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Theorem eff1olem 24098
 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 5404 . . . 4 (ℑ “ 𝐷) ⊆ dom ℑ
2 eff1olem.2 . . . 4 𝑆 = (ℑ “ 𝐷)
3 imf 13701 . . . . . 6 ℑ:ℂ⟶ℝ
43fdmi 5965 . . . . 5 dom ℑ = ℂ
54eqcomi 2619 . . . 4 ℂ = dom ℑ
61, 2, 53sstr4i 3607 . . 3 𝑆 ⊆ ℂ
7 eff2 14668 . . . . . . 7 exp:ℂ⟶(ℂ ∖ {0})
87a1i 11 . . . . . 6 (𝑆 ⊆ ℂ → exp:ℂ⟶(ℂ ∖ {0}))
98feqmptd 6159 . . . . 5 (𝑆 ⊆ ℂ → exp = (𝑦 ∈ ℂ ↦ (exp‘𝑦)))
109reseq1d 5316 . . . 4 (𝑆 ⊆ ℂ → (exp ↾ 𝑆) = ((𝑦 ∈ ℂ ↦ (exp‘𝑦)) ↾ 𝑆))
11 resmpt 5369 . . . 4 (𝑆 ⊆ ℂ → ((𝑦 ∈ ℂ ↦ (exp‘𝑦)) ↾ 𝑆) = (𝑦𝑆 ↦ (exp‘𝑦)))
1210, 11eqtrd 2644 . . 3 (𝑆 ⊆ ℂ → (exp ↾ 𝑆) = (𝑦𝑆 ↦ (exp‘𝑦)))
136, 12ax-mp 5 . 2 (exp ↾ 𝑆) = (𝑦𝑆 ↦ (exp‘𝑦))
146sseli 3564 . . . 4 (𝑦𝑆𝑦 ∈ ℂ)
157ffvelrni 6266 . . . 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 4260 . . . . . . . . . 10 (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
2018, 19sylib 207 . . . . . . . . 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 14035 . . . . . . 7 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℝ+)
24 reeff1o 24005 . . . . . . . . 9 (exp ↾ ℝ):ℝ–1-1-onto→ℝ+
25 f1ocnv 6062 . . . . . . . . 9 ((exp ↾ ℝ):ℝ–1-1-onto→ℝ+(exp ↾ ℝ):ℝ+1-1-onto→ℝ)
26 f1of 6050 . . . . . . . . 9 ((exp ↾ ℝ):ℝ+1-1-onto→ℝ → (exp ↾ ℝ):ℝ+⟶ℝ)
2724, 25, 26mp2b 10 . . . . . . . 8 (exp ↾ ℝ):ℝ+⟶ℝ
2827ffvelrni 6266 . . . . . . 7 ((abs‘𝑥) ∈ ℝ+ → ((exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ)
2923, 28syl 17 . . . . . 6 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ)
3029recnd 9947 . . . . 5 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℂ)
31 ax-icn 9874 . . . . . 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 2610 . . . . . . . . . . . 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 2610 . . . . . . . . . . . 12 (sin ↾ (-(π / 2)[,](π / 2))) = (sin ↾ (-(π / 2)[,](π / 2)))
3934, 35, 32, 36, 37, 38efif1olem4 24095 . . . . . . . . . . 11 (𝜑𝐹:𝐷1-1-onto→(abs “ {1}))
40 f1ocnv 6062 . . . . . . . . . . 11 (𝐹:𝐷1-1-onto→(abs “ {1}) → 𝐹:(abs “ {1})–1-1-onto𝐷)
41 f1of 6050 . . . . . . . . . . 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 14023 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℝ)
4544recnd 9947 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℂ)
4621, 22absne0d 14034 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ≠ 0)
4721, 45, 46divcld 10680 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝑥 / (abs‘𝑥)) ∈ ℂ)
4821, 45, 46absdivd 14042 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘(𝑥 / (abs‘𝑥))) = ((abs‘𝑥) / (abs‘(abs‘𝑥))))
49 absidm 13911 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (abs‘(abs‘𝑥)) = (abs‘𝑥))
5021, 49syl 17 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘(abs‘𝑥)) = (abs‘𝑥))
5150oveq2d 6565 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) / (abs‘(abs‘𝑥))) = ((abs‘𝑥) / (abs‘𝑥)))
5245, 46dividd 10678 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) / (abs‘𝑥)) = 1)
5348, 51, 523eqtrd 2648 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘(𝑥 / (abs‘𝑥))) = 1)
54 absf 13925 . . . . . . . . . . 11 abs:ℂ⟶ℝ
55 ffn 5958 . . . . . . . . . . 11 (abs:ℂ⟶ℝ → abs Fn ℂ)
56 fniniseg 6246 . . . . . . . . . . 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 695 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝑥 / (abs‘𝑥)) ∈ (abs “ {1}))
