MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eff1olem Structured version   Visualization version   GIF version

Theorem eff1olem 24339
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 = (𝑤𝐷 ↦ (exp‘(i · 𝑤)))
eff1olem.2 𝑆 = (ℑ “ 𝐷)
eff1olem.3 (𝜑𝐷 ⊆ ℝ)
eff1olem.4 ((𝜑 ∧ (𝑥𝐷𝑦𝐷)) → (abs‘(𝑥𝑦)) < (2 · π))
eff1olem.5 ((𝜑𝑧 ∈ ℝ) → ∃𝑦𝐷 ((𝑧𝑦) / (2 · π)) ∈ ℤ)
Assertion
Ref Expression
eff1olem (𝜑 → (exp ↾ 𝑆):𝑆1-1-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
StepHypRef Expression
1 cnvimass 5520 . . . 4 (ℑ “ 𝐷) ⊆ dom ℑ
2 eff1olem.2 . . . 4 𝑆 = (ℑ “ 𝐷)
3 imf 13897 . . . . . 6 ℑ:ℂ⟶ℝ
43fdmi 6090 . . . . 5 dom ℑ = ℂ
54eqcomi 2660 . . . 4 ℂ = dom ℑ
61, 2, 53sstr4i 3677 . . 3 𝑆 ⊆ ℂ
7 eff2 14873 . . . . . . 7 exp:ℂ⟶(ℂ ∖ {0})
87a1i 11 . . . . . 6 (𝑆 ⊆ ℂ → exp:ℂ⟶(ℂ ∖ {0}))
98feqmptd 6288 . . . . 5 (𝑆 ⊆ ℂ → exp = (𝑦 ∈ ℂ ↦ (exp‘𝑦)))
109reseq1d 5427 . . . 4 (𝑆 ⊆ ℂ → (exp ↾ 𝑆) = ((𝑦 ∈ ℂ ↦ (exp‘𝑦)) ↾ 𝑆))
11 resmpt 5484 . . . 4 (𝑆 ⊆ ℂ → ((𝑦 ∈ ℂ ↦ (exp‘𝑦)) ↾ 𝑆) = (𝑦𝑆 ↦ (exp‘𝑦)))
1210, 11eqtrd 2685 . . 3 (𝑆 ⊆ ℂ → (exp ↾ 𝑆) = (𝑦𝑆 ↦ (exp‘𝑦)))
136, 12ax-mp 5 . 2 (exp ↾ 𝑆) = (𝑦𝑆 ↦ (exp‘𝑦))
146sseli 3632 . . . 4 (𝑦𝑆𝑦 ∈ ℂ)
157ffvelrni 6398 . . . 4 (𝑦 ∈ ℂ → (exp‘𝑦) ∈ (ℂ ∖ {0}))
1614, 15syl 17 . . 3 (𝑦𝑆 → (exp‘𝑦) ∈ (ℂ ∖ {0}))
1716adantl 481 . 2 ((𝜑𝑦𝑆) → (exp‘𝑦) ∈ (ℂ ∖ {0}))
18 simpr 476 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ∈ (ℂ ∖ {0}))
19 eldifsn 4350 . . . . . . . . . 10 (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
2018, 19sylib 208 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
2120simpld 474 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
2220simprd 478 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ≠ 0)
2321, 22absrpcld 14231 . . . . . . 7 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℝ+)
24 reeff1o 24246 . . . . . . . . 9 (exp ↾ ℝ):ℝ–1-1-onto→ℝ+
25 f1ocnv 6187 . . . . . . . . 9 ((exp ↾ ℝ):ℝ–1-1-onto→ℝ+(exp ↾ ℝ):ℝ+1-1-onto→ℝ)
26 f1of 6175 . . . . . . . . 9 ((exp ↾ ℝ):ℝ+1-1-onto→ℝ → (exp ↾ ℝ):ℝ+⟶ℝ)
2724, 25, 26mp2b 10 . . . . . . . 8 (exp ↾ ℝ):ℝ+⟶ℝ
2827ffvelrni 6398 . . . . . . 7 ((abs‘𝑥) ∈ ℝ+ → ((exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ)
2923, 28syl 17 . . . . . 6 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ)
3029recnd 10106 . . . . 5 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℂ)
31 ax-icn 10033 . . . . . 6 i ∈ ℂ
32 eff1olem.3 . . . . . . . . 9 (𝜑𝐷 ⊆ ℝ)
3332adantr 480 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝐷 ⊆ ℝ)
34 eff1olem.1 . . . . . . . . . . . 12 𝐹 = (𝑤𝐷 ↦ (exp‘(i · 𝑤)))
35 eqid 2651 . . . . . . . . . . . 12 (abs “ {1}) = (abs “ {1})
36 eff1olem.4 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐷𝑦𝐷)) → (abs‘(𝑥𝑦)) < (2 · π))
