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Theorem eff1olem 26590
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 6100 . . . 4 (ℑ “ 𝐷) ⊆ dom ℑ
2 eff1olem.2 . . . 4 𝑆 = (ℑ “ 𝐷)
3 imf 15152 . . . . . 6 ℑ:ℂ⟶ℝ
43fdmi 6747 . . . . 5 dom ℑ = ℂ
54eqcomi 2746 . . . 4 ℂ = dom ℑ
61, 2, 53sstr4i 4035 . . 3 𝑆 ⊆ ℂ
7 eff2 16135 . . . . . . 7 exp:ℂ⟶(ℂ ∖ {0})
87a1i 11 . . . . . 6 (𝑆 ⊆ ℂ → exp:ℂ⟶(ℂ ∖ {0}))
98feqmptd 6977 . . . . 5 (𝑆 ⊆ ℂ → exp = (𝑦 ∈ ℂ ↦ (exp‘𝑦)))
109reseq1d 5996 . . . 4 (𝑆 ⊆ ℂ → (exp ↾ 𝑆) = ((𝑦 ∈ ℂ ↦ (exp‘𝑦)) ↾ 𝑆))
11 resmpt 6055 . . . 4 (𝑆 ⊆ ℂ → ((𝑦 ∈ ℂ ↦ (exp‘𝑦)) ↾ 𝑆) = (𝑦𝑆 ↦ (exp‘𝑦)))
1210, 11eqtrd 2777 . . 3 (𝑆 ⊆ ℂ → (exp ↾ 𝑆) = (𝑦𝑆 ↦ (exp‘𝑦)))
136, 12ax-mp 5 . 2 (exp ↾ 𝑆) = (𝑦𝑆 ↦ (exp‘𝑦))
146sseli 3979 . . . 4 (𝑦𝑆𝑦 ∈ ℂ)
157ffvelcdmi 7103 . . . 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 484 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ∈ (ℂ ∖ {0}))
19 eldifsn 4786 . . . . . . . . . 10 (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
2018, 19sylib 218 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
2120simpld 494 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
2220simprd 495 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ≠ 0)
2321, 22absrpcld 15487 . . . . . . 7 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℝ+)
24 reeff1o 26491 . . . . . . . . 9 (exp ↾ ℝ):ℝ–1-1-onto→ℝ+
25 f1ocnv 6860 . . . . . . . . 9 ((exp ↾ ℝ):ℝ–1-1-onto→ℝ+(exp ↾ ℝ):ℝ+1-1-onto→ℝ)
26 f1of 6848 . . . . . . . . 9 ((exp ↾ ℝ):ℝ+1-1-onto→ℝ → (exp ↾ ℝ):ℝ+⟶ℝ)
2724, 25, 26mp2b 10 . . . . . . . 8 (exp ↾ ℝ):ℝ+⟶ℝ
2827ffvelcdmi 7103 . . . . . . 7 ((abs‘𝑥) ∈ ℝ+ → ((exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ)
2923, 28syl 17 . . . . . 6 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ)
3029recnd 11289 . . . . 5 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℂ)
31 ax-icn 11214 . . . . . 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 2737 . . . . . . . . . . . 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 2737 . . . . . . . . . . . 12 (sin ↾ (-(π / 2)[,](π / 2))) = (sin ↾ (-(π / 2)[,](π / 2)))
3934, 35, 32, 36, 37, 38efif1olem4 26587 . . . . . . . . . . 11 (𝜑𝐹:𝐷1-1-onto→(abs “ {1}))
40 f1ocnv 6860 . . . . . . . . . . 11 (𝐹:𝐷1-1-onto→(abs “ {1}) → 𝐹:(abs “ {1})–1-1-onto𝐷)
41 f1of 6848 . . . . . . . . . . 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 15475 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℝ)
4544recnd 11289 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℂ)
4621, 22absne0d 15486 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ≠ 0)
4721, 45, 46divcld 12043 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝑥 / (abs‘𝑥)) ∈ ℂ)
4821, 45, 46absdivd 15494 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘(𝑥 / (abs‘𝑥))) = ((abs‘𝑥) / (abs‘(abs‘𝑥))))
49 absidm 15362 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (abs‘(abs‘𝑥)) = (abs‘𝑥))
5021, 49syl 17 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘(abs‘𝑥)) = (abs‘𝑥))
