . . . . . . . . . 11 ⊢ (((ℑ‘𝑥) ∈ ℝ ∧ π ∈ ℝ) → ((ℑ‘𝑥) < π → (ℑ‘𝑥) ≤ π))
55	53, 23, 54	sylancl 588	. . . . . . . . . 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 25143	. . . . . . . . 9 ⊢ (𝑥 ∈ ran log ↔ (𝑥 ∈ ℂ ∧ -π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) ≤ π))
58	47, 50, 56, 57	syl3anbrc 1339	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → 𝑥 ∈ ran log)
59		logef 25165	. . . . . . . 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 15436	. . . . . . . . . 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 10669	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) ∈ ℂ)
65		picn 25045	. . . . . . . . . . . . . 14 ⊢ π ∈ ℂ
66	65	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → π ∈ ℂ)
67		pipos 25046	. . . . . . . . . . . . . . 15 ⊢ 0 < π
68	23, 67	gt0ne0ii 11176	. . . . . . . . . . . . . 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	mulid1i 10645	. . . . . . . . . . . . . . . . . 18 ⊢ (π · 1) = π
72	70, 71	breqtrrdi 5108	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) < (π · 1))
73		1re 10641	. . . . . . . . . . . . . . . . . . 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 11515	. . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . 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 12141	. . . . . . . . . . . . . . . 16 ⊢ 1 = (0 + 1)
81	79, 80	breqtrdi 5107	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) < (0 + 1))
82	63	recoscld 15497	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (cos‘(ℑ‘𝑥)) ∈ ℝ)
83	63	resincld 15496	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (sin‘(ℑ‘𝑥)) ∈ ℝ)
84	82, 83	crimd 14591	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))) = (sin‘(ℑ‘𝑥)))
85		efeul 15515	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) = ((exp‘(ℜ‘𝑥)) · ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))))
86	85	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . 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 7171	. . . . . . . . . . . . . . . . . . . . 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 10669	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (cos‘(ℑ‘𝑥)) ∈ ℂ)
89		ax-icn 10596	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ i ∈ ℂ
90	83	recnd 10669	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (sin‘(ℑ‘𝑥)) ∈ ℂ)
91		mulcl 10621	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((i ∈ ℂ ∧ (sin‘(ℑ‘𝑥)) ∈ ℂ) → (i · (sin‘(ℑ‘𝑥))) ∈ ℂ)
92	89, 90, 91	sylancr 589	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (i · (sin‘(ℑ‘𝑥))) ∈ ℂ)
93	88, 92	addcld 10660	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥)))) ∈ ℂ)
94		recl 14469	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ ℂ → (ℜ‘𝑥) ∈ ℝ)
95	94	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℜ‘𝑥) ∈ ℝ)
96	95	recnd 10669	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℜ‘𝑥) ∈ ℂ)
97		efcl 15436	. . . . . . . . . . . . . . . . . . . . . . 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 15450	. . . . . . . . . . . . . . . . . . . . . . 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 11421	. . . . . . . . . . . . . . . . . . . . 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 2856	. . . . . . . . . . . . . . . . . . . 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 15441	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (exp‘(ℜ‘𝑥)) ∈ ℝ)
105	103, 104, 100	redivcld 11468	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((exp‘𝑥) / (exp‘(ℜ‘𝑥))) ∈ ℝ)
106	102, 105	eqeltrrd 2914	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥)))) ∈ ℝ)
107	106	reim0d 14584	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))) = 0)
