MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eff1olem Structured version   Visualization version   GIF version
Theorem eff1olem 25704
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 5989 . . . 4 (◡ℑ “ 𝐷) ⊆ dom ℑ
2 eff1olem.2 . . . 4 𝑆 = (◡ℑ “ 𝐷)
3 imf 14824 . . . . . 6 ℑ:ℂ⟶ℝ
43fdmi 6612 . . . . 5 dom ℑ = ℂ
54eqcomi 2747 . . . 4 ℂ = dom ℑ
61, 2, 53sstr4i 3964 . . 3 𝑆 ⊆ ℂ
7 eff2 15808 . . . . . . 7 exp:ℂ⟶(ℂ ∖ {0})
87a1i 11 . . . . . 6 (𝑆 ⊆ ℂ → exp:ℂ⟶(ℂ ∖ {0}))
98feqmptd 6837 . . . . 5 (𝑆 ⊆ ℂ → exp = (𝑦 ∈ ℂ ↦ (exp‘𝑦)))
109reseq1d 5890 . . . 4 (𝑆 ⊆ ℂ → (exp ↾ 𝑆) = ((𝑦 ∈ ℂ ↦ (exp‘𝑦)) ↾ 𝑆))
11 resmpt 5945 . . . 4 (𝑆 ⊆ ℂ → ((𝑦 ∈ ℂ ↦ (exp‘𝑦)) ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (exp‘𝑦)))
1210, 11eqtrd 2778 . . 3 (𝑆 ⊆ ℂ → (exp ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (exp‘𝑦)))
136, 12ax-mp 5 . 2 (exp ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (exp‘𝑦))
146sseli 3917 . . . 4 (𝑦 ∈ 𝑆 → 𝑦 ∈ ℂ)
157ffvelrni 6960 . . . 4 (𝑦 ∈ ℂ → (exp‘𝑦) ∈ (ℂ ∖ {0}))
1614, 15syl 17 . . 3 (𝑦 ∈ 𝑆 → (exp‘𝑦) ∈ (ℂ ∖ {0}))
1716adantl 482 . 2 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘𝑦) ∈ (ℂ ∖ {0}))
18 simpr 485 . . . . . . . . . 10 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ∈ (ℂ ∖ {0}))
19 eldifsn 4720 . . . . . . . . . 10 (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
2018, 19sylib 217 . . . . . . . . 9 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
2120simpld 495 . . . . . . . 8 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
2220simprd 496 . . . . . . . 8 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ≠ 0)
2321, 22absrpcld 15160 . . . . . . 7 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℝ+)
24 reeff1o 25606 . . . . . . . . 9 (exp ↾ ℝ):ℝ–1-1-onto→ℝ+
25 f1ocnv 6728 . . . . . . . . 9 ((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ → ◡(exp ↾ ℝ):ℝ+–1-1-onto→ℝ)
26 f1of 6716 . . . . . . . . 9 (◡(exp ↾ ℝ):ℝ+–1-1-onto→ℝ → ◡(exp ↾ ℝ):ℝ+⟶ℝ)
2724, 25, 26mp2b 10 . . . . . . . 8 ◡(exp ↾ ℝ):ℝ+⟶ℝ
2827ffvelrni 6960 . . . . . . 7 ((abs‘𝑥) ∈ ℝ+ → (◡(exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ)
2923, 28syl 17 . . . . . 6 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡(exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ)
3029recnd 11003 . . . . 5 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡(exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℂ)
31 ax-icn 10930 . . . . . 6 i ∈ ℂ
32 eff1olem.3 . . . . . . . . 9 (𝜑 → 𝐷 ⊆ ℝ)
3332adantr 481 . . . . . . . 8 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝐷 ⊆ ℝ)
34 eff1olem.1 . . . . . . . . . . . 12 𝐹 = (𝑤 ∈ 𝐷 ↦ (exp‘(i · 𝑤)))
35 eqid 2738 . . . . . . . . . . . 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 2738 . . . . . . . . . . . 12 (sin ↾ (-(π / 2)[,](π / 2))) = (sin ↾ (-(π / 2)[,](π / 2)))
3934, 35, 32, 36, 37, 38efif1olem4 25701 . . . . . . . . . . 11 (𝜑 → 𝐹:𝐷–1-1-onto→(◡abs “ {1}))
40 f1ocnv 6728 . . . . . . . . . . 11 (𝐹:𝐷–1-1-onto→(◡abs “ {1}) → ◡𝐹:(◡abs “ {1})–1-1-onto→𝐷)
41 f1of 6716 . . . . . . . . . . 11 (◡𝐹:(◡abs “ {1})–1-1-onto→𝐷 → ◡𝐹:(◡abs “ {1})⟶𝐷)
4239, 40, 413syl 18 . . . . . . . . . 10 (𝜑 → ◡𝐹:(◡abs “ {1})⟶𝐷)
4342adantr 481 . . . . . . . . 9 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ◡𝐹:(◡abs “ {1})⟶𝐷)
4421abscld 15148 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℝ)
4544recnd 11003 . . . . . . . . . . 11 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℂ)
4621, 22absne0d 15159 . . . . . . . . . . 11 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ≠ 0)
4721, 45, 46divcld 11751 . . . . . . . . . 10 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥 / (abs‘𝑥)) ∈ ℂ)
4821, 45, 46absdivd 15167 . . . . . . . . . . 11 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘(𝑥 / (abs‘𝑥))) = ((abs‘𝑥) / (abs‘(abs‘𝑥))))
49 absidm 15035 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (abs‘(abs‘𝑥)) = (abs‘𝑥))