5150oveq2d 7447 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) / (abs‘(abs‘𝑥))) = ((abs‘𝑥) / (abs‘𝑥)))
5245, 46dividd 12041 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) / (abs‘𝑥)) = 1)
5348, 51, 523eqtrd 2781 . . . . . . . . . 10 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (abs‘(𝑥 / (abs‘𝑥))) = 1)
54 absf 15376 . . . . . . . . . . 11 abs:ℂ⟶ℝ
55 ffn 6736 . . . . . . . . . . 11 (abs:ℂ⟶ℝ → abs Fn ℂ)
56 fniniseg 7080 . . . . . . . . . . 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 583 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝑥 / (abs‘𝑥)) ∈ (abs “ {1}))
5943, 58ffvelcdmd 7105 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(𝑥 / (abs‘𝑥))) ∈ 𝐷)
6033, 59sseldd 3984 . . . . . . 7 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℝ)
6160recnd 11289 . . . . . 6 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℂ)
62 mulcl 11239 . . . . . 6 ((i ∈ ℂ ∧ (𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℂ) → (i · (𝐹‘(𝑥 / (abs‘𝑥)))) ∈ ℂ)
6331, 61, 62sylancr 587 . . . . 5 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (i · (𝐹‘(𝑥 / (abs‘𝑥)))) ∈ ℂ)
6430, 63addcld 11280 . . . 4 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) ∈ ℂ)
6529, 60crimd 15271 . . . . 5 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (ℑ‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))) = (𝐹‘(𝑥 / (abs‘𝑥))))
6665, 59eqeltrd 2841 . . . 4 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (ℑ‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))) ∈ 𝐷)
67 ffn 6736 . . . . 5 (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
68 elpreima 7078 . . . . 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 583 . . 3 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) ∈ (ℑ “ 𝐷))
7170, 2eleqtrrdi 2852 . 2 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) ∈ 𝑆)
72 efadd 16130 . . . . . . 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 584 . . . . . 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)))))))
7429fvresd 6926 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(abs‘𝑥))) = (exp‘((exp ↾ ℝ)‘(abs‘𝑥))))
75 f1ocnvfv2 7297 . . . . . . . . 9 (((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ ∧ (abs‘𝑥) ∈ ℝ+) → ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
7624, 23, 75sylancr 587 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
7774, 76eqtr3d 2779 . . . . . . 7 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (exp‘((exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
78 oveq2 7439 . . . . . . . . . . 11 (𝑧 = (𝐹‘(𝑥 / (abs‘𝑥))) → (i · 𝑧) = (i · (𝐹‘(𝑥 / (abs‘𝑥)))))
7978fveq2d 6910 . . . . . . . . . 10 (𝑧 = (𝐹‘(𝑥 / (abs‘𝑥))) → (exp‘(i · 𝑧)) = (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥))))))
80 oveq2 7439 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (i · 𝑤) = (i · 𝑧))
8180fveq2d 6910 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (exp‘(i · 𝑤)) = (exp‘(i · 𝑧)))
8281cbvmptv 5255 . . . . . . . . . . 11 (𝑤𝐷 ↦ (exp‘(i · 𝑤))) = (𝑧𝐷 ↦ (exp‘(i · 𝑧)))
8334, 82eqtri 2765 . . . . . . . . . 10 𝐹 = (𝑧𝐷 ↦ (exp‘(i · 𝑧)))
