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Theorem eff1olem 26513
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 6041 . . . 4 (◡ℑ “ 𝐷) ⊆ dom ℑ
2 eff1olem.2 . . . 4 𝑆 = (◡ℑ “ 𝐷)
3 imf 15036 . . . . . 6 ℑ:ℂ⟶ℝ
43fdmi 6673 . . . . 5 dom ℑ = ℂ
54eqcomi 2745 . . . 4 ℂ = dom ℑ
61, 2, 53sstr4i 3985 . . 3 𝑆 ⊆ ℂ
7 eff2 16024 . . . . . . 7 exp:ℂ⟶(ℂ ∖ {0})
87a1i 11 . . . . . 6 (𝑆 ⊆ ℂ → exp:ℂ⟶(ℂ ∖ {0}))
98feqmptd 6902 . . . . 5 (𝑆 ⊆ ℂ → exp = (𝑦 ∈ ℂ ↦ (exp‘𝑦)))
109reseq1d 5937 . . . 4 (𝑆 ⊆ ℂ → (exp ↾ 𝑆) = ((𝑦 ∈ ℂ ↦ (exp‘𝑦)) ↾ 𝑆))
11 resmpt 5996 . . . 4 (𝑆 ⊆ ℂ → ((𝑦 ∈ ℂ ↦ (exp‘𝑦)) ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (exp‘𝑦)))
1210, 11eqtrd 2771 . . 3 (𝑆 ⊆ ℂ → (exp ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (exp‘𝑦)))
136, 12ax-mp 5 . 2 (exp ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (exp‘𝑦))
146sseli 3929 . . . 4 (𝑦 ∈ 𝑆 → 𝑦 ∈ ℂ)
157ffvelcdmi 7028 . . . 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 4742 . . . . . . . . . 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 15374 . . . . . . 7 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℝ+)
24 reeff1o 26413 . . . . . . . . 9 (exp ↾ ℝ):ℝ–1-1-onto→ℝ+
25 f1ocnv 6786 . . . . . . . . 9 ((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ → ◡(exp ↾ ℝ):ℝ+–1-1-onto→ℝ)
26 f1of 6774 . . . . . . . . 9 (◡(exp ↾ ℝ):ℝ+–1-1-onto→ℝ → ◡(exp ↾ ℝ):ℝ+⟶ℝ)
2724, 25, 26mp2b 10 . . . . . . . 8 ◡(exp ↾ ℝ):ℝ+⟶ℝ
2827ffvelcdmi 7028 . . . . . . 7 ((abs‘𝑥) ∈ ℝ+ → (◡(exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ)
2923, 28syl 17 . . . . . 6 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡(exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ)
3029recnd 11160 . . . . 5 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡(exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℂ)
31 ax-icn 11085 . . . . . 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 2736 . . . . . . . . . . . 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 2736 . . . . . . . . . . . 12 (sin ↾ (-(π / 2)[,](π / 2))) = (sin ↾ (-(π / 2)[,](π / 2)))
3934, 35, 32, 36, 37, 38efif1olem4 26510 . . . . . . . . . . 11 (𝜑 → 𝐹:𝐷–1-1-onto→(◡abs “ {1}))
40 f1ocnv 6786 . . . . . . . . . . 11 (𝐹:𝐷–1-1-onto→(◡abs “ {1}) → ◡𝐹:(◡abs “ {1})–1-1-onto→𝐷)
41 f1of 6774 . . . . . . . . . . 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 15362 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℝ)
4544recnd 11160 . . . . . . . . . . 11 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℂ)
4621, 22absne0d 15373 . . . . . . . . . . 11 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ≠ 0)
4721, 45, 46divcld 11917 . . . . . . . . . 10 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥 / (abs‘𝑥)) ∈ ℂ)
4821, 45, 46absdivd 15381 . . . . . . . . . . 11 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘(𝑥 / (abs‘𝑥))) = ((abs‘𝑥) / (abs‘(abs‘𝑥))))
49 absidm 15247 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (abs‘(abs‘𝑥)) = (abs‘𝑥))
5021, 49syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘(abs‘𝑥)) = (abs‘𝑥))
5150oveq2d 7374 . . . . . . . . . . 11 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) / (abs‘(abs‘𝑥))) = ((abs‘𝑥) / (abs‘𝑥)))
5245, 46dividd 11915 . . . . . . . . . . 11 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) / (abs‘𝑥)) = 1)
5348, 51, 523eqtrd 2775 . . . . . . . . . 10 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘(𝑥 / (abs‘𝑥))) = 1)
