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Theorem eff1olem 26490
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 = (𝑤 ∈ 𝐷 ↦ (exp‘(i · 𝑤)))
eff1olem.2 𝑆 = (◡ℑ “ 𝐷)
eff1olem.3 (𝜑 → 𝐷 ⊆ ℝ)
eff1olem.4 ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (abs‘(𝑥 − 𝑦)) < (2 · π))
eff1olem.5 ((𝜑 ∧ 𝑧 ∈ ℝ) → ∃𝑦 ∈ 𝐷 ((𝑧 − 𝑦) / (2 · π)) ∈ ℤ)
Assertion
Ref Expression
eff1olem (𝜑 → (exp ↾ 𝑆):𝑆–1-1-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 6042 . . . 4 (◡ℑ “ 𝐷) ⊆ dom ℑ
2 eff1olem.2 . . . 4 𝑆 = (◡ℑ “ 𝐷)
3 imf 15055 . . . . . 6 ℑ:ℂ⟶ℝ
43fdmi 6681 . . . . 5 dom ℑ = ℂ
54eqcomi 2738 . . . 4 ℂ = dom ℑ
61, 2, 53sstr4i 3995 . . 3 𝑆 ⊆ ℂ
7 eff2 16043 . . . . . . 7 exp:ℂ⟶(ℂ ∖ {0})
87a1i 11 . . . . . 6 (𝑆 ⊆ ℂ → exp:ℂ⟶(ℂ ∖ {0}))
98feqmptd 6911 . . . . 5 (𝑆 ⊆ ℂ → exp = (𝑦 ∈ ℂ ↦ (exp‘𝑦)))
109reseq1d 5938 . . . 4 (𝑆 ⊆ ℂ → (exp ↾ 𝑆) = ((𝑦 ∈ ℂ ↦ (exp‘𝑦)) ↾ 𝑆))
11 resmpt 5997 . . . 4 (𝑆 ⊆ ℂ → ((𝑦 ∈ ℂ ↦ (exp‘𝑦)) ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (exp‘𝑦)))
1210, 11eqtrd 2764 . . 3 (𝑆 ⊆ ℂ → (exp ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (exp‘𝑦)))
136, 12ax-mp 5 . 2 (exp ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (exp‘𝑦))
146sseli 3939 . . . 4 (𝑦 ∈ 𝑆 → 𝑦 ∈ ℂ)
157ffvelcdmi 7037 . . . 4 (𝑦 ∈ ℂ → (exp‘𝑦) ∈ (ℂ ∖ {0}))
1614, 15syl 17 . . 3 (𝑦 ∈ 𝑆 → (exp‘𝑦) ∈ (ℂ ∖ {0}))
1716adantl 481 . 2 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘𝑦) ∈ (ℂ ∖ {0}))
18 simpr 484 . . . . . . . . . 10 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ∈ (ℂ ∖ {0}))
19 eldifsn 4746 . . . . . . . . . 10 (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
2018, 19sylib 218 . . . . . . . . 9 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
2120simpld 494 . . . . . . . 8 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
2220simprd 495 . . . . . . . 8 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ≠ 0)
2321, 22absrpcld 15393 . . . . . . 7 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℝ+)
24 reeff1o 26390 . . . . . . . . 9 (exp ↾ ℝ):ℝ–1-1-onto→ℝ+
25 f1ocnv 6794 . . . . . . . . 9 ((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ → ◡(exp ↾ ℝ):ℝ+–1-1-onto→ℝ)
26 f1of 6782 . . . . . . . . 9 (◡(exp ↾ ℝ):ℝ+–1-1-onto→ℝ → ◡(exp ↾ ℝ):ℝ+⟶ℝ)
2724, 25, 26mp2b 10 . . . . . . . 8 ◡(exp ↾ ℝ):ℝ+⟶ℝ
2827ffvelcdmi 7037 . . . . . . 7 ((abs‘𝑥) ∈ ℝ+ → (◡(exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ)
2923, 28syl 17 . . . . . 6 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡(exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ)
3029recnd 11178 . . . . 5 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡(exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℂ)
31 ax-icn 11103 . . . . . 6 i ∈ ℂ
32 eff1olem.3 . . . . . . . . 9 (𝜑 → 𝐷 ⊆ ℝ)
3332adantr 480 . . . . . . . 8 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝐷 ⊆ ℝ)
34 eff1olem.1 . . . . . . . . . . . 12 𝐹 = (𝑤 ∈ 𝐷 ↦ (exp‘(i · 𝑤)))
35 eqid 2729 . . . . . . . . . . . 12 (◡abs “ {1}) = (◡abs “ {1})
36 eff1olem.4 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (abs‘(𝑥 − 𝑦)) < (2 · π))
37 eff1olem.5 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑧 ∈ ℝ) → ∃𝑦 ∈ 𝐷 ((𝑧 − 𝑦) / (2 · π)) ∈ ℤ)
38 eqid 2729 . . . . . . . . . . . 12 (sin ↾ (-(π / 2)[,](π / 2))) = (sin ↾ (-(π / 2)[,](π / 2)))
3934, 35, 32, 36, 37, 38efif1olem4 26487 . . . . . . . . . . 11 (𝜑 → 𝐹:𝐷–1-1-onto→(◡abs “ {1}))
40 f1ocnv 6794 . . . . . . . . . . 11 (𝐹:𝐷–1-1-onto→(◡abs “ {1}) → ◡𝐹:(◡abs “ {1})–1-1-onto→𝐷)
41 f1of 6782 . . . . . . . . . . 11 (◡𝐹:(◡abs “ {1})–1-1-onto→𝐷 → ◡𝐹:(◡abs “ {1})⟶𝐷)
4239, 40, 413syl 18 . . . . . . . . . 10 (𝜑 → ◡𝐹:(◡abs “ {1})⟶𝐷)
