Theorem eff1olem 26531

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

Step	Hyp	Ref	Expression
1		cnvimass 6035	. . . 4 ⊢ (◡ℑ “ 𝐷) ⊆ dom ℑ
2		eff1olem.2	. . . 4 ⊢ 𝑆 = (◡ℑ “ 𝐷)
3		imf 15067	. . . . . 6 ⊢ ℑ:ℂ⟶ℝ
4	3	fdmi 6667	. . . . 5 ⊢ dom ℑ = ℂ
5	4	eqcomi 2748	. . . 4 ⊢ ℂ = dom ℑ
6	1, 2, 5	3sstr4i 3966	. . 3 ⊢ 𝑆 ⊆ ℂ
7		eff2 16058	. . . . . . 7 ⊢ exp:ℂ⟶(ℂ ∖ {0})
8	7	a1i 11	. . . . . 6 ⊢ (𝑆 ⊆ ℂ → exp:ℂ⟶(ℂ ∖ {0}))
9	8	feqmptd 6896	. . . . 5 ⊢ (𝑆 ⊆ ℂ → exp = (𝑦 ∈ ℂ ↦ (exp‘𝑦)))
10	9	reseq1d 5931	. . . 4 ⊢ (𝑆 ⊆ ℂ → (exp ↾ 𝑆) = ((𝑦 ∈ ℂ ↦ (exp‘𝑦)) ↾ 𝑆))
11		resmpt 5990	. . . 4 ⊢ (𝑆 ⊆ ℂ → ((𝑦 ∈ ℂ ↦ (exp‘𝑦)) ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (exp‘𝑦)))
12	10, 11	eqtrd 2774	. . 3 ⊢ (𝑆 ⊆ ℂ → (exp ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (exp‘𝑦)))
13	6, 12	ax-mp 5	. 2 ⊢ (exp ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (exp‘𝑦))
14	6	sseli 3911	. . . 4 ⊢ (𝑦 ∈ 𝑆 → 𝑦 ∈ ℂ)
15	7	ffvelcdmi 7025	. . . 4 ⊢ (𝑦 ∈ ℂ → (exp‘𝑦) ∈ (ℂ ∖ {0}))
16	14, 15	syl 17	. . 3 ⊢ (𝑦 ∈ 𝑆 → (exp‘𝑦) ∈ (ℂ ∖ {0}))
17	16	adantl 482	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘𝑦) ∈ (ℂ ∖ {0}))
18		eldifsn 4720	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
19	18	bilani 505	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
20	19	simpld 495	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
21	19	simprd 496	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ≠ 0)
22	20, 21	absrpcld 15405	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℝ+)
23		reeff1o 26431	. . . . . . . . 9 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ+
24		f1ocnv 6780	. . . . . . . . 9 ⊢ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ → ◡(exp ↾ ℝ):ℝ+–1-1-onto→ℝ)
25		f1of 6768	. . . . . . . . 9 ⊢ (◡(exp ↾ ℝ):ℝ+–1-1-onto→ℝ → ◡(exp ↾ ℝ):ℝ+⟶ℝ)
26	23, 24, 25	mp2b 10	. . . . . . . 8 ⊢ ◡(exp ↾ ℝ):ℝ+⟶ℝ
27	26	ffvelcdmi 7025	. . . . . . 7 ⊢ ((abs‘𝑥) ∈ ℝ+ → (◡(exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ)
28	22, 27	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡(exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ)
29	28	recnd 11165	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡(exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℂ)
30		ax-icn 11089	. . . . . 6 ⊢ i ∈ ℂ
31		eff1olem.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ⊆ ℝ)
32	31	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝐷 ⊆ ℝ)
33		eff1olem.1	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝑤 ∈ 𝐷 ↦ (exp‘(i · 𝑤)))
34		eqid 2739	. . . . . . . . . . . 12 ⊢ (◡abs “ {1}) = (◡abs “ {1})
35		eff1olem.4	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (abs‘(𝑥 − 𝑦)) < (2 · π))
36		eff1olem.5	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ∃𝑦 ∈ 𝐷 ((𝑧 − 𝑦) / (2 · π)) ∈ ℤ)
37		eqid 2739	. . . . . . . . . . . 12 ⊢ (sin ↾ (-(π / 2)[,](π / 2))) = (sin ↾ (-(π / 2)[,](π / 2)))
38	33, 34, 31, 35, 36, 37	efif1olem4 26528	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐷–1-1-onto→(◡abs “ {1}))
39		f1ocnv 6780	. . . . . . . . . . 11 ⊢ (𝐹:𝐷–1-1-onto→(◡abs “ {1}) → ◡𝐹:(◡abs “ {1})–1-1-onto→𝐷)
40		f1of 6768	. . . . . . . . . . 11 ⊢ (◡𝐹:(◡abs “ {1})–1-1-onto→𝐷 → ◡𝐹:(◡abs “ {1})⟶𝐷)
