Theorem eff1olem 26614

Description: The exponential function maps the set 𝑆, of complex numbers with imaginary part in a real interval of length 2 · π, one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.) (Proof shortened by Mario Carneiro, 13-May-2014.)

Hypotheses

Ref	Expression
eff1olem.1	⊢ 𝐹 = (𝑤 ∈ 𝐷 ↦ (exp‘(i · 𝑤)))
eff1olem.2	⊢ 𝑆 = (◡ℑ “ 𝐷)
eff1olem.3	⊢ (𝜑 → 𝐷 ⊆ ℝ)
eff1olem.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (abs‘(𝑥 − 𝑦)) < (2 · π))
eff1olem.5	⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ∃𝑦 ∈ 𝐷 ((𝑧 − 𝑦) / (2 · π)) ∈ ℤ)

Assertion

Ref	Expression
eff1olem	⊢ (𝜑 → (exp ↾ 𝑆):𝑆–1-1-onto→(ℂ ∖ {0}))

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐷   𝑥,𝐹,𝑦,𝑧   𝜑,𝑤,𝑥,𝑦,𝑧   𝑥,𝑆,𝑦

Allowed substitution hints:   𝑆(𝑧,𝑤)   𝐹(𝑤)

Proof of Theorem eff1olem

Step	Hyp	Ref	Expression
1		cnvimass 6072	. . . 4 ⊢ (◡ℑ “ 𝐷) ⊆ dom ℑ
2		eff1olem.2	. . . 4 ⊢ 𝑆 = (◡ℑ “ 𝐷)
3		imf 15141	. . . . . 6 ⊢ ℑ:ℂ⟶ℝ
4	3	fdmi 6704	. . . . 5 ⊢ dom ℑ = ℂ
5	4	eqcomi 2772	. . . 4 ⊢ ℂ = dom ℑ
6	1, 2, 5	3sstr4i 3988	. . 3 ⊢ 𝑆 ⊆ ℂ
7		eff2 16132	. . . . . . 7 ⊢ exp:ℂ⟶(ℂ ∖ {0})
8	7	a1i 11	. . . . . 6 ⊢ (𝑆 ⊆ ℂ → exp:ℂ⟶(ℂ ∖ {0}))
9	8	feqmptd 6936	. . . . 5 ⊢ (𝑆 ⊆ ℂ → exp = (𝑦 ∈ ℂ ↦ (exp‘𝑦)))
10	9	reseq1d 5965	. . . 4 ⊢ (𝑆 ⊆ ℂ → (exp ↾ 𝑆) = ((𝑦 ∈ ℂ ↦ (exp‘𝑦)) ↾ 𝑆))
11		resmpt 6027	. . . 4 ⊢ (𝑆 ⊆ ℂ → ((𝑦 ∈ ℂ ↦ (exp‘𝑦)) ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (exp‘𝑦)))
12	10, 11	eqtrd 2798	. . 3 ⊢ (𝑆 ⊆ ℂ → (exp ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (exp‘𝑦)))
13	6, 12	ax-mp 5	. 2 ⊢ (exp ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (exp‘𝑦))
14	6	sseli 3933	. . . 4 ⊢ (𝑦 ∈ 𝑆 → 𝑦 ∈ ℂ)
15	7	ffvelcdmi 7065	. . . 4 ⊢ (𝑦 ∈ ℂ → (exp‘𝑦) ∈ (ℂ ∖ {0}))
16	14, 15	syl 17	. . 3 ⊢ (𝑦 ∈ 𝑆 → (exp‘𝑦) ∈ (ℂ ∖ {0}))
17	16	adantl 485	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘𝑦) ∈ (ℂ ∖ {0}))
18		eldifsn 4747	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
19	18	bilani 508	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
20	19	simpld 498	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
21	19	simprd 499	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ≠ 0)
22	20, 21	absrpcld 15479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℝ+)
23		reeff1o 26511	. . . . . . . . 9 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ+
24		f1ocnv 6820	. . . . . . . . 9 ⊢ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ → ◡(exp ↾ ℝ):ℝ+–1-1-onto→ℝ)
25		f1of 6807	. . . . . . . . 9 ⊢ (◡(exp ↾ ℝ):ℝ+–1-1-onto→ℝ → ◡(exp ↾ ℝ):ℝ+⟶ℝ)
26	23, 24, 25	mp2b 10	. . . . . . . 8 ⊢ ◡(exp ↾ ℝ):ℝ+⟶ℝ
27	26	ffvelcdmi 7065	. . . . . . 7 ⊢ ((abs‘𝑥) ∈ ℝ+ → (◡(exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ)
28	22, 27	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡(exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ)
29	28	recnd 11211	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡(exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℂ)
30		ax-icn 11133	. . . . . 6 ⊢ i ∈ ℂ
31		eff1olem.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ⊆ ℝ)
32	31	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝐷 ⊆ ℝ)
33		eff1olem.1	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝑤 ∈ 𝐷 ↦ (exp‘(i · 𝑤)))
34		eqid 2763	. . . . . . . . . . . 12 ⊢ (◡abs “ {1}) = (◡abs “ {1})
35		eff1olem.4	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (abs‘(𝑥 − 𝑦)) < (2 · π))
36		eff1olem.5	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ∃𝑦 ∈ 𝐷 ((𝑧 − 𝑦) / (2 · π)) ∈ ℤ)
37		eqid 2763	. . . . . . . . . . . 12 ⊢ (sin ↾ (-(π / 2)[,](π / 2))) = (sin ↾ (-(π / 2)[,](π / 2)))