5943, 58ffvelrnd 6268 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(𝑥 / (abs‘𝑥))) ∈ 𝐷)
6033, 59sseldd 3569 . . . . . . 7 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℝ)
6160recnd 9947 . . . . . 6 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℂ)
62 mulcl 9899 . . . . . 6 ((i ∈ ℂ ∧ (𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℂ) → (i · (𝐹‘(𝑥 / (abs‘𝑥)))) ∈ ℂ)
6331, 61, 62sylancr 694 . . . . 5 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (i · (𝐹‘(𝑥 / (abs‘𝑥)))) ∈ ℂ)
6430, 63addcld 9938 . . . 4 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) ∈ ℂ)
6529, 60crimd 13820 . . . . 5 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (ℑ‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))) = (𝐹‘(𝑥 / (abs‘𝑥))))
6665, 59eqeltrd 2688 . . . 4 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (ℑ‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))) ∈ 𝐷)
67 ffn 5958 . . . . 5 (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
68 elpreima 6245 . . . . 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 695 . . 3 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) ∈ (ℑ “ 𝐷))
7170, 2syl6eleqr 2699 . 2 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) ∈ 𝑆)
72 efadd 14663 . . . . . . 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 691 . . . . . 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 6117 . . . . . . . . 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 6433 . . . . . . . . 9 (((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ ∧ (abs‘𝑥) ∈ ℝ+) → ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
7724, 23, 76sylancr 694 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
7875, 77eqtr3d 2646 . . . . . . 7 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (exp‘((exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
79 oveq2 6557 . . . . . . . . . . 11 (𝑧 = (𝐹‘(𝑥 / (abs‘𝑥))) → (i · 𝑧) = (i · (𝐹‘(𝑥 / (abs‘𝑥)))))
8079fveq2d 6107 . . . . . . . . . 10 (𝑧 = (𝐹‘(𝑥 / (abs‘𝑥))) → (exp‘(i · 𝑧)) = (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥))))))
81 oveq2 6557 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (i · 𝑤) = (i · 𝑧))
8281fveq2d 6107 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (exp‘(i · 𝑤)) = (exp‘(i · 𝑧)))
8382cbvmptv 4678 . . . . . . . . . . 11 (𝑤𝐷 ↦ (exp‘(i · 𝑤))) = (𝑧𝐷 ↦ (exp‘(i · 𝑧)))
8434, 83eqtri 2632 . . . . . . . . . 10 𝐹 = (𝑧𝐷 ↦ (exp‘(i · 𝑧)))
85 fvex 6113 . . . . . . . . . 10 (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥))))) ∈ V
8680, 84, 85fvmpt 6191 . . . . . . . . 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 6433 . . . . . . . . 9 ((𝐹:𝐷1-1-onto→(abs “ {1}) ∧ (𝑥 / (abs‘𝑥)) ∈ (abs “ {1})) → (𝐹‘(𝐹‘(𝑥 / (abs‘𝑥)))) = (𝑥 / (abs‘𝑥)))
9088, 58, 89syl2anc 691 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(𝐹‘(𝑥 / (abs‘𝑥)))) = (𝑥 / (abs‘𝑥)))
9187, 90eqtr3d 2646 . . . . . . 7 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥))))) = (𝑥 / (abs‘𝑥)))
9278, 91oveq12d 6567 . . . . . 6 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((exp‘((exp ↾ ℝ)‘(abs‘𝑥))) · (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥)))))) = ((abs‘𝑥) · (𝑥 / (abs‘𝑥))))
9321, 45, 46divcan2d 10682 . . . . . 6 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) · (𝑥 / (abs‘𝑥))) = 𝑥)
