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Theorem eff1olem 24491
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 5641 . . . 4 (ℑ “ 𝐷) ⊆ dom ℑ
2 eff1olem.2 . . . 4 𝑆 = (ℑ “ 𝐷)
3 imf 14050 . . . . . 6 ℑ:ℂ⟶ℝ
43fdmi 6211 . . . . 5 dom ℑ = ℂ
54eqcomi 2767 . . . 4 ℂ = dom ℑ
61, 2, 53sstr4i 3783 . . 3 𝑆 ⊆ ℂ
7 eff2 15026 . . . . . . 7 exp:ℂ⟶(ℂ ∖ {0})
87a1i 11 . . . . . 6 (𝑆 ⊆ ℂ → exp:ℂ⟶(ℂ ∖ {0}))
98feqmptd 6409 . . . . 5 (𝑆 ⊆ ℂ → exp = (𝑦 ∈ ℂ ↦ (exp‘𝑦)))
109reseq1d 5548 . . . 4 (𝑆 ⊆ ℂ → (exp ↾ 𝑆) = ((𝑦 ∈ ℂ ↦ (exp‘𝑦)) ↾ 𝑆))
11 resmpt 5605 . . . 4 (𝑆 ⊆ ℂ → ((𝑦 ∈ ℂ ↦ (exp‘𝑦)) ↾ 𝑆) = (𝑦𝑆 ↦ (exp‘𝑦)))
1210, 11eqtrd 2792 . . 3 (𝑆 ⊆ ℂ → (exp ↾ 𝑆) = (𝑦𝑆 ↦ (exp‘𝑦)))
136, 12ax-mp 5 . 2 (exp ↾ 𝑆) = (𝑦𝑆 ↦ (exp‘𝑦))
146sseli 3738 . . . 4 (𝑦𝑆𝑦 ∈ ℂ)
157ffvelrni 6519 . . . 4 (𝑦 ∈ ℂ → (exp‘𝑦) ∈ (ℂ ∖ {0}))
1614, 15syl 17 . . 3 (𝑦𝑆 → (exp‘𝑦) ∈ (ℂ ∖ {0}))
1716adantl 473 . 2 ((𝜑𝑦𝑆) → (exp‘𝑦) ∈ (ℂ ∖ {0}))
18 simpr 479 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ∈ (ℂ ∖ {0}))
19 eldifsn 4460 . . . . . . . . . 10 (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
2018, 19sylib 208 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
2120simpld 477 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
2220simprd 482 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ≠ 0)
2321, 22absrpcld 14384 . . . . . . 7 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℝ+)
24 reeff1o 24398 . . . . . . . . 9 (exp ↾ ℝ):ℝ–1-1-onto→ℝ+
25 f1ocnv 6308 . . . . . . . . 9 ((exp ↾ ℝ):ℝ–1-1-onto→ℝ+(exp ↾ ℝ):ℝ+1-1-onto→ℝ)
26 f1of 6296 . . . . . . . . 9 ((exp ↾ ℝ):ℝ+1-1-onto→ℝ → (exp ↾ ℝ):ℝ+⟶ℝ)
2724, 25, 26mp2b 10 . . . . . . . 8 (exp ↾ ℝ):ℝ+⟶ℝ
2827ffvelrni 6519 . . . . . . 7 ((abs‘𝑥) ∈ ℝ+ → ((exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ)
2923, 28syl 17 . . . . . 6 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ)
3029recnd 10258 . . . . 5 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℂ)
31 ax-icn 10185 . . . . . 6 i ∈ ℂ
32 eff1olem.3 . . . . . . . . 9 (𝜑𝐷 ⊆ ℝ)
3332adantr 472 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝐷 ⊆ ℝ)
34 eff1olem.1 . . . . . . . . . . . 12 𝐹 = (𝑤𝐷 ↦ (exp‘(i · 𝑤)))
35 eqid 2758 . . . . . . . . . . . 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 2758 . . . . . . . . . . . 12 (sin ↾ (-(π / 2)[,](π / 2))) = (sin ↾ (-(π / 2)[,](π / 2)))
3934, 35, 32, 36, 37, 38efif1olem4 24488 . . . . . . . . . . 11 (𝜑𝐹:𝐷1-1-onto→(abs “ {1}))
40 f1ocnv 6308 . . . . . . . . . . 11 (𝐹:𝐷1-1-onto→(abs “ {1}) → 𝐹:(abs “ {1})–1-1-onto𝐷)
41 f1of 6296 . . . . . . . . . . 11 (𝐹:(abs “ {1})–1-1-onto𝐷𝐹:(abs “ {1})⟶𝐷)
4239, 40, 413syl 18 . . . . . . . . . 10 (𝜑𝐹:(abs “ {1})⟶𝐷)
4342adantr 472 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝐹:(abs “ {1})⟶𝐷)
4421abscld 14372 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℝ)
4544recnd 10258 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℂ)
4621, 22absne0d 14383 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ≠ 0)
4721, 45, 46divcld 10991 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝑥 / (abs‘𝑥)) ∈ ℂ)
4821, 45, 46absdivd 14391 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘(𝑥 / (abs‘𝑥))) = ((abs‘𝑥) / (abs‘(abs‘𝑥))))
49 absidm 14260 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (abs‘(abs‘𝑥)) = (abs‘𝑥))
