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Theorem eff1olem 24536
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 5636 . . . 4 (ℑ “ 𝐷) ⊆ dom ℑ
2 eff1olem.2 . . . 4 𝑆 = (ℑ “ 𝐷)
3 imf 14083 . . . . . 6 ℑ:ℂ⟶ℝ
43fdmi 6206 . . . . 5 dom ℑ = ℂ
54eqcomi 2783 . . . 4 ℂ = dom ℑ
61, 2, 53sstr4i 3800 . . 3 𝑆 ⊆ ℂ
7 eff2 15057 . . . . . . 7 exp:ℂ⟶(ℂ ∖ {0})
87a1i 11 . . . . . 6 (𝑆 ⊆ ℂ → exp:ℂ⟶(ℂ ∖ {0}))
98feqmptd 6408 . . . . 5 (𝑆 ⊆ ℂ → exp = (𝑦 ∈ ℂ ↦ (exp‘𝑦)))
109reseq1d 5545 . . . 4 (𝑆 ⊆ ℂ → (exp ↾ 𝑆) = ((𝑦 ∈ ℂ ↦ (exp‘𝑦)) ↾ 𝑆))
11 resmpt 5600 . . . 4 (𝑆 ⊆ ℂ → ((𝑦 ∈ ℂ ↦ (exp‘𝑦)) ↾ 𝑆) = (𝑦𝑆 ↦ (exp‘𝑦)))
1210, 11eqtrd 2808 . . 3 (𝑆 ⊆ ℂ → (exp ↾ 𝑆) = (𝑦𝑆 ↦ (exp‘𝑦)))
136, 12ax-mp 5 . 2 (exp ↾ 𝑆) = (𝑦𝑆 ↦ (exp‘𝑦))
146sseli 3754 . . . 4 (𝑦𝑆𝑦 ∈ ℂ)
157ffvelrni 6518 . . . 4 (𝑦 ∈ ℂ → (exp‘𝑦) ∈ (ℂ ∖ {0}))
1614, 15syl 17 . . 3 (𝑦𝑆 → (exp‘𝑦) ∈ (ℂ ∖ {0}))
1716adantl 468 . 2 ((𝜑𝑦𝑆) → (exp‘𝑦) ∈ (ℂ ∖ {0}))
18 simpr 472 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ∈ (ℂ ∖ {0}))
19 eldifsn 4464 . . . . . . . . . 10 (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
2018, 19sylib 209 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
2120simpld 483 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
2220simprd 484 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ≠ 0)
2321, 22absrpcld 14417 . . . . . . 7 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℝ+)
24 reeff1o 24442 . . . . . . . . 9 (exp ↾ ℝ):ℝ–1-1-onto→ℝ+
25 f1ocnv 6305 . . . . . . . . 9 ((exp ↾ ℝ):ℝ–1-1-onto→ℝ+(exp ↾ ℝ):ℝ+1-1-onto→ℝ)
26 f1of 6293 . . . . . . . . 9 ((exp ↾ ℝ):ℝ+1-1-onto→ℝ → (exp ↾ ℝ):ℝ+⟶ℝ)
2724, 25, 26mp2b 10 . . . . . . . 8 (exp ↾ ℝ):ℝ+⟶ℝ
2827ffvelrni 6518 . . . . . . 7 ((abs‘𝑥) ∈ ℝ+ → ((exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ)
2923, 28syl 17 . . . . . 6 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ)
3029recnd 10291 . . . . 5 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℂ)
31 ax-icn 10218 . . . . . 6 i ∈ ℂ
32 eff1olem.3 . . . . . . . . 9 (𝜑𝐷 ⊆ ℝ)
3332adantr 467 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝐷 ⊆ ℝ)
34 eff1olem.1 . . . . . . . . . . . 12 𝐹 = (𝑤𝐷 ↦ (exp‘(i · 𝑤)))
35 eqid 2774 . . . . . . . . . . . 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 2774 . . . . . . . . . . . 12 (sin ↾ (-(π / 2)[,](π / 2))) = (sin ↾ (-(π / 2)[,](π / 2)))
3934, 35, 32, 36, 37, 38efif1olem4 24533 . . . . . . . . . . 11 (𝜑𝐹:𝐷1-1-onto→(abs “ {1}))
40 f1ocnv 6305 . . . . . . . . . . 11 (𝐹:𝐷1-1-onto→(abs “ {1}) → 𝐹:(abs “ {1})–1-1-onto𝐷)
41 f1of 6293 . . . . . . . . . . 11 (𝐹:(abs “ {1})–1-1-onto𝐷𝐹:(abs “ {1})⟶𝐷)
4239, 40, 413syl 18 . . . . . . . . . 10 (𝜑𝐹:(abs “ {1})⟶𝐷)
4342adantr 467 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝐹:(abs “ {1})⟶𝐷)
4421abscld 14405 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℝ)
4544recnd 10291 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℂ)