84 fvex 6919 . . . . . . . . . 10 (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥))))) ∈ V
8579, 83, 84fvmpt 7016 . . . . . . . . 9 ((𝐹‘(𝑥 / (abs‘𝑥))) ∈ 𝐷 → (𝐹‘(𝐹‘(𝑥 / (abs‘𝑥)))) = (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥))))))
8659, 85syl 17 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(𝐹‘(𝑥 / (abs‘𝑥)))) = (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥))))))
8739adantr 480 . . . . . . . . 9 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝐹:𝐷1-1-onto→(abs “ {1}))
88 f1ocnvfv2 7297 . . . . . . . . 9 ((𝐹:𝐷1-1-onto→(abs “ {1}) ∧ (𝑥 / (abs‘𝑥)) ∈ (abs “ {1})) → (𝐹‘(𝐹‘(𝑥 / (abs‘𝑥)))) = (𝑥 / (abs‘𝑥)))
8987, 58, 88syl2anc 584 . . . . . . . 8 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(𝐹‘(𝑥 / (abs‘𝑥)))) = (𝑥 / (abs‘𝑥)))
9086, 89eqtr3d 2779 . . . . . . 7 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥))))) = (𝑥 / (abs‘𝑥)))
9177, 90oveq12d 7449 . . . . . 6 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((exp‘((exp ↾ ℝ)‘(abs‘𝑥))) · (exp‘(i · (𝐹‘(𝑥 / (abs‘𝑥)))))) = ((abs‘𝑥) · (𝑥 / (abs‘𝑥))))
9221, 45, 46divcan2d 12045 . . . . . 6 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) · (𝑥 / (abs‘𝑥))) = 𝑥)
9373, 91, 923eqtrrd 2782 . . . . 5 ((𝜑𝑥 ∈ (ℂ ∖ {0})) → 𝑥 = (exp‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
9493adantrl 716 . . . 4 ((𝜑 ∧ (𝑦𝑆𝑥 ∈ (ℂ ∖ {0}))) → 𝑥 = (exp‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
95 fveq2 6906 . . . . 5 (𝑦 = (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) → (exp‘𝑦) = (exp‘(((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
9695eqeq2d 2748 . . . 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))))))))
9794, 96syl5ibrcom 247 . . 3 ((𝜑 ∧ (𝑦𝑆𝑥 ∈ (ℂ ∖ {0}))) → (𝑦 = (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) → 𝑥 = (exp‘𝑦)))
9814adantl 481 . . . . . . 7 ((𝜑𝑦𝑆) → 𝑦 ∈ ℂ)
9998replimd 15236 . . . . . 6 ((𝜑𝑦𝑆) → 𝑦 = ((ℜ‘𝑦) + (i · (ℑ‘𝑦))))
100 absef 16233 . . . . . . . . . . 11 (𝑦 ∈ ℂ → (abs‘(exp‘𝑦)) = (exp‘(ℜ‘𝑦)))
10198, 100syl 17 . . . . . . . . . 10 ((𝜑𝑦𝑆) → (abs‘(exp‘𝑦)) = (exp‘(ℜ‘𝑦)))
10298recld 15233 . . . . . . . . . . 11 ((𝜑𝑦𝑆) → (ℜ‘𝑦) ∈ ℝ)
103102fvresd 6926 . . . . . . . . . 10 ((𝜑𝑦𝑆) → ((exp ↾ ℝ)‘(ℜ‘𝑦)) = (exp‘(ℜ‘𝑦)))
104101, 103eqtr4d 2780 . . . . . . . . 9 ((𝜑𝑦𝑆) → (abs‘(exp‘𝑦)) = ((exp ↾ ℝ)‘(ℜ‘𝑦)))
105104fveq2d 6910 . . . . . . . 8 ((𝜑𝑦𝑆) → ((exp ↾ ℝ)‘(abs‘(exp‘𝑦))) = ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))))
106 f1ocnvfv1 7296 . . . . . . . . 9 (((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ ∧ (ℜ‘𝑦) ∈ ℝ) → ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))) = (ℜ‘𝑦))
10724, 102, 106sylancr 587 . . . . . . . 8 ((𝜑𝑦𝑆) → ((exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))) = (ℜ‘𝑦))
108105, 107eqtrd 2777 . . . . . . 7 ((𝜑𝑦𝑆) → ((exp ↾ ℝ)‘(abs‘(exp‘𝑦))) = (ℜ‘𝑦))
10998imcld 15234 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝑆) → (ℑ‘𝑦) ∈ ℝ)
110109recnd 11289 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑆) → (ℑ‘𝑦) ∈ ℂ)
111 mulcl 11239 . . . . . . . . . . . . . 14 ((i ∈ ℂ ∧ (ℑ‘𝑦) ∈ ℂ) → (i · (ℑ‘𝑦)) ∈ ℂ)