54 absf 15261 . . . . . . . . . . 11 abs:ℂ⟶ℝ
55 ffn 6662 . . . . . . . . . . 11 (abs:ℂ⟶ℝ → abs Fn ℂ)
56 fniniseg 7005 . . . . . . . . . . 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 7030 . . . . . . . 8 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ 𝐷)
6033, 59sseldd 3934 . . . . . . 7 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℝ)
6160recnd 11160 . . . . . 6 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℂ)
62 mulcl 11110 . . . . . 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 11151 . . . 4 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ ℂ)
6529, 60crimd 15155 . . . . 5 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (ℑ‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))) = (◡𝐹‘(𝑥 / (abs‘𝑥))))
6665, 59eqeltrd 2836 . . . 4 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (ℑ‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))) ∈ 𝐷)
67 ffn 6662 . . . . 5 (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
68 elpreima 7003 . . . . 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 2847 . 2 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ 𝑆)
72 efadd 16017 . . . . . . 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 6854 . . . . . . . 8 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) = (exp‘(◡(exp ↾ ℝ)‘(abs‘𝑥))))
75 f1ocnvfv2 7223 . . . . . . . . 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 2773 . . . . . . 7 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (exp‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
78 oveq2 7366 . . . . . . . . . . 11 (𝑧 = (◡𝐹‘(𝑥 / (abs‘𝑥))) → (i · 𝑧) = (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))
7978fveq2d 6838 . . . . . . . . . 10 (𝑧 = (◡𝐹‘(𝑥 / (abs‘𝑥))) → (exp‘(i · 𝑧)) = (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))))
80 oveq2 7366 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (i · 𝑤) = (i · 𝑧))
8180fveq2d 6838 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (exp‘(i · 𝑤)) = (exp‘(i · 𝑧)))
8281cbvmptv 5202 . . . . . . . . . . 11 (𝑤 ∈ 𝐷 ↦ (exp‘(i · 𝑤))) = (𝑧 ∈ 𝐷 ↦ (exp‘(i · 𝑧)))
8334, 82eqtri 2759 . . . . . . . . . 10 𝐹 = (𝑧 ∈ 𝐷 ↦ (exp‘(i · 𝑧)))
84 fvex 6847 . . . . . . . . . 10 (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ V
8579, 83, 84fvmpt 6941 . . . . . . . . 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 7223 . . . . . . . . 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 2773 . . . . . . 7 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) = (𝑥 / (abs‘𝑥)))
9177, 90oveq12d 7376 . . . . . 6 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((exp‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) · (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))) = ((abs‘𝑥) · (𝑥 / (abs‘𝑥))))
9221, 45, 46divcan2d 11919 . . . . . 6 ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) · (𝑥 / (abs‘𝑥))) = 𝑥)
9373, 91, 923eqtrrd 2776 . . . . 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 6834 . . . . 5 (𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) → (exp‘𝑦) = (exp‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
9695eqeq2d 2747 . . . 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 15120 . . . . . 6 ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 = ((ℜ‘𝑦) + (i · (ℑ‘𝑦))))
100 absef 16122 . . . . . . . . . . 11 (𝑦 ∈ ℂ → (abs‘(exp‘𝑦)) = (exp‘(ℜ‘𝑦)))
10198, 100syl 17 . . . . . . . . . 10 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘(exp‘𝑦)) = (exp‘(ℜ‘𝑦)))