4342adantr 480 . . . . . . . . 9 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ◡𝐹:(◡abs “ {1})⟶𝐷)
4421abscld 15381 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℝ)
4544recnd 11178 . . . . . . . . . . 11 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℂ)
4621, 22absne0d 15392 . . . . . . . . . . 11 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ≠ 0)
4721, 45, 46divcld 11934 . . . . . . . . . 10 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥 / (abs‘𝑥)) ∈ ℂ)
4821, 45, 46absdivd 15400 . . . . . . . . . . 11 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘(𝑥 / (abs‘𝑥))) = ((abs‘𝑥) / (abs‘(abs‘𝑥))))
49 absidm 15266 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (abs‘(abs‘𝑥)) = (abs‘𝑥))
5021, 49syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘(abs‘𝑥)) = (abs‘𝑥))
5150oveq2d 7385 . . . . . . . . . . 11 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) / (abs‘(abs‘𝑥))) = ((abs‘𝑥) / (abs‘𝑥)))
5245, 46dividd 11932 . . . . . . . . . . 11 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) / (abs‘𝑥)) = 1)
5348, 51, 523eqtrd 2768 . . . . . . . . . 10 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘(𝑥 / (abs‘𝑥))) = 1)
54 absf 15280 . . . . . . . . . . 11 abs:ℂ⟶ℝ
55 ffn 6670 . . . . . . . . . . 11 (abs:ℂ⟶ℝ → abs Fn ℂ)
56 fniniseg 7014 . . . . . . . . . . 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 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥 / (abs‘𝑥)) ∈ (◡abs “ {1}))
5943, 58ffvelcdmd 7039 . . . . . . . 8 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ 𝐷)
6033, 59sseldd 3944 . . . . . . 7 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℝ)
6160recnd 11178 . . . . . 6 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℂ)
62 mulcl 11128 . . . . . 6 ((i ∈ ℂ ∧ (◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℂ) → (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))) ∈ ℂ)
6331, 61, 62sylancr 587 . . . . 5 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))) ∈ ℂ)
6430, 63addcld 11169 . . . 4 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ ℂ)
6529, 60crimd 15174 . . . . 5 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (ℑ‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))) = (◡𝐹‘(𝑥 / (abs‘𝑥))))
6665, 59eqeltrd 2828 . . . 4 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (ℑ‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))) ∈ 𝐷)
67 ffn 6670 . . . . 5 (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
68 elpreima 7012 . . . . 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 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ (◡ℑ “ 𝐷))
7170, 2eleqtrrdi 2839 . 2 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ 𝑆)
72 efadd 16036 . . . . . . 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 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (exp‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))) = ((exp‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) · (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
7429fvresd 6860 . . . . . . . 8 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) = (exp‘(◡(exp ↾ ℝ)‘(abs‘𝑥))))
75 f1ocnvfv2 7234 . . . . . . . . 9 (((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ ∧ (abs‘𝑥) ∈ ℝ+) → ((exp ↾ ℝ)‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
7624, 23, 75sylancr 587 . . . . . . . 8 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
7774, 76eqtr3d 2766 . . . . . . 7 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (exp‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
78 oveq2 7377 . . . . . . . . . . 11 (𝑧 = (◡𝐹‘(𝑥 / (abs‘𝑥))) → (i · 𝑧) = (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))