5021, 49syl 17 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘(abs‘𝑥)) = (abs‘𝑥))
5150oveq2d 6827 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) / (abs‘(abs‘𝑥))) = ((abs‘𝑥) / (abs‘𝑥)))
5245, 46dividd 10989 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) / (abs‘𝑥)) = 1)
5348, 51, 523eqtrd 2796 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘(𝑥 / (abs‘𝑥))) = 1)
54 absf 14274 . . . . . . . . . . 11 abs:ℂ⟶ℝ
55 ffn 6204 . . . . . . . . . . 11 (abs:ℂ⟶ℝ → abs Fn ℂ)
56 fniniseg 6499 . . . . . . . . . . 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 701 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝑥 / (abs‘𝑥)) ∈ (abs “ {1}))
5943, 58ffvelrnd 6521 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(𝑥 / (abs‘𝑥))) ∈ 𝐷)
6033, 59sseldd 3743 . . . . . . 7 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℝ)
6160recnd 10258 . . . . . 6 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℂ)
62 mulcl 10210 . . . . . 6 ((i ∈ ℂ ∧ (𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℂ) → (i · (𝐹‘(𝑥 / (abs‘𝑥)))) ∈ ℂ)
6331, 61, 62sylancr 698 . . . . 5 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (i · (𝐹‘(𝑥 / (abs‘𝑥)))) ∈ ℂ)
6430, 63addcld 10249 . . . 4 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) ∈ ℂ)
6529, 60crimd 14169 . . . . 5 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (ℑ‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))) = (𝐹‘(𝑥 / (abs‘𝑥))))
6665, 59eqeltrd 2837 . . . 4 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (ℑ‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))) ∈ 𝐷)
67 ffn 6204 . . . . 5 (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
68 elpreima 6498 . . . . 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 701 . . 3 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) ∈ (ℑ “ 𝐷))
7170, 2syl6eleqr 2848 . 2 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) ∈ 𝑆)
72 efadd 15021 . . . . . . 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 696 . . . . . 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 6366 . . . . . . . . 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 6694 . . . . . . . . 9 (((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ ∧ (abs‘𝑥) ∈ ℝ+) → ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
7724, 23, 76sylancr 698 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
7875, 77eqtr3d 2794 . . . . . . 7 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (exp‘((exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
79 oveq2 6819 . . . . . . . . . . 11 (𝑧 = (𝐹‘(𝑥 / (abs‘𝑥))) → (i · 𝑧) = (i · (𝐹‘(𝑥 / (abs‘𝑥)))))
8079fveq2d 6354 . . . . . . . . . 10 (𝑧 = (𝐹‘(𝑥 / (abs‘𝑥))) → (exp‘(i · 𝑧)) = (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥))))))
81 oveq2 6819 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (i · 𝑤) = (i · 𝑧))
8281fveq2d 6354 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (exp‘(i · 𝑤)) = (exp‘(i · 𝑧)))
8382cbvmptv 4900 . . . . . . . . . . 11 (𝑤𝐷 ↦ (exp‘(i · 𝑤))) = (𝑧𝐷 ↦ (exp‘(i · 𝑧)))
8434, 83eqtri 2780 . . . . . . . . . 10 𝐹 = (𝑧𝐷 ↦ (exp‘(i · 𝑧)))
85 fvex 6360 . . . . . . . . . 10 (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥))))) ∈ V