5021, 49syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘(abs‘𝑥)) = (abs‘𝑥))
5150oveq2d 7291 . . . . . . . . . . 11 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) / (abs‘(abs‘𝑥))) = ((abs‘𝑥) / (abs‘𝑥)))
5245, 46dividd 11749 . . . . . . . . . . 11 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) / (abs‘𝑥)) = 1)
5348, 51, 523eqtrd 2782 . . . . . . . . . 10 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘(𝑥 / (abs‘𝑥))) = 1)
54 absf 15049 . . . . . . . . . . 11 abs:ℂ⟶ℝ
55 ffn 6600 . . . . . . . . . . 11 (abs:ℂ⟶ℝ → abs Fn ℂ)
56 fniniseg 6937 . . . . . . . . . . 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, 58ffvelrnd 6962 . . . . . . . 8 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ 𝐷)
6033, 59sseldd 3922 . . . . . . 7 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℝ)
6160recnd 11003 . . . . . 6 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℂ)
62 mulcl 10955 . . . . . 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 10994 . . . 4 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ ℂ)
6529, 60crimd 14943 . . . . 5 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (ℑ‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))) = (◡𝐹‘(𝑥 / (abs‘𝑥))))
6665, 59eqeltrd 2839 . . . 4 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (ℑ‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))) ∈ 𝐷)
67 ffn 6600 . . . . 5 (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
68 elpreima 6935 . . . . 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 2850 . 2 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ 𝑆)
72 efadd 15803 . . . . . . 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 6794 . . . . . . . 8 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) = (exp‘(◡(exp ↾ ℝ)‘(abs‘𝑥))))
75 f1ocnvfv2 7149 . . . . . . . . 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 2780 . . . . . . 7 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (exp‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
78 oveq2 7283 . . . . . . . . . . 11 (𝑧 = (◡𝐹‘(𝑥 / (abs‘𝑥))) → (i · 𝑧) = (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))
7978fveq2d 6778 . . . . . . . . . 10 (𝑧 = (◡𝐹‘(𝑥 / (abs‘𝑥))) → (exp‘(i · 𝑧)) = (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))))
80 oveq2 7283 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (i · 𝑤) = (i · 𝑧))
8180fveq2d 6778 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (exp‘(i · 𝑤)) = (exp‘(i · 𝑧)))
8281cbvmptv 5187 . . . . . . . . . . 11 (𝑤 ∈ 𝐷 ↦ (exp‘(i · 𝑤))) = (𝑧 ∈ 𝐷 ↦ (exp‘(i · 𝑧)))
8334, 82eqtri 2766 . . . . . . . . . 10 𝐹 = (𝑧 ∈ 𝐷 ↦ (exp‘(i · 𝑧)))
84 fvex 6787 . . . . . . . . . 10 (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ V
8579, 83, 84fvmpt 6875 . . . . . . . . 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 481 . . . . . . . . 9 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝐹:𝐷–1-1-onto→(◡abs “ {1}))
88 f1ocnvfv2 7149 . . . . . . . . 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 2780 . . . . . . 7 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) = (𝑥 / (abs‘𝑥)))
9177, 90oveq12d 7293 . . . . . 6 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((exp‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) · (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))) = ((abs‘𝑥) · (𝑥 / (abs‘𝑥))))
9221, 45, 46divcan2d 11753 . . . . . 6 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) · (𝑥 / (abs‘𝑥))) = 𝑥)
9373, 91, 923eqtrrd 2783 . . . . 5 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 = (exp‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
9493adantrl 713 . . . 4 ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ (ℂ ∖ {0}))) → 𝑥 = (exp‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
95 fveq2 6774 . . . . 5 (𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) → (exp‘𝑦) = (exp‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