38	33, 34, 31, 35, 36, 37	efif1olem4 26611	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐷–1-1-onto→(◡abs “ {1}))
39		f1ocnv 6820	. . . . . . . . . . 11 ⊢ (𝐹:𝐷–1-1-onto→(◡abs “ {1}) → ◡𝐹:(◡abs “ {1})–1-1-onto→𝐷)
40		f1of 6807	. . . . . . . . . . 11 ⊢ (◡𝐹:(◡abs “ {1})–1-1-onto→𝐷 → ◡𝐹:(◡abs “ {1})⟶𝐷)
41	38, 39, 40	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ◡𝐹:(◡abs “ {1})⟶𝐷)
42	41	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ◡𝐹:(◡abs “ {1})⟶𝐷)
43	20	abscld 15467	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℝ)
44	43	recnd 11211	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℂ)
45	20, 21	absne0d 15478	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ≠ 0)
46	20, 44, 45	divcld 11968	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥 / (abs‘𝑥)) ∈ ℂ)
47	20, 44, 45	absdivd 15486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘(𝑥 / (abs‘𝑥))) = ((abs‘𝑥) / (abs‘(abs‘𝑥))))
48		absidm 15352	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (abs‘(abs‘𝑥)) = (abs‘𝑥))
49	20, 48	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘(abs‘𝑥)) = (abs‘𝑥))
50	49	oveq2d 7413	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) / (abs‘(abs‘𝑥))) = ((abs‘𝑥) / (abs‘𝑥)))
51	44, 45	dividd 11966	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) / (abs‘𝑥)) = 1)
52	47, 50, 51	3eqtrd 2802	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘(𝑥 / (abs‘𝑥))) = 1)
53		absf 15366	. . . . . . . . . . 11 ⊢ abs:ℂ⟶ℝ
54		ffn 6692	. . . . . . . . . . 11 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
55		fniniseg 7042	. . . . . . . . . . 11 ⊢ (abs Fn ℂ → ((𝑥 / (abs‘𝑥)) ∈ (◡abs “ {1}) ↔ ((𝑥 / (abs‘𝑥)) ∈ ℂ ∧ (abs‘(𝑥 / (abs‘𝑥))) = 1)))
56	53, 54, 55	mp2b 10	. . . . . . . . . 10 ⊢ ((𝑥 / (abs‘𝑥)) ∈ (◡abs “ {1}) ↔ ((𝑥 / (abs‘𝑥)) ∈ ℂ ∧ (abs‘(𝑥 / (abs‘𝑥))) = 1))
57	46, 52, 56	sylanbrc 592	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥 / (abs‘𝑥)) ∈ (◡abs “ {1}))
58	42, 57	ffvelcdmd 7067	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ 𝐷)
59	32, 58	sseldd 3938	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℝ)
60	59	recnd 11211	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℂ)
61		mulcl 11158	. . . . . 6 ⊢ ((i ∈ ℂ ∧ (◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℂ) → (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))) ∈ ℂ)
62	30, 60, 61	sylancr 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))) ∈ ℂ)
63	29, 62	addcld 11202	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ ℂ)
64	28, 59	crimd 15260	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (ℑ‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))) = (◡𝐹‘(𝑥 / (abs‘𝑥))))
65	64, 58	eqeltrd 2863	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (ℑ‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))) ∈ 𝐷)
66		ffn 6692	. . . . 5 ⊢ (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
67		elpreima 7040	. . . . 5 ⊢ (ℑ Fn ℂ → (((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ (◡ℑ “ 𝐷) ↔ (((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ ℂ ∧ (ℑ‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))) ∈ 𝐷)))
68	3, 66, 67	mp2b 10	. . . 4 ⊢ (((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ (◡ℑ “ 𝐷) ↔ (((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ ℂ ∧ (ℑ‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))) ∈ 𝐷))
69	63, 65, 68	sylanbrc 592	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ (◡ℑ “ 𝐷))
70	69, 2	eleqtrrdi 2874	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ 𝑆)