11231, 110, 111sylancr 587 . . . . . . . . . . . . 13 ((𝜑𝑦𝑆) → (i · (ℑ‘𝑦)) ∈ ℂ)
113 efcl 16118 . . . . . . . . . . . . 13 ((i · (ℑ‘𝑦)) ∈ ℂ → (exp‘(i · (ℑ‘𝑦))) ∈ ℂ)
114112, 113syl 17 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (exp‘(i · (ℑ‘𝑦))) ∈ ℂ)
115102recnd 11289 . . . . . . . . . . . . 13 ((𝜑𝑦𝑆) → (ℜ‘𝑦) ∈ ℂ)
116 efcl 16118 . . . . . . . . . . . . 13 ((ℜ‘𝑦) ∈ ℂ → (exp‘(ℜ‘𝑦)) ∈ ℂ)
117115, 116syl 17 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (exp‘(ℜ‘𝑦)) ∈ ℂ)
118 efne0 16133 . . . . . . . . . . . . 13 ((ℜ‘𝑦) ∈ ℂ → (exp‘(ℜ‘𝑦)) ≠ 0)
119115, 118syl 17 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (exp‘(ℜ‘𝑦)) ≠ 0)
120114, 117, 119divcan3d 12048 . . . . . . . . . . 11 ((𝜑𝑦𝑆) → (((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))) / (exp‘(ℜ‘𝑦))) = (exp‘(i · (ℑ‘𝑦))))
12199fveq2d 6910 . . . . . . . . . . . . 13 ((𝜑𝑦𝑆) → (exp‘𝑦) = (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))))
122 efadd 16130 . . . . . . . . . . . . . 14 (((ℜ‘𝑦) ∈ ℂ ∧ (i · (ℑ‘𝑦)) ∈ ℂ) → (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
123115, 112, 122syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑦𝑆) → (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
124121, 123eqtrd 2777 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (exp‘𝑦) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
125124, 101oveq12d 7449 . . . . . . . . . . 11 ((𝜑𝑦𝑆) → ((exp‘𝑦) / (abs‘(exp‘𝑦))) = (((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))) / (exp‘(ℜ‘𝑦))))
126 elpreima 7078 . . . . . . . . . . . . . . . 16 (ℑ Fn ℂ → (𝑦 ∈ (ℑ “ 𝐷) ↔ (𝑦 ∈ ℂ ∧ (ℑ‘𝑦) ∈ 𝐷)))
1273, 67, 126mp2b 10 . . . . . . . . . . . . . . 15 (𝑦 ∈ (ℑ “ 𝐷) ↔ (𝑦 ∈ ℂ ∧ (ℑ‘𝑦) ∈ 𝐷))
128127simprbi 496 . . . . . . . . . . . . . 14 (𝑦 ∈ (ℑ “ 𝐷) → (ℑ‘𝑦) ∈ 𝐷)
129128, 2eleq2s 2859 . . . . . . . . . . . . 13 (𝑦𝑆 → (ℑ‘𝑦) ∈ 𝐷)
130129adantl 481 . . . . . . . . . . . 12 ((𝜑𝑦𝑆) → (ℑ‘𝑦) ∈ 𝐷)
131 oveq2 7439 . . . . . . . . . . . . . 14 (𝑤 = (ℑ‘𝑦) → (i · 𝑤) = (i · (ℑ‘𝑦)))
132131fveq2d 6910 . . . . . . . . . . . . 13 (𝑤 = (ℑ‘𝑦) → (exp‘(i · 𝑤)) = (exp‘(i · (ℑ‘𝑦))))
133 fvex 6919 . . . . . . . . . . . . 13 (exp‘(i · (ℑ‘𝑦))) ∈ V
134132, 34, 133fvmpt 7016 . . . . . . . . . . . 12 ((ℑ‘𝑦) ∈ 𝐷 → (𝐹‘(ℑ‘𝑦)) = (exp‘(i · (ℑ‘𝑦))))
135130, 134syl 17 . . . . . . . . . . 11 ((𝜑𝑦𝑆) → (𝐹‘(ℑ‘𝑦)) = (exp‘(i · (ℑ‘𝑦))))
136120, 125, 1353eqtr4d 2787 . . . . . . . . . 10 ((𝜑𝑦𝑆) → ((exp‘𝑦) / (abs‘(exp‘𝑦))) = (𝐹‘(ℑ‘𝑦)))
137136fveq2d 6910 . . . . . . . . 9 ((𝜑𝑦𝑆) → (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))) = (𝐹‘(𝐹‘(ℑ‘𝑦))))
138 f1ocnvfv1 7296 . . . . . . . . . 10 ((𝐹:𝐷1-1-onto→(abs “ {1}) ∧ (ℑ‘𝑦) ∈ 𝐷) → (𝐹‘(𝐹‘(ℑ‘𝑦))) = (ℑ‘𝑦))
13939, 129, 138syl2an 596 . . . . . . . . 9 ((𝜑𝑦𝑆) → (𝐹‘(𝐹‘(ℑ‘𝑦))) = (ℑ‘𝑦))
140137, 139eqtrd 2777 . . . . . . . 8 ((𝜑𝑦𝑆) → (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))) = (ℑ‘𝑦))
141140oveq2d 7447 . . . . . . 7 ((𝜑𝑦𝑆) → (i · (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦))))) = (i · (ℑ‘𝑦)))
142108, 141oveq12d 7449 . . . . . 6 ((𝜑𝑦𝑆) → (((exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))) = ((ℜ‘𝑦) + (i · (ℑ‘𝑦))))