37 eff1olem.5 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℝ) → ∃𝑦𝐷 ((𝑧𝑦) / (2 · π)) ∈ ℤ)
38 eqid 2651 . . . . . . . . . . . 12 (sin ↾ (-(π / 2)[,](π / 2))) = (sin ↾ (-(π / 2)[,](π / 2)))
3934, 35, 32, 36, 37, 38efif1olem4 24336 . . . . . . . . . . 11 (𝜑𝐹:𝐷1-1-onto→(abs “ {1}))
40 f1ocnv 6187 . . . . . . . . . . 11 (𝐹:𝐷1-1-onto→(abs “ {1}) → 𝐹:(abs “ {1})–1-1-onto𝐷)
41 f1of 6175 . . . . . . . . . . 11 (𝐹:(abs “ {1})–1-1-onto𝐷𝐹:(abs “ {1})⟶𝐷)
4239, 40, 413syl 18 . . . . . . . . . 10 (𝜑𝐹:(abs “ {1})⟶𝐷)
4342adantr 480 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝐹:(abs “ {1})⟶𝐷)
4421abscld 14219 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℝ)
4544recnd 10106 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℂ)
4621, 22absne0d 14230 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ≠ 0)
4721, 45, 46divcld 10839 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝑥 / (abs‘𝑥)) ∈ ℂ)
4821, 45, 46absdivd 14238 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘(𝑥 / (abs‘𝑥))) = ((abs‘𝑥) / (abs‘(abs‘𝑥))))
49 absidm 14107 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (abs‘(abs‘𝑥)) = (abs‘𝑥))
5021, 49syl 17 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘(abs‘𝑥)) = (abs‘𝑥))
5150oveq2d 6706 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) / (abs‘(abs‘𝑥))) = ((abs‘𝑥) / (abs‘𝑥)))
5245, 46dividd 10837 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) / (abs‘𝑥)) = 1)
5348, 51, 523eqtrd 2689 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘(𝑥 / (abs‘𝑥))) = 1)
54 absf 14121 . . . . . . . . . . 11 abs:ℂ⟶ℝ
55 ffn 6083 . . . . . . . . . . 11 (abs:ℂ⟶ℝ → abs Fn ℂ)
56 fniniseg 6378 . . . . . . . . . . 11 (abs Fn ℂ → ((𝑥 / (abs‘𝑥)) ∈ (abs “ {1}) ↔ ((𝑥 / (abs‘𝑥)) ∈ ℂ ∧ (abs‘(𝑥 / (abs‘𝑥))) = 1)))
5754, 55, 56mp2b 10 . . . . . . . . . 10 ((𝑥 / (abs‘𝑥)) ∈ (abs “ {1}) ↔ ((𝑥 / (abs‘𝑥)) ∈ ℂ ∧ (abs‘(𝑥 / (abs‘𝑥))) = 1))
5847, 53, 57sylanbrc 699 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝑥 / (abs‘𝑥)) ∈ (abs “ {1}))
5943, 58ffvelrnd 6400 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(𝑥 / (abs‘𝑥))) ∈ 𝐷)
6033, 59sseldd 3637 . . . . . . 7 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℝ)
6160recnd 10106 . . . . . 6 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℂ)
62 mulcl 10058 . . . . . 6 ((i ∈ ℂ ∧ (𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℂ) → (i · (𝐹‘(𝑥 / (abs‘𝑥)))) ∈ ℂ)
6331, 61, 62sylancr 696 . . . . 5 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (i · (𝐹‘(𝑥 / (abs‘𝑥)))) ∈ ℂ)
6430, 63addcld 10097 . . . 4 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) ∈ ℂ)
6529, 60crimd 14016 . . . . 5 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (ℑ‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))) = (𝐹‘(𝑥 / (abs‘𝑥))))