10298recld 15117 . . . . . . . . . . 11 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℜ‘𝑦) ∈ ℝ)
103102fvresd 6854 . . . . . . . . . 10 ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((exp ↾ ℝ)‘(ℜ‘𝑦)) = (exp‘(ℜ‘𝑦)))
104101, 103eqtr4d 2774 . . . . . . . . 9 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘(exp‘𝑦)) = ((exp ↾ ℝ)‘(ℜ‘𝑦)))
105104fveq2d 6838 . . . . . . . 8 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) = (◡(exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))))
106 f1ocnvfv1 7222 . . . . . . . . 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 2771 . . . . . . 7 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) = (ℜ‘𝑦))
10998imcld 15118 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℑ‘𝑦) ∈ ℝ)
110109recnd 11160 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℑ‘𝑦) ∈ ℂ)
111 mulcl 11110 . . . . . . . . . . . . . 14 ((i ∈ ℂ ∧ (ℑ‘𝑦) ∈ ℂ) → (i · (ℑ‘𝑦)) ∈ ℂ)
11231, 110, 111sylancr 587 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (i · (ℑ‘𝑦)) ∈ ℂ)
113 efcl 16005 . . . . . . . . . . . . 13 ((i · (ℑ‘𝑦)) ∈ ℂ → (exp‘(i · (ℑ‘𝑦))) ∈ ℂ)
114112, 113syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘(i · (ℑ‘𝑦))) ∈ ℂ)
115102recnd 11160 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℜ‘𝑦) ∈ ℂ)
116 efcl 16005 . . . . . . . . . . . . 13 ((ℜ‘𝑦) ∈ ℂ → (exp‘(ℜ‘𝑦)) ∈ ℂ)
117115, 116syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘(ℜ‘𝑦)) ∈ ℂ)
118 efne0 16021 . . . . . . . . . . . . 13 ((ℜ‘𝑦) ∈ ℂ → (exp‘(ℜ‘𝑦)) ≠ 0)
119115, 118syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘(ℜ‘𝑦)) ≠ 0)
120114, 117, 119divcan3d 11922 . . . . . . . . . . 11 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))) / (exp‘(ℜ‘𝑦))) = (exp‘(i · (ℑ‘𝑦))))
12199fveq2d 6838 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘𝑦) = (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))))
122 efadd 16017 . . . . . . . . . . . . . 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 2771 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘𝑦) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
125124, 101oveq12d 7376 . . . . . . . . . . 11 ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((exp‘𝑦) / (abs‘(exp‘𝑦))) = (((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))) / (exp‘(ℜ‘𝑦))))
126 elpreima 7003 . . . . . . . . . . . . . . . 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 2854 . . . . . . . . . . . . 13 (𝑦 ∈ 𝑆 → (ℑ‘𝑦) ∈ 𝐷)
130129adantl 481 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℑ‘𝑦) ∈ 𝐷)
131 oveq2 7366 . . . . . . . . . . . . . 14 (𝑤 = (ℑ‘𝑦) → (i · 𝑤) = (i · (ℑ‘𝑦)))
132131fveq2d 6838 . . . . . . . . . . . . 13 (𝑤 = (ℑ‘𝑦) → (exp‘(i · 𝑤)) = (exp‘(i · (ℑ‘𝑦))))
133 fvex 6847 . . . . . . . . . . . . 13 (exp‘(i · (ℑ‘𝑦))) ∈ V
134132, 34, 133fvmpt 6941 . . . . . . . . . . . 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 2781 . . . . . . . . . 10 ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((exp‘𝑦) / (abs‘(exp‘𝑦))) = (𝐹‘(ℑ‘𝑦)))
137136fveq2d 6838 . . . . . . . . 9 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))) = (◡𝐹‘(𝐹‘(ℑ‘𝑦))))
138 f1ocnvfv1 7222 . . . . . . . . . 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 2771 . . . . . . . 8 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))) = (ℑ‘𝑦))
141140oveq2d 7374 . . . . . . 7 ((𝜑 ∧ 𝑦 ∈ 𝑆) → (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦))))) = (i · (ℑ‘𝑦)))