4621, 22absne0d 14416 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ≠ 0)
4721, 45, 46divcld 11024 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝑥 / (abs‘𝑥)) ∈ ℂ)
4821, 45, 46absdivd 14424 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘(𝑥 / (abs‘𝑥))) = ((abs‘𝑥) / (abs‘(abs‘𝑥))))
49 absidm 14293 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (abs‘(abs‘𝑥)) = (abs‘𝑥))
5021, 49syl 17 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘(abs‘𝑥)) = (abs‘𝑥))
5150oveq2d 6828 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) / (abs‘(abs‘𝑥))) = ((abs‘𝑥) / (abs‘𝑥)))
5245, 46dividd 11022 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) / (abs‘𝑥)) = 1)
5348, 51, 523eqtrd 2812 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘(𝑥 / (abs‘𝑥))) = 1)
54 absf 14307 . . . . . . . . . . 11 abs:ℂ⟶ℝ
55 ffn 6196 . . . . . . . . . . 11 (abs:ℂ⟶ℝ → abs Fn ℂ)
56 fniniseg 6498 . . . . . . . . . . 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 573 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝑥 / (abs‘𝑥)) ∈ (abs “ {1}))
5943, 58ffvelrnd 6520 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(𝑥 / (abs‘𝑥))) ∈ 𝐷)
6033, 59sseldd 3759 . . . . . . 7 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℝ)
6160recnd 10291 . . . . . 6 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℂ)
62 mulcl 10243 . . . . . 6 ((i ∈ ℂ ∧ (𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℂ) → (i · (𝐹‘(𝑥 / (abs‘𝑥)))) ∈ ℂ)
6331, 61, 62sylancr 576 . . . . 5 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (i · (𝐹‘(𝑥 / (abs‘𝑥)))) ∈ ℂ)
6430, 63addcld 10282 . . . 4 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) ∈ ℂ)
6529, 60crimd 14202 . . . . 5 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (ℑ‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))) = (𝐹‘(𝑥 / (abs‘𝑥))))
6665, 59eqeltrd 2853 . . . 4 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (ℑ‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))) ∈ 𝐷)
67 ffn 6196 . . . . 5 (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
68 elpreima 6497 . . . . 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 573 . . 3 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) ∈ (ℑ “ 𝐷))
7170, 2syl6eleqr 2864 . 2 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) ∈ 𝑆)
72 efadd 15052 . . . . . . 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 574 . . . . . 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 6365 . . . . . . . . 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 6695 . . . . . . . . 9 (((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ ∧ (abs‘𝑥) ∈ ℝ+) → ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
7724, 23, 76sylancr 576 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