108	84, 107	eqtr3d 2858	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (sin‘(ℑ‘𝑥)) = 0)
109		sineq0 25109	. . . . . . . . . . . . . . . . . 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 11993	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
113		zleltp1 12034	. . . . . . . . . . . . . . . 16 ⊢ ((((ℑ‘𝑥) / π) ∈ ℤ ∧ 0 ∈ ℤ) → (((ℑ‘𝑥) / π) ≤ 0 ↔ ((ℑ‘𝑥) / π) < (0 + 1)))
114	111, 112, 113	sylancl 588	. . . . . . . . . . . . . . 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 10873	. . . . . . . . . . . . . . . 16 ⊢ -1 = (0 − 1)
117	65	mulm1i 11085	. . . . . . . . . . . . . . . . . 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 5101	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (-1 · π) < (ℑ‘𝑥))
120	73	renegcli 10947	. . . . . . . . . . . . . . . . . . 19 ⊢ -1 ∈ ℝ
121	120	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → -1 ∈ ℝ)
122		ltmuldiv 11513	. . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . 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 5102	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (0 − 1) < ((ℑ‘𝑥) / π))
126		zlem1lt 12035	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℤ ∧ ((ℑ‘𝑥) / π) ∈ ℤ) → (0 ≤ ((ℑ‘𝑥) / π) ↔ (0 − 1) < ((ℑ‘𝑥) / π)))
127	112, 111, 126	sylancr 589	. . . . . . . . . . . . . . 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 11468	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) ∈ ℝ)
130		0re 10643	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
131		letri3 10726	. . . . . . . . . . . . . . 15 ⊢ ((((ℑ‘𝑥) / π) ∈ ℝ ∧ 0 ∈ ℝ) → (((ℑ‘𝑥) / π) = 0 ↔ (((ℑ‘𝑥) / π) ≤ 0 ∧ 0 ≤ ((ℑ‘𝑥) / π))))
132	129, 130, 131	sylancl 588	. . . . . . . . . . . . . 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 711	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) = 0)
134	64, 66, 69, 133	diveq0d 11423	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) = 0)
135		reim0b 14478	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (𝑥 ∈ ℝ ↔ (ℑ‘𝑥) = 0))
136	135	ad2antrr 724	. . . . . . . . . . . 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 15458	. . . . . . . . . 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 25222	. . . . . . . . 9 ⊢ ((exp‘𝑥) ∈ 𝐷 ↔ ((exp‘𝑥) ∈ ℂ ∧ ((exp‘𝑥) ∈ ℝ → (exp‘𝑥) ∈ ℝ⁺)))
141	62, 139, 140	sylanbrc 585	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (exp‘𝑥) ∈ 𝐷)
142		funfvima2 6993	. . . . . . . . 9 ⊢ ((Fun log ∧ 𝐷 ⊆ dom log) → ((exp‘𝑥) ∈ 𝐷 → (log‘(exp‘𝑥)) ∈ (log “ 𝐷)))
143	9, 13, 142	mp2an 690	. . . . . . . 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 2914	. . . . . 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 3971	. . . 4 ⊢ (^◡ℑ “ (-π(,)π)) ⊆ (log “ 𝐷)
148	44, 147	eqssi 3983	. . 3 ⊢ (log “ 𝐷) = (^◡ℑ “ (-π(,)π))
149		f1oeq3 6606	. . 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 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∖ cdif 3933 ⊆ wss 3936 {csn 4567 class class class wbr 5066 ^◡ccnv 5554 dom cdm 5555 ran crn 5556 ↾ cres 5557 “ cima 5558 Fun wfun 6349 Fn wfn 6350 ⟶wf 6351 –1-1→wf1 6352 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 ici 10539 + caddc 10540 · cmul 10542 -∞cmnf 10673 ℝ^*cxr 10674 < clt 10675 ≤ cle 10676 − cmin 10870 -cneg 10871 / cdiv 11297 ℤcz 11982 ℝ⁺crp 12390 (,)cioo 12739 (,]cioc 12740 ℜcre 14456 ℑcim 14457 expce 15415 sincsin 15417 cosccos 15418 πcpi 15420 logclog 25138

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-sin 15423 df-cos 15424 df-pi 15426 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 df-log 25140

This theorem is referenced by: efopnlem2 25240

W3C validator