41	38, 39, 40	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ◡𝐹:(◡abs “ {1})⟶𝐷)
42	41	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ◡𝐹:(◡abs “ {1})⟶𝐷)
43	20	abscld 15393	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℝ)
44	43	recnd 11165	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℂ)
45	20, 21	absne0d 15404	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ≠ 0)
46	20, 44, 45	divcld 11923	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥 / (abs‘𝑥)) ∈ ℂ)
47	20, 44, 45	absdivd 15412	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘(𝑥 / (abs‘𝑥))) = ((abs‘𝑥) / (abs‘(abs‘𝑥))))
48		absidm 15278	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (abs‘(abs‘𝑥)) = (abs‘𝑥))
49	20, 48	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘(abs‘𝑥)) = (abs‘𝑥))
50	49	oveq2d 7373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) / (abs‘(abs‘𝑥))) = ((abs‘𝑥) / (abs‘𝑥)))
51	44, 45	dividd 11921	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) / (abs‘𝑥)) = 1)
52	47, 50, 51	3eqtrd 2778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘(𝑥 / (abs‘𝑥))) = 1)
53		absf 15292	. . . . . . . . . . 11 ⊢ abs:ℂ⟶ℝ
54		ffn 6656	. . . . . . . . . . 11 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
55		fniniseg 7002	. . . . . . . . . . 11 ⊢ (abs Fn ℂ → ((𝑥 / (abs‘𝑥)) ∈ (◡abs “ {1}) ↔ ((𝑥 / (abs‘𝑥)) ∈ ℂ ∧ (abs‘(𝑥 / (abs‘𝑥))) = 1)))
56	53, 54, 55	mp2b 10	. . . . . . . . . 10 ⊢ ((𝑥 / (abs‘𝑥)) ∈ (◡abs “ {1}) ↔ ((𝑥 / (abs‘𝑥)) ∈ ℂ ∧ (abs‘(𝑥 / (abs‘𝑥))) = 1))
57	46, 52, 56	sylanbrc 589	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥 / (abs‘𝑥)) ∈ (◡abs “ {1}))
58	42, 57	ffvelcdmd 7027	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ 𝐷)
59	32, 58	sseldd 3916	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℝ)
60	59	recnd 11165	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℂ)
61		mulcl 11114	. . . . . 6 ⊢ ((i ∈ ℂ ∧ (◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℂ) → (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))) ∈ ℂ)
62	30, 60, 61	sylancr 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))) ∈ ℂ)
63	29, 62	addcld 11156	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ ℂ)
64	28, 59	crimd 15186	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (ℑ‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))) = (◡𝐹‘(𝑥 / (abs‘𝑥))))
65	64, 58	eqeltrd 2839	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (ℑ‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))) ∈ 𝐷)
66		ffn 6656	. . . . 5 ⊢ (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
67		elpreima 7000	. . . . 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)))
68	3, 66, 67	mp2b 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))
69	63, 65, 68	sylanbrc 589	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ (◡ℑ “ 𝐷))
70	69, 2	eleqtrrdi 2850	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ 𝑆)
71		efadd 16051	. . . . . . 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)))))))
72	29, 62, 71	syl2anc 590	. . . . . 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)))))))
73	28	fvresd 6848	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) = (exp‘(◡(exp ↾ ℝ)‘(abs‘𝑥))))