6665, 59eqeltrd 2730 . . . 4 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (ℑ‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))) ∈ 𝐷)
67 ffn 6083 . . . . 5 (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
68 elpreima 6377 . . . . 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)))
693, 67, 68mp2b 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))
7064, 66, 69sylanbrc 699 . . 3 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) ∈ (ℑ “ 𝐷))
7170, 2syl6eleqr 2741 . 2 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) ∈ 𝑆)
72 efadd 14868 . . . . . . 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)))))))
7330, 63, 72syl2anc 694 . . . . . 6 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (exp‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))) = ((exp‘((exp ↾ ℝ)‘(abs‘𝑥))) · (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
74 fvres 6245 . . . . . . . . 9 (((exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ → ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(abs‘𝑥))) = (exp‘((exp ↾ ℝ)‘(abs‘𝑥))))
7529, 74syl 17 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(abs‘𝑥))) = (exp‘((exp ↾ ℝ)‘(abs‘𝑥))))
76 f1ocnvfv2 6573 . . . . . . . . 9 (((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ ∧ (abs‘𝑥) ∈ ℝ+) → ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
7724, 23, 76sylancr 696 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
7875, 77eqtr3d 2687 . . . . . . 7 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (exp‘((exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
79 oveq2 6698 . . . . . . . . . . 11 (𝑧 = (𝐹‘(𝑥 / (abs‘𝑥))) → (i · 𝑧) = (i · (𝐹‘(𝑥 / (abs‘𝑥)))))
8079fveq2d 6233 . . . . . . . . . 10 (𝑧 = (𝐹‘(𝑥 / (abs‘𝑥))) → (exp‘(i · 𝑧)) = (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥))))))
81 oveq2 6698 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (i · 𝑤) = (i · 𝑧))
8281fveq2d 6233 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (exp‘(i · 𝑤)) = (exp‘(i · 𝑧)))
8382cbvmptv 4783 . . . . . . . . . . 11 (𝑤𝐷 ↦ (exp‘(i · 𝑤))) = (𝑧𝐷 ↦ (exp‘(i · 𝑧)))
8434, 83eqtri 2673 . . . . . . . . . 10 𝐹 = (𝑧𝐷 ↦ (exp‘(i · 𝑧)))
85 fvex 6239 . . . . . . . . . 10 (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥))))) ∈ V
8680, 84, 85fvmpt 6321 . . . . . . . . 9 ((𝐹‘(𝑥 / (abs‘𝑥))) ∈ 𝐷 → (𝐹‘(𝐹‘(𝑥 / (abs‘𝑥)))) = (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥))))))
8759, 86syl 17 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(𝐹‘(𝑥 / (abs‘𝑥)))) = (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥))))))
8839adantr 480 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝐹:𝐷1-1-onto→(abs “ {1}))