142108, 141oveq12d 7376 . . . . . 6 ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))) = ((ℜ‘𝑦) + (i · (ℑ‘𝑦))))
14399, 142eqtr4d 2774 . . . . 5 ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 = ((◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))))
144 fveq2 6834 . . . . . . . 8 (𝑥 = (exp‘𝑦) → (abs‘𝑥) = (abs‘(exp‘𝑦)))
145144fveq2d 6838 . . . . . . 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 7376 . . . . . . . . 9 (𝑥 = (exp‘𝑦) → (𝑥 / (abs‘𝑥)) = ((exp‘𝑦) / (abs‘(exp‘𝑦))))
148147fveq2d 6838 . . . . . . . 8 (𝑥 = (exp‘𝑦) → (◡𝐹‘(𝑥 / (abs‘𝑥))) = (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))
149148oveq2d 7374 . . . . . . 7 (𝑥 = (exp‘𝑦) → (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))) = (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦))))))
150145, 149oveq12d 7376 . . . . . 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 2747 . . . . 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 7612 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 1541   ∈ wcel 2113   ≠ wne 2932  ∃wrex 3060   ∖ cdif 3898   ⊆ wss 3901  {csn 4580   class class class wbr 5098   ↦ cmpt 5179  ◡ccnv 5623  dom cdm 5624   ↾ cres 5626   “ cima 5627   Fn wfn 6487  ⟶wf 6488  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7358  ℂcc 11024  ℝcr 11025  0cc0 11026  1c1 11027  ici 11028   + caddc 11029   · cmul 11031   < clt 11166   − cmin 11364  -cneg 11365   / cdiv 11794  2c2 12200  ℤcz 12488  ℝ+crp 12905  [,]cicc 13264  ℜcre 15020  ℑcim 15021  abscabs 15157  expce 15984  sincsin 15986  πcpi 15989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104  ax-addf 11105
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-map 8765  df-pm 8766  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-fi 9314  df-sup 9345  df-inf 9346  df-oi 9415  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-q 12862  df-rp 12906  df-xneg 13026  df-xadd 13027  df-xmul 13028  df-ioo 13265  df-ioc 13266  df-ico 13267  df-icc 13268  df-fz 13424  df-fzo 13571  df-fl 13712  df-mod 13790  df-seq 13925  df-exp 13985  df-fac 14197  df-bc 14226  df-hash 14254  df-shft 14990  df-cj 15022  df-re 15023  df-im 15024  df-sqrt 15158  df-abs 15159  df-limsup 15394  df-clim 15411  df-rlim 15412  df-sum 15610  df-ef 15990  df-sin 15992  df-cos 15993  df-pi 15995  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-mulr 17191  df-starv 17192  df-sca 17193  df-vsca 17194  df-ip 17195  df-tset 17196  df-ple 17197  df-ds 17199  df-unif 17200  df-hom 17201  df-cco 17202  df-rest 17342  df-topn 17343  df-0g 17361  df-gsum 17362  df-topgen 17363  df-pt 17364  df-prds 17367  df-xrs 17423  df-qtop 17428  df-imas 17429  df-xps 17431  df-mre 17505  df-mrc 17506  df-acs 17508  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18709  df-mulg 18998  df-cntz 19246  df-cmn 19711  df-psmet 21301  df-xmet 21302  df-met 21303  df-bl 21304  df-mopn 21305  df-fbas 21306  df-fg 21307  df-cnfld 21310  df-top 22838  df-topon 22855  df-topsp 22877  df-bases 22890  df-cld 22963  df-ntr 22964  df-cls 22965  df-nei 23042  df-lp 23080  df-perf 23081  df-cn 23171  df-cnp 23172  df-haus 23259  df-tx 23506  df-hmeo 23699  df-fil 23790  df-fm 23882  df-flim 23883  df-flf 23884  df-xms 24264  df-ms 24265  df-tms 24266  df-cncf 24827  df-limc 25823  df-dv 25824
This theorem is referenced by:  eff1o  26514
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