7875, 77eqtr3d 2810 . . . . . . 7 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (exp‘((exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
79 oveq2 6820 . . . . . . . . . . 11 (𝑧 = (𝐹‘(𝑥 / (abs‘𝑥))) → (i · 𝑧) = (i · (𝐹‘(𝑥 / (abs‘𝑥)))))
8079fveq2d 6352 . . . . . . . . . 10 (𝑧 = (𝐹‘(𝑥 / (abs‘𝑥))) → (exp‘(i · 𝑧)) = (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥))))))
81 oveq2 6820 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (i · 𝑤) = (i · 𝑧))
8281fveq2d 6352 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (exp‘(i · 𝑤)) = (exp‘(i · 𝑧)))
8382cbvmptv 4897 . . . . . . . . . . 11 (𝑤𝐷 ↦ (exp‘(i · 𝑤))) = (𝑧𝐷 ↦ (exp‘(i · 𝑧)))
8434, 83eqtri 2796 . . . . . . . . . 10 𝐹 = (𝑧𝐷 ↦ (exp‘(i · 𝑧)))
85 fvex 6359 . . . . . . . . . 10 (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥))))) ∈ V
8680, 84, 85fvmpt 6441 . . . . . . . . 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 467 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝐹:𝐷1-1-onto→(abs “ {1}))
89 f1ocnvfv2 6695 . . . . . . . . 9 ((𝐹:𝐷1-1-onto→(abs “ {1}) ∧ (𝑥 / (abs‘𝑥)) ∈ (abs “ {1})) → (𝐹‘(𝐹‘(𝑥 / (abs‘𝑥)))) = (𝑥 / (abs‘𝑥)))
9088, 58, 89syl2anc 574 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(𝐹‘(𝑥 / (abs‘𝑥)))) = (𝑥 / (abs‘𝑥)))
9187, 90eqtr3d 2810 . . . . . . 7 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥))))) = (𝑥 / (abs‘𝑥)))
9278, 91oveq12d 6830 . . . . . 6 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((exp‘((exp ↾ ℝ)‘(abs‘𝑥))) · (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥)))))) = ((abs‘𝑥) · (𝑥 / (abs‘𝑥))))
9321, 45, 46divcan2d 11026 . . . . . 6 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) · (𝑥 / (abs‘𝑥))) = 𝑥)
9473, 92, 933eqtrrd 2813 . . . . 5 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝑥 = (exp‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
9594adantrl 696 . . . 4 ((𝜑 ∧ (𝑦𝑆𝑥 ∈ (ℂ ∖ {0}))) → 𝑥 = (exp‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
96 fveq2 6348 . . . . 5 (𝑦 = (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) → (exp‘𝑦) = (exp‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
9796eqeq2d 2784 . . . 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 238 . . 3 ((𝜑 ∧ (𝑦𝑆𝑥 ∈ (ℂ ∖ {0}))) → (𝑦 = (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) → 𝑥 = (exp‘𝑦)))
9914adantl 468 . . . . . . 7 ((𝜑𝑦𝑆) → 𝑦 ∈ ℂ)
10099replimd 14167 . . . . . 6 ((𝜑𝑦𝑆) → 𝑦 = ((ℜ‘𝑦) + (i · (ℑ‘𝑦))))
101 absef 15155 . . . . . . . . . . 11 (𝑦 ∈ ℂ → (abs‘(exp‘𝑦)) = (exp‘(ℜ‘𝑦)))
10299, 101syl 17 . . . . . . . . . 10 ((𝜑𝑦𝑆) → (abs‘(exp‘𝑦)) = (exp‘(ℜ‘𝑦)))
10399recld 14164 . . . . . . . . . . 11 ((𝜑𝑦𝑆) → (ℜ‘𝑦) ∈ ℝ)
104 fvres 6365 . . . . . . . . . . 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 2811 . . . . . . . . 9 ((𝜑𝑦𝑆) → (abs‘(exp‘𝑦)) = ((exp ↾ ℝ)‘(ℜ‘𝑦)))
107106fveq2d 6352 . . . . . . . 8 ((𝜑𝑦𝑆) → ((exp ↾ ℝ)‘(abs‘(exp‘𝑦))) = ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))))