89 f1ocnvfv2 6573 . . . . . . . . 9 ((𝐹:𝐷1-1-onto→(abs “ {1}) ∧ (𝑥 / (abs‘𝑥)) ∈ (abs “ {1})) → (𝐹‘(𝐹‘(𝑥 / (abs‘𝑥)))) = (𝑥 / (abs‘𝑥)))
9088, 58, 89syl2anc 694 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(𝐹‘(𝑥 / (abs‘𝑥)))) = (𝑥 / (abs‘𝑥)))
9187, 90eqtr3d 2687 . . . . . . 7 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥))))) = (𝑥 / (abs‘𝑥)))
9278, 91oveq12d 6708 . . . . . 6 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((exp‘((exp ↾ ℝ)‘(abs‘𝑥))) · (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥)))))) = ((abs‘𝑥) · (𝑥 / (abs‘𝑥))))
9321, 45, 46divcan2d 10841 . . . . . 6 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) · (𝑥 / (abs‘𝑥))) = 𝑥)
9473, 92, 933eqtrrd 2690 . . . . 5 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝑥 = (exp‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
9594adantrl 752 . . . 4 ((𝜑 ∧ (𝑦𝑆𝑥 ∈ (ℂ ∖ {0}))) → 𝑥 = (exp‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
96 fveq2 6229 . . . . 5 (𝑦 = (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) → (exp‘𝑦) = (exp‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
9796eqeq2d 2661 . . . 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))))))))
9895, 97syl5ibrcom 237 . . 3 ((𝜑 ∧ (𝑦𝑆𝑥 ∈ (ℂ ∖ {0}))) → (𝑦 = (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) → 𝑥 = (exp‘𝑦)))
9914adantl 481 . . . . . . 7 ((𝜑𝑦𝑆) → 𝑦 ∈ ℂ)
10099replimd 13981 . . . . . 6 ((𝜑𝑦𝑆) → 𝑦 = ((ℜ‘𝑦) + (i · (ℑ‘𝑦))))
101 absef 14971 . . . . . . . . . . 11 (𝑦 ∈ ℂ → (abs‘(exp‘𝑦)) = (exp‘(ℜ‘𝑦)))
10299, 101syl 17 . . . . . . . . . 10 ((𝜑𝑦𝑆) → (abs‘(exp‘𝑦)) = (exp‘(ℜ‘𝑦)))
10399recld 13978 . . . . . . . . . . 11 ((𝜑𝑦𝑆) → (ℜ‘𝑦) ∈ ℝ)
104 fvres 6245 . . . . . . . . . . 11 ((ℜ‘𝑦) ∈ ℝ → ((exp ↾ ℝ)‘(ℜ‘𝑦)) = (exp‘(ℜ‘𝑦)))
105103, 104syl 17 . . . . . . . . . 10 ((𝜑𝑦𝑆) → ((exp ↾ ℝ)‘(ℜ‘𝑦)) = (exp‘(ℜ‘𝑦)))
106102, 105eqtr4d 2688 . . . . . . . . 9 ((𝜑𝑦𝑆) → (abs‘(exp‘𝑦)) = ((exp ↾ ℝ)‘(ℜ‘𝑦)))
107106fveq2d 6233 . . . . . . . 8 ((𝜑𝑦𝑆) → ((exp ↾ ℝ)‘(abs‘(exp‘𝑦))) = ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))))
108 f1ocnvfv1 6572 . . . . . . . . 9 (((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ ∧ (ℜ‘𝑦) ∈ ℝ) → ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))) = (ℜ‘𝑦))
10924, 103, 108sylancr 696 . . . . . . . 8 ((𝜑𝑦𝑆) → ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))) = (ℜ‘𝑦))
110107, 109eqtrd 2685 . . . . . . 7 ((𝜑𝑦𝑆) → ((exp ↾ ℝ)‘(abs‘(exp‘𝑦))) = (ℜ‘𝑦))
11199imcld 13979 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝑆) → (ℑ‘𝑦) ∈ ℝ)
112111recnd 10106 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑆) → (ℑ‘𝑦) ∈ ℂ)
113 mulcl 10058 . . . . . . . . . . . . . 14 ((i ∈ ℂ ∧ (ℑ‘𝑦) ∈ ℂ) → (i · (ℑ‘𝑦)) ∈ ℂ)
11431, 112, 113sylancr 696 . . . . . . . . . . . . 13 ((𝜑𝑦𝑆) → (i · (ℑ‘𝑦)) ∈ ℂ)
