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Theorem logf1o2 26394

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 26309	. . . 4 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
2		f1of1 6831	. . . 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 26386	. . 3 ⊢ 𝐷 ⊆ (ℂ ∖ {0})
6		f1ores 6846	. . 3 ⊢ ((log:(ℂ ∖ {0})–1-1→ran log ∧ 𝐷 ⊆ (ℂ ∖ {0})) → (log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷))
7	3, 5, 6	mp2an 688	. 2 ⊢ (log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷)
8		f1ofun 6834	. . . . . . 7 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → Fun log)
9	1, 8	ax-mp 5	. . . . . 6 ⊢ Fun log
10		f1of 6832	. . . . . . . . 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 6728	. . . . . . 7 ⊢ dom log = (ℂ ∖ {0})
13	5, 12	sseqtrri 4018	. . . . . 6 ⊢ 𝐷 ⊆ dom log
14		funimass4 6955	. . . . . 6 ⊢ ((Fun log ∧ 𝐷 ⊆ dom log) → ((log “ 𝐷) ⊆ (^◡ℑ “ (-π(,)π)) ↔ ∀𝑥 ∈ 𝐷 (log‘𝑥) ∈ (^◡ℑ “ (-π(,)π))))
15	9, 13, 14	mp2an 688	. . . . 5 ⊢ ((log “ 𝐷) ⊆ (^◡ℑ “ (-π(,)π)) ↔ ∀𝑥 ∈ 𝐷 (log‘𝑥) ∈ (^◡ℑ “ (-π(,)π)))
16	4	ellogdm 26383	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺)))
17	16	simplbi 496	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ)
18	4	logdmn0 26384	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0)
19	17, 18	logcld 26315	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (log‘𝑥) ∈ ℂ)
20	19	imcld 15146	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ∈ ℝ)
21	17, 18	logimcld 26316	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → (-π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) ≤ π))
22	21	simpld 493	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → -π < (ℑ‘(log‘𝑥)))
23		pire 26204	. . . . . . . . 9 ⊢ π ∈ ℝ
24	23	a1i 11	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → π ∈ ℝ)
25	21	simprd 494	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ≤ π)
26	4	logdmnrp 26385	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → ¬ -𝑥 ∈ ℝ⁺)
27		lognegb 26334	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (-𝑥 ∈ ℝ⁺ ↔ (ℑ‘(log‘𝑥)) = π))
28	17, 18, 27	syl2anc 582	. . . . . . . . . . 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 231	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ≠ π)
31	30	necomd 2994	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → π ≠ (ℑ‘(log‘𝑥)))
32	20, 24, 25, 31	leneltd 11372	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) < π)
33	23	renegcli 11525	. . . . . . . . 9 ⊢ -π ∈ ℝ
34	33	rexri 11276	. . . . . . . 8 ⊢ -π ∈ ℝ^*
35	23	rexri 11276	. . . . . . . 8 ⊢ π ∈ ℝ^*
36		elioo2 13369	. . . . . . . 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 688	. . . . . . 7 ⊢ ((ℑ‘(log‘𝑥)) ∈ (-π(,)π) ↔ ((ℑ‘(log‘𝑥)) ∈ ℝ ∧ -π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) < π))
38	20, 22, 32, 37	syl3anbrc 1341	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ∈ (-π(,)π))
39		imf 15064	. . . . . . 7 ⊢ ℑ:ℂ⟶ℝ
40		ffn 6716	. . . . . . 7 ⊢ (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
41		elpreima 7058	. . . . . . 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 581	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (log‘𝑥) ∈ (^◡ℑ “ (-π(,)π)))
44	15, 43	mprgbir 3066	. . . 4 ⊢ (log “ 𝐷) ⊆ (^◡ℑ “ (-π(,)π))
45		elpreima 7058	. . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → 𝑥 ∈ ℂ)
48		eliooord 13387	. . . . . . . . . . 11 ⊢ ((ℑ‘𝑥) ∈ (-π(,)π) → (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π))
49	48	adantl 480	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π))
50	49	simpld 493	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → -π < (ℑ‘𝑥))
51	49	simprd 494	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (ℑ‘𝑥) < π)
52		imcl 15062	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (ℑ‘𝑥) ∈ ℝ)
53	52	adantr 479	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (ℑ‘𝑥) ∈ ℝ)
54		ltle 11306	. . . . . . . . . . 11 ⊢ (((ℑ‘𝑥) ∈ ℝ ∧ π ∈ ℝ) → ((ℑ‘𝑥) < π → (ℑ‘𝑥) ≤ π))
55	53, 23, 54	sylancl 584	. . . . . . . . . 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 26304	. . . . . . . . 9 ⊢ (𝑥 ∈ ran log ↔ (𝑥 ∈ ℂ ∧ -π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) ≤ π))
58	47, 50, 56, 57	syl3anbrc 1341	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → 𝑥 ∈ ran log)
59		logef 26326	. . . . . . . 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 16030	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) ∈ ℂ)