108 f1ocnvfv1 6694 . . . . . . . . 9 (((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ ∧ (ℜ‘𝑦) ∈ ℝ) → ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))) = (ℜ‘𝑦))
10924, 103, 108sylancr 576 . . . . . . . 8 ((𝜑𝑦𝑆) → ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))) = (ℜ‘𝑦))
110107, 109eqtrd 2808 . . . . . . 7 ((𝜑𝑦𝑆) → ((exp ↾ ℝ)‘(abs‘(exp‘𝑦))) = (ℜ‘𝑦))
11199imcld 14165 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝑆) → (ℑ‘𝑦) ∈ ℝ)
112111recnd 10291 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑆) → (ℑ‘𝑦) ∈ ℂ)
113 mulcl 10243 . . . . . . . . . . . . . 14 ((i ∈ ℂ ∧ (ℑ‘𝑦) ∈ ℂ) → (i · (ℑ‘𝑦)) ∈ ℂ)
11431, 112, 113sylancr 576 . . . . . . . . . . . . 13 ((𝜑𝑦𝑆) → (i · (ℑ‘𝑦)) ∈ ℂ)
115 efcl 15041 . . . . . . . . . . . . 13 ((i · (ℑ‘𝑦)) ∈ ℂ → (exp‘(i · (ℑ‘𝑦))) ∈ ℂ)
116114, 115syl 17 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (exp‘(i · (ℑ‘𝑦))) ∈ ℂ)
117103recnd 10291 . . . . . . . . . . . . 13 ((𝜑𝑦𝑆) → (ℜ‘𝑦) ∈ ℂ)
118 efcl 15041 . . . . . . . . . . . . 13 ((ℜ‘𝑦) ∈ ℂ → (exp‘(ℜ‘𝑦)) ∈ ℂ)
119117, 118syl 17 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (exp‘(ℜ‘𝑦)) ∈ ℂ)
120 efne0 15055 . . . . . . . . . . . . 13 ((ℜ‘𝑦) ∈ ℂ → (exp‘(ℜ‘𝑦)) ≠ 0)
121117, 120syl 17 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (exp‘(ℜ‘𝑦)) ≠ 0)
122116, 119, 121divcan3d 11029 . . . . . . . . . . 11 ((𝜑𝑦𝑆) → (((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))) / (exp‘(ℜ‘𝑦))) = (exp‘(i · (ℑ‘𝑦))))
123100fveq2d 6352 . . . . . . . . . . . . 13 ((𝜑𝑦𝑆) → (exp‘𝑦) = (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))))
124 efadd 15052 . . . . . . . . . . . . . 14 (((ℜ‘𝑦) ∈ ℂ ∧ (i · (ℑ‘𝑦)) ∈ ℂ) → (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
125117, 114, 124syl2anc 574 . . . . . . . . . . . . 13 ((𝜑𝑦𝑆) → (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
126123, 125eqtrd 2808 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (exp‘𝑦) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
127126, 102oveq12d 6830 . . . . . . . . . . 11 ((𝜑𝑦𝑆) → ((exp‘𝑦) / (abs‘(exp‘𝑦))) = (((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))) / (exp‘(ℜ‘𝑦))))
128 elpreima 6497 . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . 14 (𝑦 ∈ (ℑ “ 𝐷) → (ℑ‘𝑦) ∈ 𝐷)
131130, 2eleq2s 2871 . . . . . . . . . . . . 13 (𝑦𝑆 → (ℑ‘𝑦) ∈ 𝐷)
132131adantl 468 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (ℑ‘𝑦) ∈ 𝐷)
133 oveq2 6820 . . . . . . . . . . . . . 14 (𝑤 = (ℑ‘𝑦) → (i · 𝑤) = (i · (ℑ‘𝑦)))
134133fveq2d 6352 . . . . . . . . . . . . 13 (𝑤 = (ℑ‘𝑦) → (exp‘(i · 𝑤)) = (exp‘(i · (ℑ‘𝑦))))
135 fvex 6359 . . . . . . . . . . . . 13 (exp‘(i · (ℑ‘𝑦))) ∈ V
136134, 34, 135fvmpt 6441 . . . . . . . . . . . 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 2818 . . . . . . . . . 10 ((𝜑𝑦𝑆) → ((exp‘𝑦) / (abs‘(exp‘𝑦))) = (𝐹‘(ℑ‘𝑦)))