8680, 84, 85fvmpt 6442 . . . . . . . . 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 472 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝐹:𝐷1-1-onto→(abs “ {1}))
89 f1ocnvfv2 6694 . . . . . . . . 9 ((𝐹:𝐷1-1-onto→(abs “ {1}) ∧ (𝑥 / (abs‘𝑥)) ∈ (abs “ {1})) → (𝐹‘(𝐹‘(𝑥 / (abs‘𝑥)))) = (𝑥 / (abs‘𝑥)))
9088, 58, 89syl2anc 696 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(𝐹‘(𝑥 / (abs‘𝑥)))) = (𝑥 / (abs‘𝑥)))
9187, 90eqtr3d 2794 . . . . . . 7 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥))))) = (𝑥 / (abs‘𝑥)))
9278, 91oveq12d 6829 . . . . . 6 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((exp‘((exp ↾ ℝ)‘(abs‘𝑥))) · (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥)))))) = ((abs‘𝑥) · (𝑥 / (abs‘𝑥))))
9321, 45, 46divcan2d 10993 . . . . . 6 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) · (𝑥 / (abs‘𝑥))) = 𝑥)
9473, 92, 933eqtrrd 2797 . . . . 5 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝑥 = (exp‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
9594adantrl 754 . . . 4 ((𝜑 ∧ (𝑦𝑆𝑥 ∈ (ℂ ∖ {0}))) → 𝑥 = (exp‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
96 fveq2 6350 . . . . 5 (𝑦 = (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) → (exp‘𝑦) = (exp‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
9796eqeq2d 2768 . . . 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 473 . . . . . . 7 ((𝜑𝑦𝑆) → 𝑦 ∈ ℂ)
10099replimd 14134 . . . . . 6 ((𝜑𝑦𝑆) → 𝑦 = ((ℜ‘𝑦) + (i · (ℑ‘𝑦))))
101 absef 15124 . . . . . . . . . . 11 (𝑦 ∈ ℂ → (abs‘(exp‘𝑦)) = (exp‘(ℜ‘𝑦)))
10299, 101syl 17 . . . . . . . . . 10 ((𝜑𝑦𝑆) → (abs‘(exp‘𝑦)) = (exp‘(ℜ‘𝑦)))
10399recld 14131 . . . . . . . . . . 11 ((𝜑𝑦𝑆) → (ℜ‘𝑦) ∈ ℝ)
104 fvres 6366 . . . . . . . . . . 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 2795 . . . . . . . . 9 ((𝜑𝑦𝑆) → (abs‘(exp‘𝑦)) = ((exp ↾ ℝ)‘(ℜ‘𝑦)))
107106fveq2d 6354 . . . . . . . 8 ((𝜑𝑦𝑆) → ((exp ↾ ℝ)‘(abs‘(exp‘𝑦))) = ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))))
108 f1ocnvfv1 6693 . . . . . . . . 9 (((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ ∧ (ℜ‘𝑦) ∈ ℝ) → ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))) = (ℜ‘𝑦))
10924, 103, 108sylancr 698 . . . . . . . 8 ((𝜑𝑦𝑆) → ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))) = (ℜ‘𝑦))
110107, 109eqtrd 2792 . . . . . . 7 ((𝜑𝑦𝑆) → ((exp ↾ ℝ)‘(abs‘(exp‘𝑦))) = (ℜ‘𝑦))
11199imcld 14132 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝑆) → (ℑ‘𝑦) ∈ ℝ)
112111recnd 10258 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑆) → (ℑ‘𝑦) ∈ ℂ)
113 mulcl 10210 . . . . . . . . . . . . . 14 ((i ∈ ℂ ∧ (ℑ‘𝑦) ∈ ℂ) → (i · (ℑ‘𝑦)) ∈ ℂ)
11431, 112, 113sylancr 698 . . . . . . . . . . . . 13 ((𝜑𝑦𝑆) → (i · (ℑ‘𝑦)) ∈ ℂ)
115 efcl 15010 . . . . . . . . . . . . 13 ((i · (ℑ‘𝑦)) ∈ ℂ → (exp‘(i · (ℑ‘𝑦))) ∈ ℂ)
116114, 115syl 17 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (exp‘(i · (ℑ‘𝑦))) ∈ ℂ)
117103recnd 10258 . . . . . . . . . . . . 13 ((𝜑𝑦𝑆) → (ℜ‘𝑦) ∈ ℂ)
118 efcl 15010 . . . . . . . . . . . . 13 ((ℜ‘𝑦) ∈ ℂ → (exp‘(ℜ‘𝑦)) ∈ ℂ)
119117, 118syl 17 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (exp‘(ℜ‘𝑦)) ∈ ℂ)
120 efne0 15024 . . . . . . . . . . . . 13 ((ℜ‘𝑦) ∈ ℂ → (exp‘(ℜ‘𝑦)) ≠ 0)
121117, 120syl 17 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (exp‘(ℜ‘𝑦)) ≠ 0)