7978fveq2d 6844 . . . . . . . . . 10 (𝑧 = (◡𝐹‘(𝑥 / (abs‘𝑥))) → (exp‘(i · 𝑧)) = (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))))
80 oveq2 7377 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (i · 𝑤) = (i · 𝑧))
8180fveq2d 6844 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (exp‘(i · 𝑤)) = (exp‘(i · 𝑧)))
8281cbvmptv 5206 . . . . . . . . . . 11 (𝑤 ∈ 𝐷 ↦ (exp‘(i · 𝑤))) = (𝑧 ∈ 𝐷 ↦ (exp‘(i · 𝑧)))
8334, 82eqtri 2752 . . . . . . . . . 10 𝐹 = (𝑧 ∈ 𝐷 ↦ (exp‘(i · 𝑧)))
84 fvex 6853 . . . . . . . . . 10 (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ V
8579, 83, 84fvmpt 6950 . . . . . . . . 9 ((◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ 𝐷 → (𝐹‘(◡𝐹‘(𝑥 / (abs‘𝑥)))) = (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))))
8659, 85syl 17 . . . . . . . 8 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(◡𝐹‘(𝑥 / (abs‘𝑥)))) = (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))))
8739adantr 480 . . . . . . . . 9 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝐹:𝐷–1-1-onto→(◡abs “ {1}))
88 f1ocnvfv2 7234 . . . . . . . . 9 ((𝐹:𝐷–1-1-onto→(◡abs “ {1}) ∧ (𝑥 / (abs‘𝑥)) ∈ (◡abs “ {1})) → (𝐹‘(◡𝐹‘(𝑥 / (abs‘𝑥)))) = (𝑥 / (abs‘𝑥)))
8987, 58, 88syl2anc 584 . . . . . . . 8 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(◡𝐹‘(𝑥 / (abs‘𝑥)))) = (𝑥 / (abs‘𝑥)))
9086, 89eqtr3d 2766 . . . . . . 7 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) = (𝑥 / (abs‘𝑥)))
9177, 90oveq12d 7387 . . . . . 6 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((exp‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) · (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))) = ((abs‘𝑥) · (𝑥 / (abs‘𝑥))))
9221, 45, 46divcan2d 11936 . . . . . 6 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) · (𝑥 / (abs‘𝑥))) = 𝑥)
9373, 91, 923eqtrrd 2769 . . . . 5 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 = (exp‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
9493adantrl 716 . . . 4 ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ (ℂ ∖ {0}))) → 𝑥 = (exp‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
95 fveq2 6840 . . . . 5 (𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) → (exp‘𝑦) = (exp‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
9695eqeq2d 2740 . . . 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 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ (ℂ ∖ {0}))) → (𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) → 𝑥 = (exp‘𝑦)))
9814adantl 481 . . . . . . 7 ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ℂ)
9998replimd 15139 . . . . . 6 ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 = ((ℜ‘𝑦) + (i · (ℑ‘𝑦))))
100 absef 16141 . . . . . . . . . . 11 (𝑦 ∈ ℂ → (abs‘(exp‘𝑦)) = (exp‘(ℜ‘𝑦)))
10198, 100syl 17 . . . . . . . . . 10 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘(exp‘𝑦)) = (exp‘(ℜ‘𝑦)))
10298recld 15136 . . . . . . . . . . 11 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℜ‘𝑦) ∈ ℝ)
103102fvresd 6860 . . . . . . . . . 10 ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((exp ↾ ℝ)‘(ℜ‘𝑦)) = (exp‘(ℜ‘𝑦)))
104101, 103eqtr4d 2767 . . . . . . . . 9 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘(exp‘𝑦)) = ((exp ↾ ℝ)‘(ℜ‘𝑦)))
105104fveq2d 6844 . . . . . . . 8 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) = (◡(exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))))
106 f1ocnvfv1 7233 . . . . . . . . 9 (((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ ∧ (ℜ‘𝑦) ∈ ℝ) → (◡(exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))) = (ℜ‘𝑦))
10724, 102, 106sylancr 587 . . . . . . . 8 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡(exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))) = (ℜ‘𝑦))