115 efcl 14857 . . . . . . . . . . . . 13 ((i · (ℑ‘𝑦)) ∈ ℂ → (exp‘(i · (ℑ‘𝑦))) ∈ ℂ)
116114, 115syl 17 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (exp‘(i · (ℑ‘𝑦))) ∈ ℂ)
117103recnd 10106 . . . . . . . . . . . . 13 ((𝜑𝑦𝑆) → (ℜ‘𝑦) ∈ ℂ)
118 efcl 14857 . . . . . . . . . . . . 13 ((ℜ‘𝑦) ∈ ℂ → (exp‘(ℜ‘𝑦)) ∈ ℂ)
119117, 118syl 17 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (exp‘(ℜ‘𝑦)) ∈ ℂ)
120 efne0 14871 . . . . . . . . . . . . 13 ((ℜ‘𝑦) ∈ ℂ → (exp‘(ℜ‘𝑦)) ≠ 0)
121117, 120syl 17 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (exp‘(ℜ‘𝑦)) ≠ 0)
122116, 119, 121divcan3d 10844 . . . . . . . . . . 11 ((𝜑𝑦𝑆) → (((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))) / (exp‘(ℜ‘𝑦))) = (exp‘(i · (ℑ‘𝑦))))
123100fveq2d 6233 . . . . . . . . . . . . 13 ((𝜑𝑦𝑆) → (exp‘𝑦) = (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))))
124 efadd 14868 . . . . . . . . . . . . . 14 (((ℜ‘𝑦) ∈ ℂ ∧ (i · (ℑ‘𝑦)) ∈ ℂ) → (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
125117, 114, 124syl2anc 694 . . . . . . . . . . . . 13 ((𝜑𝑦𝑆) → (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
126123, 125eqtrd 2685 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (exp‘𝑦) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
127126, 102oveq12d 6708 . . . . . . . . . . 11 ((𝜑𝑦𝑆) → ((exp‘𝑦) / (abs‘(exp‘𝑦))) = (((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))) / (exp‘(ℜ‘𝑦))))
128 elpreima 6377 . . . . . . . . . . . . . . . 16 (ℑ Fn ℂ → (𝑦 ∈ (ℑ “ 𝐷) ↔ (𝑦 ∈ ℂ ∧ (ℑ‘𝑦) ∈ 𝐷)))
1293, 67, 128mp2b 10 . . . . . . . . . . . . . . 15 (𝑦 ∈ (ℑ “ 𝐷) ↔ (𝑦 ∈ ℂ ∧ (ℑ‘𝑦) ∈ 𝐷))
130129simprbi 479 . . . . . . . . . . . . . 14 (𝑦 ∈ (ℑ “ 𝐷) → (ℑ‘𝑦) ∈ 𝐷)
131130, 2eleq2s 2748 . . . . . . . . . . . . 13 (𝑦𝑆 → (ℑ‘𝑦) ∈ 𝐷)
132131adantl 481 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (ℑ‘𝑦) ∈ 𝐷)
133 oveq2 6698 . . . . . . . . . . . . . 14 (𝑤 = (ℑ‘𝑦) → (i · 𝑤) = (i · (ℑ‘𝑦)))
134133fveq2d 6233 . . . . . . . . . . . . 13 (𝑤 = (ℑ‘𝑦) → (exp‘(i · 𝑤)) = (exp‘(i · (ℑ‘𝑦))))
135 fvex 6239 . . . . . . . . . . . . 13 (exp‘(i · (ℑ‘𝑦))) ∈ V
136134, 34, 135fvmpt 6321 . . . . . . . . . . . 12 ((ℑ‘𝑦) ∈ 𝐷 → (𝐹‘(ℑ‘𝑦)) = (exp‘(i · (ℑ‘𝑦))))
137132, 136syl 17 . . . . . . . . . . 11 ((𝜑𝑦𝑆) → (𝐹‘(ℑ‘𝑦)) = (exp‘(i · (ℑ‘𝑦))))
138122, 127, 1373eqtr4d 2695 . . . . . . . . . 10 ((𝜑𝑦𝑆) → ((exp‘𝑦) / (abs‘(exp‘𝑦))) = (𝐹‘(ℑ‘𝑦)))
139138fveq2d 6233 . . . . . . . . 9 ((𝜑𝑦𝑆) → (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))) = (𝐹‘(𝐹‘(ℑ‘𝑦))))
140 f1ocnvfv1 6572 . . . . . . . . . 10 ((𝐹:𝐷1-1-onto→(abs “ {1}) ∧ (ℑ‘𝑦) ∈ 𝐷) → (𝐹‘(𝐹‘(ℑ‘𝑦))) = (ℑ‘𝑦))
14139, 131, 140syl2an 493 . . . . . . . . 9 ((𝜑𝑦𝑆) → (𝐹‘(𝐹‘(ℑ‘𝑦))) = (ℑ‘𝑦))
142139, 141eqtrd 2685 . . . . . . . 8 ((𝜑𝑦𝑆) → (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))) = (ℑ‘𝑦))