71		efadd 16125	. . . . . . 7 ⊢ (((◡(exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℂ ∧ (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))) ∈ ℂ) → (exp‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))) = ((exp‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) · (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
72	29, 62, 71	syl2anc 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (exp‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))) = ((exp‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) · (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
73	28	fvresd 6888	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) = (exp‘(◡(exp ↾ ℝ)‘(abs‘𝑥))))
74		f1ocnvfv2 7262	. . . . . . . . 9 ⊢ (((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ ∧ (abs‘𝑥) ∈ ℝ+) → ((exp ↾ ℝ)‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
75	23, 22, 74	sylancr 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
76	73, 75	eqtr3d 2800	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (exp‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
77		oveq2 7405	. . . . . . . . . . 11 ⊢ (𝑧 = (◡𝐹‘(𝑥 / (abs‘𝑥))) → (i · 𝑧) = (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))
78	77	fveq2d 6872	. . . . . . . . . 10 ⊢ (𝑧 = (◡𝐹‘(𝑥 / (abs‘𝑥))) → (exp‘(i · 𝑧)) = (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))))
79		oveq2 7405	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → (i · 𝑤) = (i · 𝑧))
80	79	fveq2d 6872	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (exp‘(i · 𝑤)) = (exp‘(i · 𝑧)))
81	80	cbvmptv 5205	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝐷 ↦ (exp‘(i · 𝑤))) = (𝑧 ∈ 𝐷 ↦ (exp‘(i · 𝑧)))
82	33, 81	eqtri 2786	. . . . . . . . . 10 ⊢ 𝐹 = (𝑧 ∈ 𝐷 ↦ (exp‘(i · 𝑧)))
83		fvex 6881	. . . . . . . . . 10 ⊢ (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ V
84	78, 82, 83	fvmpt 6976	. . . . . . . . 9 ⊢ ((◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ 𝐷 → (𝐹‘(◡𝐹‘(𝑥 / (abs‘𝑥)))) = (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))))
85	58, 84	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(◡𝐹‘(𝑥 / (abs‘𝑥)))) = (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))))
86	38	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝐹:𝐷–1-1-onto→(◡abs “ {1}))
87		f1ocnvfv2 7262	. . . . . . . . 9 ⊢ ((𝐹:𝐷–1-1-onto→(◡abs “ {1}) ∧ (𝑥 / (abs‘𝑥)) ∈ (◡abs “ {1})) → (𝐹‘(◡𝐹‘(𝑥 / (abs‘𝑥)))) = (𝑥 / (abs‘𝑥)))
88	86, 57, 87	syl2anc 593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(◡𝐹‘(𝑥 / (abs‘𝑥)))) = (𝑥 / (abs‘𝑥)))
89	85, 88	eqtr3d 2800	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) = (𝑥 / (abs‘𝑥)))
90	76, 89	oveq12d 7415	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((exp‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) · (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))) = ((abs‘𝑥) · (𝑥 / (abs‘𝑥))))
91	20, 44, 45	divcan2d 11970	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) · (𝑥 / (abs‘𝑥))) = 𝑥)
92	72, 90, 91	3eqtrrd 2803	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 = (exp‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
93	92	adantrl 726	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ (ℂ ∖ {0}))) → 𝑥 = (exp‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
94		fveq2 6868	. . . . 5 ⊢ (𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) → (exp‘𝑦) = (exp‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
95	94	eqeq2d 2774	. . . 4 ⊢ (𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) → (𝑥 = (exp‘𝑦) ↔ 𝑥 = (exp‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))))))
96	93, 95	syl5ibrcom 249	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ (ℂ ∖ {0}))) → (𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) → 𝑥 = (exp‘𝑦)))