139138fveq2d 6352 . . . . . . . . 9 ((𝜑𝑦𝑆) → (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))) = (𝐹‘(𝐹‘(ℑ‘𝑦))))
140 f1ocnvfv1 6694 . . . . . . . . . 10 ((𝐹:𝐷1-1-onto→(abs “ {1}) ∧ (ℑ‘𝑦) ∈ 𝐷) → (𝐹‘(𝐹‘(ℑ‘𝑦))) = (ℑ‘𝑦))
14139, 131, 140syl2an 584 . . . . . . . . 9 ((𝜑𝑦𝑆) → (𝐹‘(𝐹‘(ℑ‘𝑦))) = (ℑ‘𝑦))
142139, 141eqtrd 2808 . . . . . . . 8 ((𝜑𝑦𝑆) → (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))) = (ℑ‘𝑦))
143142oveq2d 6828 . . . . . . 7 ((𝜑𝑦𝑆) → (i · (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦))))) = (i · (ℑ‘𝑦)))
144110, 143oveq12d 6830 . . . . . 6 ((𝜑𝑦𝑆) → (((exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))) = ((ℜ‘𝑦) + (i · (ℑ‘𝑦))))
145100, 144eqtr4d 2811 . . . . 5 ((𝜑𝑦𝑆) → 𝑦 = (((exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))))
146 fveq2 6348 . . . . . . . 8 (𝑥 = (exp‘𝑦) → (abs‘𝑥) = (abs‘(exp‘𝑦)))
147146fveq2d 6352 . . . . . . 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 6830 . . . . . . . . 9 (𝑥 = (exp‘𝑦) → (𝑥 / (abs‘𝑥)) = ((exp‘𝑦) / (abs‘(exp‘𝑦))))
150149fveq2d 6352 . . . . . . . 8 (𝑥 = (exp‘𝑦) → (𝐹‘(𝑥 / (abs‘𝑥))) = (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))
151150oveq2d 6828 . . . . . . 7 (𝑥 = (exp‘𝑦) → (i · (𝐹‘(𝑥 / (abs‘𝑥)))) = (i · (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦))))))
152147, 151oveq12d 6830 . . . . . 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 2784 . . . . 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 238 . . . 4 ((𝜑𝑦𝑆) → (𝑥 = (exp‘𝑦) → 𝑦 = (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
155154adantrr 697 . . 3 ((𝜑 ∧ (𝑦𝑆𝑥 ∈ (ℂ ∖ {0}))) → (𝑥 = (exp‘𝑦) → 𝑦 = (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
15698, 155impbid 203 . 2 ((𝜑 ∧ (𝑦𝑆𝑥 ∈ (ℂ ∖ {0}))) → (𝑦 = (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) ↔ 𝑥 = (exp‘𝑦)))
15713, 17, 71, 156f1o2d 7055 1 (𝜑 → (exp ↾ 𝑆):𝑆1-1-onto→(ℂ ∖ {0}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 383   = wceq 1634  wcel 2148  wne 2946  wrex 3065  cdif 3726  wss 3729  {csn 4326   class class class wbr 4797  cmpt 4876  ccnv 5262  dom cdm 5263  cres 5265  cima 5266   Fn wfn 6037  wf 6038  1-1-ontowf1o 6041  cfv 6042  (class class class)co 6812  cc 10157  cr 10158  0cc0 10159  1c1 10160  ici 10161   + caddc 10162   · cmul 10164   < clt 10297  cmin 10489  -cneg 10490   / cdiv 10907  2c2 11293  cz 11601  +crp 12052  [,]cicc 12402  cre 14067  cim 14068  abscabs 14204  expce 15020  sincsin 15022  πcpi 15025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-rep 4917  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117  ax-inf2 8723  ax-cnex 10215  ax-resscn 10216  ax-1cn 10217  ax-icn 10218  ax-addcl 10219  ax-addrcl 10220  ax-mulcl 10221  ax-mulrcl 10222  ax-mulcom 10223  ax-addass 10224  ax-mulass 10225  ax-distr 10226  ax-i2m1 10227  ax-1ne0 10228  ax-1rid 10229  ax-rnegex 10230  ax-rrecex 10231  ax-cnre 10232  ax-pre-lttri 10233  ax-pre-lttrn 10234  ax-pre-ltadd 10235  ax-pre-mulgt0 10236  ax-pre-sup 10237  ax-addf 10238  ax-mulf 10239