62	61	adantr 479	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (exp‘𝑥) ∈ ℂ)
63	53	adantr 479	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) ∈ ℝ)
64	63	recnd 11246	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) ∈ ℂ)
65		picn 26205	. . . . . . . . . . . . . 14 ⊢ π ∈ ℂ
66	65	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → π ∈ ℂ)
67		pipos 26206	. . . . . . . . . . . . . . 15 ⊢ 0 < π
68	23, 67	gt0ne0ii 11754	. . . . . . . . . . . . . 14 ⊢ π ≠ 0
69	68	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → π ≠ 0)
70	51	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) < π)
71	65	mulridi 11222	. . . . . . . . . . . . . . . . . 18 ⊢ (π · 1) = π
72	70, 71	breqtrrdi 5189	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) < (π · 1))
73		1re 11218	. . . . . . . . . . . . . . . . . . 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 12093	. . . . . . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (((ℑ‘𝑥) / π) < 1 ↔ (ℑ‘𝑥) < (π · 1)))
79	72, 78	mpbird 256	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) < 1)
80		1e0p1 12723	. . . . . . . . . . . . . . . 16 ⊢ 1 = (0 + 1)
81	79, 80	breqtrdi 5188	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) < (0 + 1))
82	63	recoscld 16091	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (cos‘(ℑ‘𝑥)) ∈ ℝ)
83	63	resincld 16090	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (sin‘(ℑ‘𝑥)) ∈ ℝ)
84	82, 83	crimd 15183	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))) = (sin‘(ℑ‘𝑥)))
85		efeul 16109	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) = ((exp‘(ℜ‘𝑥)) · ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))))
86	85	ad2antrr 722	. . . . . . . . . . . . . . . . . . . . . 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 7426	. . . . . . . . . . . . . . . . . . . . 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 11246	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (cos‘(ℑ‘𝑥)) ∈ ℂ)
89		ax-icn 11171	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ i ∈ ℂ
90	83	recnd 11246	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (sin‘(ℑ‘𝑥)) ∈ ℂ)
91		mulcl 11196	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((i ∈ ℂ ∧ (sin‘(ℑ‘𝑥)) ∈ ℂ) → (i · (sin‘(ℑ‘𝑥))) ∈ ℂ)
92	89, 90, 91	sylancr 585	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (i · (sin‘(ℑ‘𝑥))) ∈ ℂ)
93	88, 92	addcld 11237	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥)))) ∈ ℂ)
94		recl 15061	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ ℂ → (ℜ‘𝑥) ∈ ℝ)
95	94	ad2antrr 722	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℜ‘𝑥) ∈ ℝ)
96	95	recnd 11246	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℜ‘𝑥) ∈ ℂ)
97		efcl 16030	. . . . . . . . . . . . . . . . . . . . . . 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 16044	. . . . . . . . . . . . . . . . . . . . . . 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 11999	. . . . . . . . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((exp‘𝑥) / (exp‘(ℜ‘𝑥))) = ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥)))))
103		simpr 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (exp‘𝑥) ∈ ℝ)
104	95	reefcld 16035	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (exp‘(ℜ‘𝑥)) ∈ ℝ)
105	103, 104, 100	redivcld 12046	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((exp‘𝑥) / (exp‘(ℜ‘𝑥))) ∈ ℝ)
106	102, 105	eqeltrrd 2832	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥)))) ∈ ℝ)
107	106	reim0d 15176	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘((cos‘(ℑ‘𝑥)) + (i · (sin‘(ℑ‘𝑥))))) = 0)
108	84, 107	eqtr3d 2772	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (sin‘(ℑ‘𝑥)) = 0)
109		sineq0 26269	. . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) ∈ ℤ)
112		0z 12573	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
113		zleltp1 12617	. . . . . . . . . . . . . . . 16 ⊢ ((((ℑ‘𝑥) / π) ∈ ℤ ∧ 0 ∈ ℤ) → (((ℑ‘𝑥) / π) ≤ 0 ↔ ((ℑ‘𝑥) / π) < (0 + 1)))
114	111, 112, 113	sylancl 584	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (((ℑ‘𝑥) / π) ≤ 0 ↔ ((ℑ‘𝑥) / π) < (0 + 1)))
115	81, 114	mpbird 256	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) ≤ 0)
116		df-neg 11451	. . . . . . . . . . . . . . . 16 ⊢ -1 = (0 − 1)
117	65	mulm1i 11663	. . . . . . . . . . . . . . . . . 18 ⊢ (-1 · π) = -π
118	50	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → -π < (ℑ‘𝑥))
119	117, 118	eqbrtrid 5182	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (-1 · π) < (ℑ‘𝑥))