108105, 107eqtrd 2764 . . . . . . 7 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) = (ℜ‘𝑦))
10998imcld 15137 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℑ‘𝑦) ∈ ℝ)
110109recnd 11178 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℑ‘𝑦) ∈ ℂ)
111 mulcl 11128 . . . . . . . . . . . . . 14 ((i ∈ ℂ ∧ (ℑ‘𝑦) ∈ ℂ) → (i · (ℑ‘𝑦)) ∈ ℂ)
11231, 110, 111sylancr 587 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (i · (ℑ‘𝑦)) ∈ ℂ)
113 efcl 16024 . . . . . . . . . . . . 13 ((i · (ℑ‘𝑦)) ∈ ℂ → (exp‘(i · (ℑ‘𝑦))) ∈ ℂ)
114112, 113syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘(i · (ℑ‘𝑦))) ∈ ℂ)
115102recnd 11178 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℜ‘𝑦) ∈ ℂ)
116 efcl 16024 . . . . . . . . . . . . 13 ((ℜ‘𝑦) ∈ ℂ → (exp‘(ℜ‘𝑦)) ∈ ℂ)
117115, 116syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘(ℜ‘𝑦)) ∈ ℂ)
118 efne0 16040 . . . . . . . . . . . . 13 ((ℜ‘𝑦) ∈ ℂ → (exp‘(ℜ‘𝑦)) ≠ 0)
119115, 118syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘(ℜ‘𝑦)) ≠ 0)
120114, 117, 119divcan3d 11939 . . . . . . . . . . 11 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))) / (exp‘(ℜ‘𝑦))) = (exp‘(i · (ℑ‘𝑦))))
12199fveq2d 6844 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘𝑦) = (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))))
122 efadd 16036 . . . . . . . . . . . . . 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 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
124121, 123eqtrd 2764 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘𝑦) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
125124, 101oveq12d 7387 . . . . . . . . . . 11 ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((exp‘𝑦) / (abs‘(exp‘𝑦))) = (((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))) / (exp‘(ℜ‘𝑦))))
126 elpreima 7012 . . . . . . . . . . . . . . . 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 2846 . . . . . . . . . . . . 13 (𝑦 ∈ 𝑆 → (ℑ‘𝑦) ∈ 𝐷)
130129adantl 481 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℑ‘𝑦) ∈ 𝐷)
131 oveq2 7377 . . . . . . . . . . . . . 14 (𝑤 = (ℑ‘𝑦) → (i · 𝑤) = (i · (ℑ‘𝑦)))
132131fveq2d 6844 . . . . . . . . . . . . 13 (𝑤 = (ℑ‘𝑦) → (exp‘(i · 𝑤)) = (exp‘(i · (ℑ‘𝑦))))
133 fvex 6853 . . . . . . . . . . . . 13 (exp‘(i · (ℑ‘𝑦))) ∈ V
134132, 34, 133fvmpt 6950 . . . . . . . . . . . 12 ((ℑ‘𝑦) ∈ 𝐷 → (𝐹‘(ℑ‘𝑦)) = (exp‘(i · (ℑ‘𝑦))))
135130, 134syl 17 . . . . . . . . . . 11 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝐹‘(ℑ‘𝑦)) = (exp‘(i · (ℑ‘𝑦))))
136120, 125, 1353eqtr4d 2774 . . . . . . . . . 10 ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((exp‘𝑦) / (abs‘(exp‘𝑦))) = (𝐹‘(ℑ‘𝑦)))
137136fveq2d 6844 . . . . . . . . 9 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))) = (◡𝐹‘(𝐹‘(ℑ‘𝑦))))
138 f1ocnvfv1 7233 . . . . . . . . . 10 ((𝐹:𝐷–1-1-onto→(◡abs “ {1}) ∧ (ℑ‘𝑦) ∈ 𝐷) → (◡𝐹‘(𝐹‘(ℑ‘𝑦))) = (ℑ‘𝑦))
13939, 129, 138syl2an 596 . . . . . . . . 9 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡𝐹‘(𝐹‘(ℑ‘𝑦))) = (ℑ‘𝑦))
140137, 139eqtrd 2764 . . . . . . . 8 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))) = (ℑ‘𝑦))
141140oveq2d 7385 . . . . . . 7 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦))))) = (i · (ℑ‘𝑦)))
142108, 141oveq12d 7387 . . . . . 6 ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))) = ((ℜ‘𝑦) + (i · (ℑ‘𝑦))))
14399, 142eqtr4d 2767 . . . . 5 ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 = ((◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))))
144 fveq2 6840 . . . . . . . 8 (𝑥 = (exp‘𝑦) → (abs‘𝑥) = (abs‘(exp‘𝑦)))
145144fveq2d 6844 . . . . . . 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 7387 . . . . . . . . 9 (𝑥 = (exp‘𝑦) → (𝑥 / (abs‘𝑥)) = ((exp‘𝑦) / (abs‘(exp‘𝑦))))