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-fal 1640  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-int 4623  df-iun 4667  df-iin 4668  df-br 4798  df-opab 4860  df-mpt 4877  df-tr 4900  df-id 5171  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-se 5223  df-we 5224  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-pred 5834  df-ord 5880  df-on 5881  df-lim 5882  df-suc 5883  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-isom 6051  df-riota 6773  df-ov 6815  df-oprab 6816  df-mpt2 6817  df-of 7065  df-om 7234  df-1st 7336  df-2nd 7337  df-supp 7468  df-wrecs 7580  df-recs 7642  df-rdg 7680  df-1o 7734  df-2o 7735  df-oadd 7738  df-er 7917  df-map 8032  df-pm 8033  df-ixp 8084  df-en 8131  df-dom 8132  df-sdom 8133  df-fin 8134  df-fsupp 8453  df-fi 8494  df-sup 8525  df-inf 8526  df-oi 8592  df-card 8986  df-cda 9213  df-pnf 10299  df-mnf 10300  df-xr 10301  df-ltxr 10302  df-le 10303  df-sub 10491  df-neg 10492  df-div 10908  df-nn 11244  df-2 11302  df-3 11303  df-4 11304  df-5 11305  df-6 11306  df-7 11307  df-8 11308  df-9 11309  df-n0 11517  df-z 11602  df-dec 11718  df-uz 11911  df-q 12014  df-rp 12053  df-xneg 12168  df-xadd 12169  df-xmul 12170  df-ioo 12403  df-ioc 12404  df-ico 12405  df-icc 12406  df-fz 12556  df-fzo 12696  df-fl 12823  df-mod 12899  df-seq 13031  df-exp 13090  df-fac 13287  df-bc 13316  df-hash 13344  df-shft 14037  df-cj 14069  df-re 14070  df-im 14071  df-sqrt 14205  df-abs 14206  df-limsup 14432  df-clim 14449  df-rlim 14450  df-sum 14647  df-ef 15026  df-sin 15028  df-cos 15029  df-pi 15031  df-struct 16086  df-ndx 16087  df-slot 16088  df-base 16090  df-sets 16091  df-ress 16092  df-plusg 16182  df-mulr 16183  df-starv 16184  df-sca 16185  df-vsca 16186  df-ip 16187  df-tset 16188  df-ple 16189  df-ds 16192  df-unif 16193  df-hom 16194  df-cco 16195  df-rest 16311  df-topn 16312  df-0g 16330  df-gsum 16331  df-topgen 16332  df-pt 16333  df-prds 16336  df-xrs 16390  df-qtop 16395  df-imas 16396  df-xps 16398  df-mre 16474  df-mrc 16475  df-acs 16477  df-mgm 17470  df-sgrp 17512  df-mnd 17523  df-submnd 17564  df-mulg 17769  df-cntz 17977  df-cmn 18422  df-psmet 19973  df-xmet 19974  df-met 19975  df-bl 19976  df-mopn 19977  df-fbas 19978  df-fg 19979  df-cnfld 19982  df-top 20939  df-topon 20956  df-topsp 20978  df-bases 20991  df-cld 21064  df-ntr 21065  df-cls 21066  df-nei 21143  df-lp 21181  df-perf 21182  df-cn 21272  df-cnp 21273  df-haus 21360  df-tx 21606  df-hmeo 21799  df-fil 21890  df-fm 21982  df-flim 21983  df-flf 21984  df-xms 22365  df-ms 22366  df-tms 22367  df-cncf 22921  df-limc 23871  df-dv 23872
This theorem is referenced by:  eff1o  24537
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