9473, 92, 933eqtrrd 2649 . . . . 5 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝑥 = (exp‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
9594adantrl 748 . . . 4 ((𝜑 ∧ (𝑦𝑆𝑥 ∈ (ℂ ∖ {0}))) → 𝑥 = (exp‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
96 fveq2 6103 . . . . 5 (𝑦 = (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) → (exp‘𝑦) = (exp‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
9796eqeq2d 2620 . . . 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 236 . . 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 13785 . . . . . 6 ((𝜑𝑦𝑆) → 𝑦 = ((ℜ‘𝑦) + (i · (ℑ‘𝑦))))
101 absef 14766 . . . . . . . . . . 11 (𝑦 ∈ ℂ → (abs‘(exp‘𝑦)) = (exp‘(ℜ‘𝑦)))
10299, 101syl 17 . . . . . . . . . 10 ((𝜑𝑦𝑆) → (abs‘(exp‘𝑦)) = (exp‘(ℜ‘𝑦)))
10399recld 13782 . . . . . . . . . . 11 ((𝜑𝑦𝑆) → (ℜ‘𝑦) ∈ ℝ)
104 fvres 6117 . . . . . . . . . . 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 2647 . . . . . . . . 9 ((𝜑𝑦𝑆) → (abs‘(exp‘𝑦)) = ((exp ↾ ℝ)‘(ℜ‘𝑦)))
107106fveq2d 6107 . . . . . . . 8 ((𝜑𝑦𝑆) → ((exp ↾ ℝ)‘(abs‘(exp‘𝑦))) = ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))))
108 f1ocnvfv1 6432 . . . . . . . . 9 (((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ ∧ (ℜ‘𝑦) ∈ ℝ) → ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))) = (ℜ‘𝑦))
10924, 103, 108sylancr 694 . . . . . . . 8 ((𝜑𝑦𝑆) → ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))) = (ℜ‘𝑦))
110107, 109eqtrd 2644 . . . . . . 7 ((𝜑𝑦𝑆) → ((exp ↾ ℝ)‘(abs‘(exp‘𝑦))) = (ℜ‘𝑦))
11199imcld 13783 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝑆) → (ℑ‘𝑦) ∈ ℝ)
112111recnd 9947 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑆) → (ℑ‘𝑦) ∈ ℂ)
113 mulcl 9899 . . . . . . . . . . . . . 14 ((i ∈ ℂ ∧ (ℑ‘𝑦) ∈ ℂ) → (i · (ℑ‘𝑦)) ∈ ℂ)
11431, 112, 113sylancr 694 . . . . . . . . . . . . 13 ((𝜑𝑦𝑆) → (i · (ℑ‘𝑦)) ∈ ℂ)
115 efcl 14652 . . . . . . . . . . . . 13 ((i · (ℑ‘𝑦)) ∈ ℂ → (exp‘(i · (ℑ‘𝑦))) ∈ ℂ)
116114, 115syl 17 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (exp‘(i · (ℑ‘𝑦))) ∈ ℂ)
117103recnd 9947 . . . . . . . . . . . . 13 ((𝜑𝑦𝑆) → (ℜ‘𝑦) ∈ ℂ)
118 efcl 14652 . . . . . . . . . . . . 13 ((ℜ‘𝑦) ∈ ℂ → (exp‘(ℜ‘𝑦)) ∈ ℂ)
119117, 118syl 17 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (exp‘(ℜ‘𝑦)) ∈ ℂ)
120 efne0 14666 . . . . . . . . . . . . 13 ((ℜ‘𝑦) ∈ ℂ → (exp‘(ℜ‘𝑦)) ≠ 0)
121117, 120syl 17 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (exp‘(ℜ‘𝑦)) ≠ 0)
122116, 119, 121divcan3d 10685 . . . . . . . . . . 11 ((𝜑𝑦𝑆) → (((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))) / (exp‘(ℜ‘𝑦))) = (exp‘(i · (ℑ‘𝑦))))
123100fveq2d 6107 . . . . . . . . . . . . 13 ((𝜑𝑦𝑆) → (exp‘𝑦) = (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))))
124 efadd 14663 . . . . . . . . . . . . . 14 (((ℜ‘𝑦) ∈ ℂ ∧ (i · (ℑ‘𝑦)) ∈ ℂ) → (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
125117, 114, 124syl2anc 691 . . . . . . . . . . . . 13 ((𝜑𝑦𝑆) → (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
126123, 125eqtrd 2644 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (exp‘𝑦) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
127126, 102oveq12d 6567 . . . . . . . . . . 11 ((𝜑𝑦𝑆) → ((exp‘𝑦) / (abs‘(exp‘𝑦))) = (((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))) / (exp‘(ℜ‘𝑦))))
128 elpreima 6245 . . . . . . . . . . . . . . . 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 2706 . . . . . . . . . . . . 13 (𝑦𝑆 → (ℑ‘𝑦) ∈ 𝐷)