148147fveq2d 6844 . . . . . . . 8 (𝑥 = (exp‘𝑦) → (◡𝐹‘(𝑥 / (abs‘𝑥))) = (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))
149148oveq2d 7385 . . . . . . 7 (𝑥 = (exp‘𝑦) → (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))) = (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦))))))
150145, 149oveq12d 7387 . . . . . 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 2740 . . . . 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 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝑥 = (exp‘𝑦) → 𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
153152adantrr 717 . . 3 ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ (ℂ ∖ {0}))) → (𝑥 = (exp‘𝑦) → 𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
15497, 153impbid 212 . 2 ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ (ℂ ∖ {0}))) → (𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ↔ 𝑥 = (exp‘𝑦)))
15513, 17, 71, 154f1o2d 7623 1 (𝜑 → (exp ↾ 𝑆):𝑆–1-1-onto→(ℂ ∖ {0}))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053   ∖ cdif 3908   ⊆ wss 3911  {csn 4585   class class class wbr 5102   ↦ cmpt 5183  ◡ccnv 5630  dom cdm 5631   ↾ cres 5633   “ cima 5634   Fn wfn 6494  ⟶wf 6495  –1-1-onto→wf1o 6498  ‘cfv 6499  (class class class)co 7369  ℂcc 11042  ℝcr 11043  0cc0 11044  1c1 11045  ici 11046   + caddc 11047   · cmul 11049   < clt 11184   − cmin 11381  -cneg 11382   / cdiv 11811  2c2 12217  ℤcz 12505  ℝ+crp 12927  [,]cicc 13285  ℜcre 15039  ℑcim 15040  abscabs 15176  expce 16003  sincsin 16005  πcpi 16008
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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-inf2 9570  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-pre-sup 11122  ax-addf 11123
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633  df-om 7823  df-1st 7947  df-2nd 7948  df-supp 8117  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-er 8648  df-map 8778  df-pm 8779  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9289  df-fi 9338  df-sup 9369  df-inf 9370  df-oi 9439  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-uz 12770  df-q 12884  df-rp 12928  df-xneg 13048  df-xadd 13049  df-xmul 13050  df-ioo 13286  df-ioc 13287  df-ico 13288  df-icc 13289  df-fz 13445  df-fzo 13592  df-fl 13730  df-mod 13808  df-seq 13943  df-exp 14003  df-fac 14215  df-bc 14244  df-hash 14272  df-shft 15009  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-limsup 15413  df-clim 15430  df-rlim 15431  df-sum 15629  df-ef 16009  df-sin 16011  df-cos 16012  df-pi 16014  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-mulr 17210  df-starv 17211  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-unif 17219  df-hom 17220  df-cco 17221  df-rest 17361  df-topn 17362  df-0g 17380  df-gsum 17381  df-topgen 17382  df-pt 17383  df-prds 17386  df-xrs 17441  df-qtop 17446  df-imas 17447  df-xps 17449  df-mre 17523  df-mrc 17524  df-acs 17526  df-mgm 18549  df-sgrp 18628  df-mnd 18644  df-submnd 18693  df-mulg 18982  df-cntz 19231  df-cmn 19696  df-psmet 21288  df-xmet 21289  df-met 21290  df-bl 21291  df-mopn 21292  df-fbas 21293  df-fg 21294  df-cnfld 21297  df-top 22814  df-topon 22831  df-topsp 22853  df-bases 22866  df-cld 22939  df-ntr 22940  df-cls 22941  df-nei 23018  df-lp 23056  df-perf 23057  df-cn 23147  df-cnp 23148  df-haus 23235  df-tx 23482  df-hmeo 23675  df-fil 23766  df-fm 23858  df-flim 23859  df-flf 23860  df-xms 24241  df-ms 24242  df-tms 24243  df-cncf 24804  df-limc 25800  df-dv 25801
This theorem is referenced by:  eff1o  26491
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