97	14	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ℂ)
98	97	replimd 15225	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 = ((ℜ‘𝑦) + (i · (ℑ‘𝑦))))
99		absef 16230	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ → (abs‘(exp‘𝑦)) = (exp‘(ℜ‘𝑦)))
100	97, 99	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘(exp‘𝑦)) = (exp‘(ℜ‘𝑦)))
101	97	recld 15222	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℜ‘𝑦) ∈ ℝ)
102	101	fvresd 6888	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((exp ↾ ℝ)‘(ℜ‘𝑦)) = (exp‘(ℜ‘𝑦)))
103	100, 102	eqtr4d 2801	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘(exp‘𝑦)) = ((exp ↾ ℝ)‘(ℜ‘𝑦)))
104	103	fveq2d 6872	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) = (◡(exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))))
105		f1ocnvfv1 7261	. . . . . . . . 9 ⊢ (((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ ∧ (ℜ‘𝑦) ∈ ℝ) → (◡(exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))) = (ℜ‘𝑦))
106	23, 101, 105	sylancr 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡(exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))) = (ℜ‘𝑦))
107	104, 106	eqtrd 2798	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) = (ℜ‘𝑦))
108	97	imcld 15223	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℑ‘𝑦) ∈ ℝ)
109	108	recnd 11211	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℑ‘𝑦) ∈ ℂ)
110		mulcl 11158	. . . . . . . . . . . . . 14 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝑦) ∈ ℂ) → (i · (ℑ‘𝑦)) ∈ ℂ)
111	30, 109, 110	sylancr 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (i · (ℑ‘𝑦)) ∈ ℂ)
112		efcl 16113	. . . . . . . . . . . . 13 ⊢ ((i · (ℑ‘𝑦)) ∈ ℂ → (exp‘(i · (ℑ‘𝑦))) ∈ ℂ)
113	111, 112	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘(i · (ℑ‘𝑦))) ∈ ℂ)
114	101	recnd 11211	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℜ‘𝑦) ∈ ℂ)
115		efcl 16113	. . . . . . . . . . . . 13 ⊢ ((ℜ‘𝑦) ∈ ℂ → (exp‘(ℜ‘𝑦)) ∈ ℂ)
116	114, 115	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘(ℜ‘𝑦)) ∈ ℂ)
117		efne0 16129	. . . . . . . . . . . . 13 ⊢ ((ℜ‘𝑦) ∈ ℂ → (exp‘(ℜ‘𝑦)) ≠ 0)
118	114, 117	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘(ℜ‘𝑦)) ≠ 0)
119	113, 116, 118	divcan3d 11973	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))) / (exp‘(ℜ‘𝑦))) = (exp‘(i · (ℑ‘𝑦))))
120	98	fveq2d 6872	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘𝑦) = (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))))
121		efadd 16125	. . . . . . . . . . . . . 14 ⊢ (((ℜ‘𝑦) ∈ ℂ ∧ (i · (ℑ‘𝑦)) ∈ ℂ) → (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
122	114, 111, 121	syl2anc 593	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
123	120, 122	eqtrd 2798	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘𝑦) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
124	123, 100	oveq12d 7415	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((exp‘𝑦) / (abs‘(exp‘𝑦))) = (((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))) / (exp‘(ℜ‘𝑦))))
125		elpreima 7040	. . . . . . . . . . . . . . . 16 ⊢ (ℑ Fn ℂ → (𝑦 ∈ (◡ℑ “ 𝐷) ↔ (𝑦 ∈ ℂ ∧ (ℑ‘𝑦) ∈ 𝐷)))
126	3, 66, 125	mp2b 10	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (◡ℑ “ 𝐷) ↔ (𝑦 ∈ ℂ ∧ (ℑ‘𝑦) ∈ 𝐷))
127	126	simprbi 501	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (◡ℑ “ 𝐷) → (ℑ‘𝑦) ∈ 𝐷)
128	127, 2	eleq2s 2881	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑆 → (ℑ‘𝑦) ∈ 𝐷)
129	128	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℑ‘𝑦) ∈ 𝐷)
130		oveq2 7405	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (ℑ‘𝑦) → (i · 𝑤) = (i · (ℑ‘𝑦)))