120	73	renegcli 11525	. . . . . . . . . . . . . . . . . . 19 ⊢ -1 ∈ ℝ
121	120	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → -1 ∈ ℝ)
122		ltmuldiv 12091	. . . . . . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((-1 · π) < (ℑ‘𝑥) ↔ -1 < ((ℑ‘𝑥) / π)))
124	119, 123	mpbid 231	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → -1 < ((ℑ‘𝑥) / π))
125	116, 124	eqbrtrrid 5183	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (0 − 1) < ((ℑ‘𝑥) / π))
126		zlem1lt 12618	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℤ ∧ ((ℑ‘𝑥) / π) ∈ ℤ) → (0 ≤ ((ℑ‘𝑥) / π) ↔ (0 − 1) < ((ℑ‘𝑥) / π)))
127	112, 111, 126	sylancr 585	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (0 ≤ ((ℑ‘𝑥) / π) ↔ (0 − 1) < ((ℑ‘𝑥) / π)))
128	125, 127	mpbird 256	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → 0 ≤ ((ℑ‘𝑥) / π))
129	63, 75, 69	redivcld 12046	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) ∈ ℝ)
130		0re 11220	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
131		letri3 11303	. . . . . . . . . . . . . . 15 ⊢ ((((ℑ‘𝑥) / π) ∈ ℝ ∧ 0 ∈ ℝ) → (((ℑ‘𝑥) / π) = 0 ↔ (((ℑ‘𝑥) / π) ≤ 0 ∧ 0 ≤ ((ℑ‘𝑥) / π))))
132	129, 130, 131	sylancl 584	. . . . . . . . . . . . . 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 709	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → ((ℑ‘𝑥) / π) = 0)
134	64, 66, 69, 133	diveq0d 12001	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (ℑ‘𝑥) = 0)
135		reim0b 15070	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (𝑥 ∈ ℝ ↔ (ℑ‘𝑥) = 0))
136	135	ad2antrr 722	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (𝑥 ∈ ℝ ↔ (ℑ‘𝑥) = 0))
137	134, 136	mpbird 256	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → 𝑥 ∈ ℝ)
138	137	rpefcld 16052	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) ∧ (exp‘𝑥) ∈ ℝ) → (exp‘𝑥) ∈ ℝ⁺)
139	138	ex 411	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → ((exp‘𝑥) ∈ ℝ → (exp‘𝑥) ∈ ℝ⁺))
140	4	ellogdm 26383	. . . . . . . . 9 ⊢ ((exp‘𝑥) ∈ 𝐷 ↔ ((exp‘𝑥) ∈ ℂ ∧ ((exp‘𝑥) ∈ ℝ → (exp‘𝑥) ∈ ℝ⁺)))
141	62, 139, 140	sylanbrc 581	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (exp‘𝑥) ∈ 𝐷)
142		funfvima2 7234	. . . . . . . . 9 ⊢ ((Fun log ∧ 𝐷 ⊆ dom log) → ((exp‘𝑥) ∈ 𝐷 → (log‘(exp‘𝑥)) ∈ (log “ 𝐷)))
143	9, 13, 142	mp2an 688	. . . . . . . 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 2832	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → 𝑥 ∈ (log “ 𝐷))
146	46, 145	sylbi 216	. . . . 5 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → 𝑥 ∈ (log “ 𝐷))
147	146	ssriv 3985	. . . 4 ⊢ (^◡ℑ “ (-π(,)π)) ⊆ (log “ 𝐷)
148	44, 147	eqssi 3997	. . 3 ⊢ (log “ 𝐷) = (^◡ℑ “ (-π(,)π))
149		f1oeq3 6822	. . 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 229	1 ⊢ (log ↾ 𝐷):𝐷–1-1-onto→(^◡ℑ “ (-π(,)π))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∀wral 3059 ∖ cdif 3944 ⊆ wss 3947 {csn 4627 class class class wbr 5147 ^◡ccnv 5674 dom cdm 5675 ran crn 5676 ↾ cres 5677 “ cima 5678 Fun wfun 6536 Fn wfn 6537 ⟶wf 6538 –1-1→wf1 6539 –1-1-onto→wf1o 6541 ‘cfv 6542 (class class class)co 7411 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 ici 11114 + caddc 11115 · cmul 11117 -∞cmnf 11250 ℝ^*cxr 11251 < clt 11252 ≤ cle 11253 − cmin 11448 -cneg 11449 / cdiv 11875 ℤcz 12562 ℝ⁺crp 12978 (,)cioo 13328 (,]cioc 13329 ℜcre 15048 ℑcim 15049 expce 16009 sincsin 16011 cosccos 16012 πcpi 16014 logclog 26299

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ioo 13332 df-ioc 13333 df-ico 13334 df-icc 13335 df-fz 13489 df-fzo 13632 df-fl 13761 df-mod 13839 df-seq 13971 df-exp 14032 df-fac 14238 df-bc 14267 df-hash 14295 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-ef 16015 df-sin 16017 df-cos 16018 df-pi 16020 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-prds 17397 df-xrs 17452 df-qtop 17457 df-imas 17458 df-xps 17460 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-mulg 18987 df-cntz 19222 df-cmn 19691 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-fbas 21141 df-fg 21142 df-cnfld 21145 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cld 22743 df-ntr 22744 df-cls 22745 df-nei 22822 df-lp 22860 df-perf 22861 df-cn 22951 df-cnp 22952 df-haus 23039 df-tx 23286 df-hmeo 23479 df-fil 23570 df-fm 23662 df-flim 23663 df-flf 23664 df-xms 24046 df-ms 24047 df-tms 24048 df-cncf 24618 df-limc 25615 df-dv 25616 df-log 26301

This theorem is referenced by: efopnlem2 26401

W3C validator