143142oveq2d 6706 . . . . . . 7 ((𝜑𝑦𝑆) → (i · (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦))))) = (i · (ℑ‘𝑦)))
144110, 143oveq12d 6708 . . . . . 6 ((𝜑𝑦𝑆) → (((exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))) = ((ℜ‘𝑦) + (i · (ℑ‘𝑦))))
145100, 144eqtr4d 2688 . . . . 5 ((𝜑𝑦𝑆) → 𝑦 = (((exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))))
146 fveq2 6229 . . . . . . . 8 (𝑥 = (exp‘𝑦) → (abs‘𝑥) = (abs‘(exp‘𝑦)))
147146fveq2d 6233 . . . . . . 7 (𝑥 = (exp‘𝑦) → ((exp ↾ ℝ)‘(abs‘𝑥)) = ((exp ↾ ℝ)‘(abs‘(exp‘𝑦))))
148 id 22 . . . . . . . . . 10 (𝑥 = (exp‘𝑦) → 𝑥 = (exp‘𝑦))
149148, 146oveq12d 6708 . . . . . . . . 9 (𝑥 = (exp‘𝑦) → (𝑥 / (abs‘𝑥)) = ((exp‘𝑦) / (abs‘(exp‘𝑦))))
150149fveq2d 6233 . . . . . . . 8 (𝑥 = (exp‘𝑦) → (𝐹‘(𝑥 / (abs‘𝑥))) = (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))
151150oveq2d 6706 . . . . . . 7 (𝑥 = (exp‘𝑦) → (i · (𝐹‘(𝑥 / (abs‘𝑥)))) = (i · (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦))))))
152147, 151oveq12d 6708 . . . . . 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)))))))
153152eqeq2d 2661 . . . . 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))))))))
154145, 153syl5ibrcom 237 . . . 4 ((𝜑𝑦𝑆) → (𝑥 = (exp‘𝑦) → 𝑦 = (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
155154adantrr 753 . . 3 ((𝜑 ∧ (𝑦𝑆𝑥 ∈ (ℂ ∖ {0}))) → (𝑥 = (exp‘𝑦) → 𝑦 = (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
15698, 155impbid 202 . 2 ((𝜑 ∧ (𝑦𝑆𝑥 ∈ (ℂ ∖ {0}))) → (𝑦 = (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) ↔ 𝑥 = (exp‘𝑦)))
15713, 17, 71, 156f1o2d 6929 1 (𝜑 → (exp ↾ 𝑆):𝑆1-1-onto→(ℂ ∖ {0}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wrex 2942  cdif 3604  wss 3607  {csn 4210   class class class wbr 4685  cmpt 4762  ccnv 5142  dom cdm 5143  cres 5145  cima 5146   Fn wfn 5921  wf 5922  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  cc 9972  cr 9973  0cc0 9974  1c1 9975  ici 9976   + caddc 9977   · cmul 9979   < clt 10112  cmin 10304  -cneg 10305   / cdiv 10722  2c2 11108  cz 11415  +crp 11870  [,]cicc 12216  cre 13881  cim 13882  abscabs 14018  expce 14836  sincsin 14838  πcpi 14841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ioc 12218  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-shft 13851  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246  df-clim 14263  df-rlim 14264  df-sum 14461  df-ef 14842  df-sin 14844  df-cos 14845  df-pi 14847  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-lp 20988  df-perf 20989  df-cn 21079  df-cnp 21080  df-haus 21167  df-tx 21413  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-xms 22172  df-ms 22173  df-tms 22174  df-cncf 22728  df-limc 23675  df-dv 23676
This theorem is referenced by:  eff1o  24340
  Copyright terms: Public domain W3C validator