131	130	fveq2d 6872	. . . . . . . . . . . . 13 ⊢ (𝑤 = (ℑ‘𝑦) → (exp‘(i · 𝑤)) = (exp‘(i · (ℑ‘𝑦))))
132		fvex 6881	. . . . . . . . . . . . 13 ⊢ (exp‘(i · (ℑ‘𝑦))) ∈ V
133	131, 33, 132	fvmpt 6976	. . . . . . . . . . . 12 ⊢ ((ℑ‘𝑦) ∈ 𝐷 → (𝐹‘(ℑ‘𝑦)) = (exp‘(i · (ℑ‘𝑦))))
134	129, 133	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝐹‘(ℑ‘𝑦)) = (exp‘(i · (ℑ‘𝑦))))
135	119, 124, 134	3eqtr4d 2808	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((exp‘𝑦) / (abs‘(exp‘𝑦))) = (𝐹‘(ℑ‘𝑦)))
136	135	fveq2d 6872	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))) = (◡𝐹‘(𝐹‘(ℑ‘𝑦))))
137		f1ocnvfv1 7261	. . . . . . . . . 10 ⊢ ((𝐹:𝐷–1-1-onto→(◡abs “ {1}) ∧ (ℑ‘𝑦) ∈ 𝐷) → (◡𝐹‘(𝐹‘(ℑ‘𝑦))) = (ℑ‘𝑦))
138	38, 128, 137	syl2an 605	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡𝐹‘(𝐹‘(ℑ‘𝑦))) = (ℑ‘𝑦))
139	136, 138	eqtrd 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))) = (ℑ‘𝑦))
140	139	oveq2d 7413	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦))))) = (i · (ℑ‘𝑦)))
141	107, 140	oveq12d 7415	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))) = ((ℜ‘𝑦) + (i · (ℑ‘𝑦))))
142	98, 141	eqtr4d 2801	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 = ((◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))))
143		fveq2 6868	. . . . . . . 8 ⊢ (𝑥 = (exp‘𝑦) → (abs‘𝑥) = (abs‘(exp‘𝑦)))
144	143	fveq2d 6872	. . . . . . 7 ⊢ (𝑥 = (exp‘𝑦) → (◡(exp ↾ ℝ)‘(abs‘𝑥)) = (◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))))
145		id 22	. . . . . . . . . 10 ⊢ (𝑥 = (exp‘𝑦) → 𝑥 = (exp‘𝑦))
146	145, 143	oveq12d 7415	. . . . . . . . 9 ⊢ (𝑥 = (exp‘𝑦) → (𝑥 / (abs‘𝑥)) = ((exp‘𝑦) / (abs‘(exp‘𝑦))))
147	146	fveq2d 6872	. . . . . . . 8 ⊢ (𝑥 = (exp‘𝑦) → (◡𝐹‘(𝑥 / (abs‘𝑥))) = (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))
148	147	oveq2d 7413	. . . . . . 7 ⊢ (𝑥 = (exp‘𝑦) → (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))) = (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦))))))
149	144, 148	oveq12d 7415	. . . . . 6 ⊢ (𝑥 = (exp‘𝑦) → ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) = ((◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))))
150	149	eqeq2d 2774	. . . . 5 ⊢ (𝑥 = (exp‘𝑦) → (𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ↔ 𝑦 = ((◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦))))))))
151	142, 150	syl5ibrcom 249	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝑥 = (exp‘𝑦) → 𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
152	151	adantrr 727	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ (ℂ ∖ {0}))) → (𝑥 = (exp‘𝑦) → 𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
153	96, 152	impbid 214	. 2 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ (ℂ ∖ {0}))) → (𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ↔ 𝑥 = (exp‘𝑦)))
154	13, 17, 70, 153	f1o2d 7651	1 ⊢ (𝜑 → (exp ↾ 𝑆):𝑆–1-1-onto→(ℂ ∖ {0}))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1561   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3087   ∖ cdif 3902   ⊆ wss 3905  {csn 4583   class class class wbr 5101   ↦ cmpt 5182  ^◡ccnv 5647  dom cdm 5648   ↾ cres 5650   “ cima 5651   Fn wfn 6517  ⟶wf 6518  –1-1-onto→wf1o 6521  ‘cfv 6522  (class class class)co 7397  ℂcc 11072  ℝcr 11073  0cc0 11074  1c1 11075  ici 11076   + caddc 11077   · cmul 11079   < clt 11217   − cmin 11415  -cneg 11416   / cdiv 11845  2c2 12273  ℤcz 12569  ℝ⁺crp 12994  [,]cicc 13353  ℜcre 15125  ℑcim 15126  abscabs 15262  expce 16092  sincsin 16094  πcpi 16097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719  ax-inf2 9597  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151  ax-pre-sup 11152  ax-addf 11153