9695eqeq2d 2749 . . . 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 246 . . 3 ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ (ℂ ∖ {0}))) → (𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) → 𝑥 = (exp‘𝑦)))
9814adantl 482 . . . . . . 7 ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ℂ)
9998replimd 14908 . . . . . 6 ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 = ((ℜ‘𝑦) + (i · (ℑ‘𝑦))))
100 absef 15906 . . . . . . . . . . 11 (𝑦 ∈ ℂ → (abs‘(exp‘𝑦)) = (exp‘(ℜ‘𝑦)))
10198, 100syl 17 . . . . . . . . . 10 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘(exp‘𝑦)) = (exp‘(ℜ‘𝑦)))
10298recld 14905 . . . . . . . . . . 11 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℜ‘𝑦) ∈ ℝ)
103102fvresd 6794 . . . . . . . . . 10 ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((exp ↾ ℝ)‘(ℜ‘𝑦)) = (exp‘(ℜ‘𝑦)))
104101, 103eqtr4d 2781 . . . . . . . . 9 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘(exp‘𝑦)) = ((exp ↾ ℝ)‘(ℜ‘𝑦)))
105104fveq2d 6778 . . . . . . . 8 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) = (◡(exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))))
106 f1ocnvfv1 7148 . . . . . . . . 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 2778 . . . . . . 7 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) = (ℜ‘𝑦))
10998imcld 14906 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℑ‘𝑦) ∈ ℝ)
110109recnd 11003 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℑ‘𝑦) ∈ ℂ)
111 mulcl 10955 . . . . . . . . . . . . . 14 ((i ∈ ℂ ∧ (ℑ‘𝑦) ∈ ℂ) → (i · (ℑ‘𝑦)) ∈ ℂ)
11231, 110, 111sylancr 587 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (i · (ℑ‘𝑦)) ∈ ℂ)
113 efcl 15792 . . . . . . . . . . . . 13 ((i · (ℑ‘𝑦)) ∈ ℂ → (exp‘(i · (ℑ‘𝑦))) ∈ ℂ)
114112, 113syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘(i · (ℑ‘𝑦))) ∈ ℂ)
115102recnd 11003 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℜ‘𝑦) ∈ ℂ)
116 efcl 15792 . . . . . . . . . . . . 13 ((ℜ‘𝑦) ∈ ℂ → (exp‘(ℜ‘𝑦)) ∈ ℂ)
117115, 116syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘(ℜ‘𝑦)) ∈ ℂ)
118 efne0 15806 . . . . . . . . . . . . 13 ((ℜ‘𝑦) ∈ ℂ → (exp‘(ℜ‘𝑦)) ≠ 0)
119115, 118syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘(ℜ‘𝑦)) ≠ 0)
120114, 117, 119divcan3d 11756 . . . . . . . . . . 11 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))) / (exp‘(ℜ‘𝑦))) = (exp‘(i · (ℑ‘𝑦))))
12199fveq2d 6778 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘𝑦) = (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))))
122 efadd 15803 . . . . . . . . . . . . . 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 2778 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘𝑦) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
125124, 101oveq12d 7293 . . . . . . . . . . 11 ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((exp‘𝑦) / (abs‘(exp‘𝑦))) = (((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))) / (exp‘(ℜ‘𝑦))))
126 elpreima 6935 . . . . . . . . . . . . . . . 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 497 . . . . . . . . . . . . . 14 (𝑦 ∈ (◡ℑ “ 𝐷) → (ℑ‘𝑦) ∈ 𝐷)
129128, 2eleq2s 2857 . . . . . . . . . . . . 13 (𝑦 ∈ 𝑆 → (ℑ‘𝑦) ∈ 𝐷)
130129adantl 482 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℑ‘𝑦) ∈ 𝐷)
131 oveq2 7283 . . . . . . . . . . . . . 14 (𝑤 = (ℑ‘𝑦) → (i · 𝑤) = (i · (ℑ‘𝑦)))
132131fveq2d 6778 . . . . . . . . . . . . 13 (𝑤 = (ℑ‘𝑦) → (exp‘(i · 𝑤)) = (exp‘(i · (ℑ‘𝑦))))
133 fvex 6787 . . . . . . . . . . . . 13 (exp‘(i · (ℑ‘𝑦))) ∈ V
134132, 34, 133fvmpt 6875 . . . . . . . . . . . 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 2788 . . . . . . . . . 10 ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((exp‘𝑦) / (abs‘(exp‘𝑦))) = (𝐹‘(ℑ‘𝑦)))
137136fveq2d 6778 . . . . . . . . 9 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))) = (◡𝐹‘(𝐹‘(ℑ‘𝑦))))