122116, 119, 121divcan3d 10996 . . . . . . . . . . 11 ((𝜑𝑦𝑆) → (((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))) / (exp‘(ℜ‘𝑦))) = (exp‘(i · (ℑ‘𝑦))))
123100fveq2d 6354 . . . . . . . . . . . . 13 ((𝜑𝑦𝑆) → (exp‘𝑦) = (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))))
124 efadd 15021 . . . . . . . . . . . . . 14 (((ℜ‘𝑦) ∈ ℂ ∧ (i · (ℑ‘𝑦)) ∈ ℂ) → (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
125117, 114, 124syl2anc 696 . . . . . . . . . . . . 13 ((𝜑𝑦𝑆) → (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
126123, 125eqtrd 2792 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (exp‘𝑦) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
127126, 102oveq12d 6829 . . . . . . . . . . 11 ((𝜑𝑦𝑆) → ((exp‘𝑦) / (abs‘(exp‘𝑦))) = (((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))) / (exp‘(ℜ‘𝑦))))
128 elpreima 6498 . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . 14 (𝑦 ∈ (ℑ “ 𝐷) → (ℑ‘𝑦) ∈ 𝐷)
131130, 2eleq2s 2855 . . . . . . . . . . . . 13 (𝑦𝑆 → (ℑ‘𝑦) ∈ 𝐷)
132131adantl 473 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (ℑ‘𝑦) ∈ 𝐷)
133 oveq2 6819 . . . . . . . . . . . . . 14 (𝑤 = (ℑ‘𝑦) → (i · 𝑤) = (i · (ℑ‘𝑦)))
134133fveq2d 6354 . . . . . . . . . . . . 13 (𝑤 = (ℑ‘𝑦) → (exp‘(i · 𝑤)) = (exp‘(i · (ℑ‘𝑦))))
135 fvex 6360 . . . . . . . . . . . . 13 (exp‘(i · (ℑ‘𝑦))) ∈ V
136134, 34, 135fvmpt 6442 . . . . . . . . . . . 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 2802 . . . . . . . . . 10 ((𝜑𝑦𝑆) → ((exp‘𝑦) / (abs‘(exp‘𝑦))) = (𝐹‘(ℑ‘𝑦)))
139138fveq2d 6354 . . . . . . . . 9 ((𝜑𝑦𝑆) → (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))) = (𝐹‘(𝐹‘(ℑ‘𝑦))))
140 f1ocnvfv1 6693 . . . . . . . . . 10 ((𝐹:𝐷1-1-onto→(abs “ {1}) ∧ (ℑ‘𝑦) ∈ 𝐷) → (𝐹‘(𝐹‘(ℑ‘𝑦))) = (ℑ‘𝑦))
14139, 131, 140syl2an 495 . . . . . . . . 9 ((𝜑𝑦𝑆) → (𝐹‘(𝐹‘(ℑ‘𝑦))) = (ℑ‘𝑦))
142139, 141eqtrd 2792 . . . . . . . 8 ((𝜑𝑦𝑆) → (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))) = (ℑ‘𝑦))
143142oveq2d 6827 . . . . . . 7 ((𝜑𝑦𝑆) → (i · (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦))))) = (i · (ℑ‘𝑦)))
144110, 143oveq12d 6829 . . . . . 6 ((𝜑𝑦𝑆) → (((exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))) = ((ℜ‘𝑦) + (i · (ℑ‘𝑦))))
145100, 144eqtr4d 2795 . . . . 5 ((𝜑𝑦𝑆) → 𝑦 = (((exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))))
146 fveq2 6350 . . . . . . . 8 (𝑥 = (exp‘𝑦) → (abs‘𝑥) = (abs‘(exp‘𝑦)))
147146fveq2d 6354 . . . . . . 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 6829 . . . . . . . . 9 (𝑥 = (exp‘𝑦) → (𝑥 / (abs‘𝑥)) = ((exp‘𝑦) / (abs‘(exp‘𝑦))))
150149fveq2d 6354 . . . . . . . 8 (𝑥 = (exp‘𝑦) → (𝐹‘(𝑥 / (abs‘𝑥))) = (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))
151150oveq2d 6827 . . . . . . 7 (𝑥 = (exp‘𝑦) → (i · (𝐹‘(𝑥 / (abs‘𝑥)))) = (i · (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦))))))
152147, 151oveq12d 6829 . . . . . 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 2768 . . . . 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 755 . . 