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-se 5602  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-isom 6531  df-riota 7354  df-ov 7400  df-oprab 7401  df-mpo 7402  df-of 7661  df-om 7848  df-1st 7971  df-2nd 7972  df-supp 8142  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-1o 8438  df-2o 8439  df-er 8679  df-map 8811  df-pm 8812  df-ixp 8881  df-en 8929  df-dom 8930  df-sdom 8931  df-fin 8932  df-fsupp 9309  df-fi 9358  df-sup 9389  df-inf 9390  df-oi 9459  df-card 9898  df-pnf 11219  df-mnf 11220  df-xr 11221  df-ltxr 11222  df-le 11223  df-sub 11417  df-neg 11418  df-div 11846  df-nn 12212  df-2 12281  df-3 12282  df-4 12283  df-5 12284  df-6 12285  df-7 12286  df-8 12287  df-9 12288  df-n0 12483  df-z 12570  df-dec 12690  df-uz 12841  df-q 12951  df-rp 12995  df-xneg 13115  df-xadd 13116  df-xmul 13117  df-ioo 13354  df-ioc 13355  df-ico 13356  df-icc 13357  df-fz 13514  df-fzo 13661  df-fl 13803  df-mod 13881  df-seq 14016  df-exp 14076  df-fac 14288  df-bc 14317  df-hash 14345  df-shft 15081  df-cj 15127  df-re 15128  df-im 15129  df-sqrt 15263  df-abs 15264  df-limsup 15499  df-clim 15516  df-rlim 15517  df-sum 15715  df-ef 16098  df-sin 16100  df-cos 16101  df-pi 16103  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17247  df-ress 17268  df-plusg 17300  df-mulr 17301  df-starv 17302  df-sca 17303  df-vsca 17304  df-ip 17305  df-tset 17306  df-ple 17307  df-ds 17309  df-unif 17310  df-hom 17311  df-cco 17312  df-rest 17452  df-topn 17453  df-0g 17471  df-gsum 17472  df-topgen 17473  df-pt 17474  df-prds 17477  df-xrs 17533  df-qtop 17538  df-imas 17539  df-xps 17541  df-mre 17615  df-mrc 17616  df-acs 17618  df-mgm 18675  df-sgrp 18754  df-mnd 18770  df-submnd 18819  df-mulg 19111  df-cntz 19358  df-cmn 19823  df-psmet 21417  df-xmet 21418  df-met 21419  df-bl 21420  df-mopn 21421  df-fbas 21422  df-fg 21423  df-cnfld 21426  df-top 22955  df-topon 22972  df-topsp 22994  df-bases 23007  df-cld 23080  df-ntr 23081  df-cls 23082  df-nei 23159  df-lp 23197  df-perf 23198  df-cn 23288  df-cnp 23289  df-haus 23376  df-tx 23623  df-hmeo 23816  df-fil 23907  df-fm 23999  df-flim 24000  df-flf 24001  df-xms 24381  df-ms 24382  df-tms 24383  df-cncf 24941  df-limc 25929  df-dv 25930

This theorem is referenced by:  eff1o  26615