132131adantl 481 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (ℑ‘𝑦) ∈ 𝐷)
133 oveq2 6557 . . . . . . . . . . . . . 14 (𝑤 = (ℑ‘𝑦) → (i · 𝑤) = (i · (ℑ‘𝑦)))
134133fveq2d 6107 . . . . . . . . . . . . 13 (𝑤 = (ℑ‘𝑦) → (exp‘(i · 𝑤)) = (exp‘(i · (ℑ‘𝑦))))
135 fvex 6113 . . . . . . . . . . . . 13 (exp‘(i · (ℑ‘𝑦))) ∈ V
136134, 34, 135fvmpt 6191 . . . . . . . . . . . 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 2654 . . . . . . . . . 10 ((𝜑𝑦𝑆) → ((exp‘𝑦) / (abs‘(exp‘𝑦))) = (𝐹‘(ℑ‘𝑦)))
139138fveq2d 6107 . . . . . . . . 9 ((𝜑𝑦𝑆) → (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))) = (𝐹‘(𝐹‘(ℑ‘𝑦))))
140 f1ocnvfv1 6432 . . . . . . . . . 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 2644 . . . . . . . 8 ((𝜑𝑦𝑆) → (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))) = (ℑ‘𝑦))
143142oveq2d 6565 . . . . . . 7 ((𝜑𝑦𝑆) → (i · (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦))))) = (i · (ℑ‘𝑦)))
144110, 143oveq12d 6567 . . . . . 6 ((𝜑𝑦𝑆) → (((exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))) = ((ℜ‘𝑦) + (i · (ℑ‘𝑦))))
145100, 144eqtr4d 2647 . . . . 5 ((𝜑𝑦𝑆) → 𝑦 = (((exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))))
146 fveq2 6103 . . . . . . . 8 (𝑥 = (exp‘𝑦) → (abs‘𝑥) = (abs‘(exp‘𝑦)))
147146fveq2d 6107 . . . . . . 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 6567 . . . . . . . . 9 (𝑥 = (exp‘𝑦) → (𝑥 / (abs‘𝑥)) = ((exp‘𝑦) / (abs‘(exp‘𝑦))))
150149fveq2d 6107 . . . . . . . 8 (𝑥 = (exp‘𝑦) → (𝐹‘(𝑥 / (abs‘𝑥))) = (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))
151150oveq2d 6565 . . . . . . 7 (𝑥 = (exp‘𝑦) → (i · (𝐹‘(𝑥 / (abs‘𝑥)))) = (i · (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦))))))
152147, 151oveq12d 6567 . . . . . 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 2620 . . . . 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 236 . . . 4 ((𝜑𝑦𝑆) → (𝑥 = (exp‘𝑦) → 𝑦 = (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
155154adantrr 749 . . 3 ((𝜑 ∧ (𝑦𝑆𝑥 ∈ (ℂ ∖ {0}))) → (𝑥 = (exp‘𝑦) → 𝑦 = (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
15698, 155impbid 201 . 2 ((𝜑 ∧ (𝑦𝑆𝑥 ∈ (ℂ ∖ {0}))) → (𝑦 = (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) ↔ 𝑥 = (exp‘𝑦)))
15713, 17, 71, 156f1o2d 6785 1 (𝜑 → (exp ↾ 𝑆):𝑆1-1-onto→(ℂ ∖ {0}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   ∖ cdif 3537   ⊆ wss 3540  {csn 4125   class class class wbr 4583   ↦ cmpt 4643  ◡ccnv 5037  dom cdm 5038   ↾ cres 5040   “ cima 5041   Fn wfn 5799  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816  ici 9817   + caddc 9818   · cmul 9820   < clt 9953   − cmin 10145  -cneg 10146   / cdiv 10563  2c2 10947  ℤcz 11254  ℝ+crp 11708  [,]cicc 12049  ℜcre 13685  ℑcim 13686  abscabs 13822  expce 14631  sincsin 14633  πcpi 14636 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ioc 12051  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-shft 13655  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-sum 14265  df-ef 14637  df-sin 14639  df-cos 14640  df-pi 14642  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-perf 20751  df-cn 20841  df-cnp 20842  df-haus 20929  df-tx 21175  df-hmeo 21368  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-xms 21935  df-ms 21936  df-tms 21937  df-cncf 22489  df-limc 23436  df-dv 23437 This theorem is referenced by:  eff1o  24099
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