138 f1ocnvfv1 7148 . . . . . . . . . 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 2778 . . . . . . . 8 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))) = (ℑ‘𝑦))
141140oveq2d 7291 . . . . . . 7 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦))))) = (i · (ℑ‘𝑦)))
142108, 141oveq12d 7293 . . . . . 6 ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))) = ((ℜ‘𝑦) + (i · (ℑ‘𝑦))))
14399, 142eqtr4d 2781 . . . . 5 ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 = ((◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))))
144 fveq2 6774 . . . . . . . 8 (𝑥 = (exp‘𝑦) → (abs‘𝑥) = (abs‘(exp‘𝑦)))
145144fveq2d 6778 . . . . . . 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 7293 . . . . . . . . 9 (𝑥 = (exp‘𝑦) → (𝑥 / (abs‘𝑥)) = ((exp‘𝑦) / (abs‘(exp‘𝑦))))
148147fveq2d 6778 . . . . . . . 8 (𝑥 = (exp‘𝑦) → (◡𝐹‘(𝑥 / (abs‘𝑥))) = (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))
149148oveq2d 7291 . . . . . . 7 (𝑥 = (exp‘𝑦) → (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))) = (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦))))))
150145, 149oveq12d 7293 . . . . . 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 2749 . . . . 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 246 . . . 4 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝑥 = (exp‘𝑦) → 𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
153152adantrr 714 . . 3 ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ (ℂ ∖ {0}))) → (𝑥 = (exp‘𝑦) → 𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
15497, 153impbid 211 . 2 ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ (ℂ ∖ {0}))) → (𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ↔ 𝑥 = (exp‘𝑦)))
15513, 17, 71, 154f1o2d 7523 1 (𝜑 → (exp ↾ 𝑆):𝑆–1-1-onto→(ℂ ∖ {0}))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3065   ∖ cdif 3884   ⊆ wss 3887  {csn 4561   class class class wbr 5074   ↦ cmpt 5157  ◡ccnv 5588  dom cdm 5589   ↾ cres 5591   “ cima 5592   Fn wfn 6428  ⟶wf 6429  –1-1-onto→wf1o 6432  ‘cfv 6433  (class class class)co 7275  ℂcc 10869  ℝcr 10870  0cc0 10871  1c1 10872  ici 10873   + caddc 10874   · cmul 10876   < clt 11009   − cmin 11205  -cneg 11206   / cdiv 11632  2c2 12028  ℤcz 12319  ℝ+crp 12730  [,]cicc 13082  ℜcre 14808  ℑcim 14809  abscabs 14945  expce 15771  sincsin 15773  πcpi 15776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949  ax-addf 10950  ax-mulf 10951
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-fi 9170  df-sup 9201  df-inf 9202  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ioo 13083  df-ioc 13084  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-fl 13512  df-mod 13590  df-seq 13722  df-exp 13783  df-fac 13988  df-bc 14017  df-hash 14045  df-shft 14778  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-limsup 15180  df-clim 15197  df-rlim 15198  df-sum 15398  df-ef 15777  df-sin 15779  df-cos 15780  df-pi 15782  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-starv 16977  df-sca 16978  df-vsca 16979  df-ip 16980  df-tset 16981  df-ple 16982  df-ds 16984  df-unif 16985  df-hom 16986  df-cco 16987  df-rest 17133  df-topn 17134  df-0g 17152  df-gsum 17153  df-topgen 17154  df-pt 17155  df-prds 17158  df-xrs 17213  df-qtop 17218  df-imas 17219  df-xps 17221  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-submnd 18431  df-mulg 18701  df-cntz 18923  df-cmn 19388  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-fbas 20594  df-fg 20595  df-cnfld 20598  df-top 22043  df-topon 22060  df-topsp 22082  df-bases 22096  df-cld 22170  df-ntr 22171  df-cls 22172  df-nei 22249  df-lp 22287  df-perf 22288  df-cn 22378  df-cnp 22379  df-haus 22466  df-tx 22713  df-hmeo 22906  df-fil 22997  df-fm 23089  df-flim 23090  df-flf 23091  df-xms 23473  df-ms 23474  df-tms 23475  df-cncf 24041  df-limc 25030  df-dv 25031
This theorem is referenced by:  eff1o  25705
  Copyright terms: Public domain W3C validator