14399, 142eqtr4d 2780 . . . . 5 ((𝜑𝑦𝑆) → 𝑦 = (((exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))))
144 fveq2 6906 . . . . . . . 8 (𝑥 = (exp‘𝑦) → (abs‘𝑥) = (abs‘(exp‘𝑦)))
145144fveq2d 6910 . . . . . . 7 (𝑥 = (exp‘𝑦) → ((exp ↾ ℝ)‘(abs‘𝑥)) = ((exp ↾ ℝ)‘(abs‘(exp‘𝑦))))
146 id 22 . . . . . . . . . 10 (𝑥 = (exp‘𝑦) → 𝑥 = (exp‘𝑦))
147146, 144oveq12d 7449 . . . . . . . . 9 (𝑥 = (exp‘𝑦) → (𝑥 / (abs‘𝑥)) = ((exp‘𝑦) / (abs‘(exp‘𝑦))))
148147fveq2d 6910 . . . . . . . 8 (𝑥 = (exp‘𝑦) → (𝐹‘(𝑥 / (abs‘𝑥))) = (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))
149148oveq2d 7447 . . . . . . 7 (𝑥 = (exp‘𝑦) → (i · (𝐹‘(𝑥 / (abs‘𝑥)))) = (i · (𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦))))))
150145, 149oveq12d 7449 . . . . . 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)))))))
151150eqeq2d 2748 . . . . 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))))))))
152143, 151syl5ibrcom 247 . . . 4 ((𝜑𝑦𝑆) → (𝑥 = (exp‘𝑦) → 𝑦 = (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
153152adantrr 717 . . 3 ((𝜑 ∧ (𝑦𝑆𝑥 ∈ (ℂ ∖ {0}))) → (𝑥 = (exp‘𝑦) → 𝑦 = (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥)))))))
15497, 153impbid 212 . 2 ((𝜑 ∧ (𝑦𝑆𝑥 ∈ (ℂ ∖ {0}))) → (𝑦 = (((exp ↾ ℝ)‘(abs‘𝑥)) + (i · (𝐹‘(𝑥 / (abs‘𝑥))))) ↔ 𝑥 = (exp‘𝑦)))
15513, 17, 71, 154f1o2d 7687 1 (𝜑 → (exp ↾ 𝑆):𝑆1-1-onto→(ℂ ∖ {0}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wrex 3070  cdif 3948  wss 3951  {csn 4626   class class class wbr 5143  cmpt 5225  ccnv 5684  dom cdm 5685  cres 5687  cima 5688   Fn wfn 6556  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  cc 11153  cr 11154  0cc0 11155  1c1 11156  ici 11157   + caddc 11158   · cmul 11160   < clt 11295  cmin 11492  -cneg 11493   / cdiv 11920  2c2 12321  cz 12613  +crp 13034  [,]cicc 13390  cre 15136  cim 15137  abscabs 15273  expce 16097  sincsin 16099  πcpi 16102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ioc 13392  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-fac 14313  df-bc 14342  df-hash 14370  df-shft 15106  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-limsup 15507  df-clim 15524  df-rlim 15525  df-sum 15723  df-ef 16103  df-sin 16105  df-cos 16106  df-pi 16108  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17467  df-topn 17468  df-0g 17486  df-gsum 17487  df-topgen 17488  df-pt 17489  df-prds 17492  df-xrs 17547  df-qtop 17552  df-imas 17553  df-xps 17555  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-mulg 19086  df-cntz 19335  df-cmn 19800  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-fbas 21361  df-fg 21362  df-cnfld 21365  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-lp 23144  df-perf 23145  df-cn 23235  df-cnp 23236  df-haus 23323  df-tx 23570  df-hmeo 23763  df-fil 23854  df-fm 23946  df-flim 23947  df-flf 23948  df-xms 24330  df-ms 24331  df-tms 24332  df-cncf 24904  df-limc 25901  df-dv 25902
This theorem is referenced by:  eff1o  26591
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