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 7050 1 (𝜑 → (exp ↾ 𝑆):𝑆1-1-onto→(ℂ ∖ {0}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1630  wcel 2137  wne 2930  wrex 3049  cdif 3710  wss 3713  {csn 4319   class class class wbr 4802  cmpt 4879  ccnv 5263  dom cdm 5264  cres 5266  cima 5267   Fn wfn 6042  wf 6043  1-1-ontowf1o 6046  cfv 6047  (class class class)co 6811  cc 10124  cr 10125  0cc0 10126  1c1 10127  ici 10128   + caddc 10129   · cmul 10131   < clt 10264  cmin 10456  -cneg 10457   / cdiv 10874  2c2 11260  cz 11567  +crp 12023  [,]cicc 12369  cre 14034  cim 14035  abscabs 14171  expce 14989  sincsin 14991  πcpi 14994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-inf2 8709  ax-cnex 10182  ax-resscn 10183  ax-1cn 10184  ax-icn 10185  ax-addcl 10186  ax-addrcl 10187  ax-mulcl 10188  ax-mulrcl 10189  ax-mulcom 10190  ax-addass 10191  ax-mulass 10192  ax-distr 10193  ax-i2m1 10194  ax-1ne0 10195  ax-1rid 10196  ax-rnegex 10197  ax-rrecex 10198  ax-cnre 10199  ax-pre-lttri 10200  ax-pre-lttrn 10201  ax-pre-ltadd 10202  ax-pre-mulgt0 10203  ax-pre-sup 10204  ax-addf 10205  ax-mulf 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-iin 4673  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-se 5224  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-isom 6056  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-of 7060  df-om 7229  df-1st 7331  df-2nd 7332  df-supp 7462  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-1o 7727  df-2o 7728  df-oadd 7731  df-er 7909  df-map 8023  df-pm 8024  df-ixp 8073  df-en 8120  df-dom 8121  df-sdom 8122  df-fin 8123  df-fsupp 8439  df-fi 8480  df-sup 8511  df-inf 8512  df-oi 8578  df-card 8953  df-cda 9180  df-pnf 10266  df-mnf 10267  df-xr 10268  df-ltxr 10269  df-le 10270  df-sub 10458  df-neg 10459  df-div 10875  df-nn 11211  df-2 11269  df-3 11270  df-4 11271  df-5 11272  df-6 11273  df-7 11274  df-8 11275  df-9 11276  df-n0 11483  df-z 11568  df-dec 11684  df-uz 11878  df-q 11980  df-rp 12024  df-xneg 12137  df-xadd 12138  df-xmul 12139  df-ioo 12370  df-ioc 12371  df-ico 12372  df-icc 12373  df-fz 12518  df-fzo 12658  df-fl 12785  df-mod 12861  df-seq 12994  df-exp 13053  df-fac 13253  df-bc 13282  df-hash 13310  df-shft 14004  df-cj 14036  df-re 14037  df-im 14038  df-sqrt 14172  df-abs 14173  df-limsup 14399  df-clim 14416  df-rlim 14417  df-sum 14614  df-ef 14995  df-sin 14997  df-cos 14998  df-pi 15000  df-struct 16059  df-ndx 16060  df-slot 16061  df-base 16063  df-sets 16064  df-ress 16065  df-plusg 16154  df-mulr 16155  df-starv 16156  df-sca 16157  df-vsca 16158  df-ip 16159  df-tset 16160  df-ple 16161  df-ds 16164  df-unif 16165  df-hom 16166  df-cco 16167  df-rest 16283  df-topn 16284  df-0g 16302  df-gsum 16303  df-topgen 16304  df-pt 16305  df-prds 16308  df-xrs 16362  df-qtop 16367  df-imas 16368  df-xps 16370  df-mre 16446  df-mrc 16447  df-acs 16449  df-mgm 17441  df-sgrp 17483  df-mnd 17494  df-submnd 17535  df-mulg 17740  df-cntz 17948  df-cmn 18393  df-psmet 19938  df-xmet 19939  df-met 19940  df-bl 19941  df-mopn 19942  df-fbas 19943  df-fg 19944  df-cnfld 19947  df-top 20899  df-topon 20916  df-topsp 20937  df-bases 20950  df-cld 21023  df-ntr 21024  df-cls 21025  df-nei 21102  df-lp 21140  df-perf 21141  df-cn 21231  df-cnp 21232  df-haus 21319  df-tx 21565  df-hmeo 21758  df-fil 21849  df-fm 21941  df-flim 21942  df-flf 21943  df-xms 22324  df-ms 22325  df-tms 22326  df-cncf 22880  df-limc 23827  df-dv 23828
This theorem is referenced by:  eff1o  24492
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