74		f1ocnvfv2 7222	. . . . . . . . 9 ⊢ (((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ ∧ (abs‘𝑥) ∈ ℝ+) → ((exp ↾ ℝ)‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
75	23, 22, 74	sylancr 593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
76	73, 75	eqtr3d 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (exp‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
77		oveq2 7365	. . . . . . . . . . 11 ⊢ (𝑧 = (◡𝐹‘(𝑥 / (abs‘𝑥))) → (i · 𝑧) = (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))
78	77	fveq2d 6832	. . . . . . . . . 10 ⊢ (𝑧 = (◡𝐹‘(𝑥 / (abs‘𝑥))) → (exp‘(i · 𝑧)) = (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))))
79		oveq2 7365	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → (i · 𝑤) = (i · 𝑧))
80	79	fveq2d 6832	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (exp‘(i · 𝑤)) = (exp‘(i · 𝑧)))
81	80	cbvmptv 5177	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝐷 ↦ (exp‘(i · 𝑤))) = (𝑧 ∈ 𝐷 ↦ (exp‘(i · 𝑧)))
82	33, 81	eqtri 2762	. . . . . . . . . 10 ⊢ 𝐹 = (𝑧 ∈ 𝐷 ↦ (exp‘(i · 𝑧)))
83		fvex 6841	. . . . . . . . . 10 ⊢ (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ V
84	78, 82, 83	fvmpt 6936	. . . . . . . . 9 ⊢ ((◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ 𝐷 → (𝐹‘(◡𝐹‘(𝑥 / (abs‘𝑥)))) = (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))))
85	58, 84	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(◡𝐹‘(𝑥 / (abs‘𝑥)))) = (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))))
86	38	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝐹:𝐷–1-1-onto→(◡abs “ {1}))
87		f1ocnvfv2 7222	. . . . . . . . 9 ⊢ ((𝐹:𝐷–1-1-onto→(◡abs “ {1}) ∧ (𝑥 / (abs‘𝑥)) ∈ (◡abs “ {1})) → (𝐹‘(◡𝐹‘(𝑥 / (abs‘𝑥)))) = (𝑥 / (abs‘𝑥)))
88	86, 57, 87	syl2anc 590	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(◡𝐹‘(𝑥 / (abs‘𝑥)))) = (𝑥 / (abs‘𝑥)))
89	85, 88	eqtr3d 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) = (𝑥 / (abs‘𝑥)))
90	76, 89	oveq12d 7375	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((exp‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) · (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))) = ((abs‘𝑥) · (𝑥 / (abs‘𝑥))))
91	20, 44, 45	divcan2d 11925	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) · (𝑥 / (abs‘𝑥))) = 𝑥)
92	72, 90, 91	3eqtrrd 2779	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 = (exp‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
93	92	adantrl 722	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ (ℂ ∖ {0}))) → 𝑥 = (exp‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
94		fveq2 6828	. . . . 5 ⊢ (𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) → (exp‘𝑦) = (exp‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
95	94	eqeq2d 2750	. . . 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))))))))
96	93, 95	syl5ibrcom 248	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ (ℂ ∖ {0}))) → (𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) → 𝑥 = (exp‘𝑦)))
97	14	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ℂ)
98	97	replimd 15151	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 = ((ℜ‘𝑦) + (i · (ℑ‘𝑦))))
99		absef 16156	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ → (abs‘(exp‘𝑦)) = (exp‘(ℜ‘𝑦)))
100	97, 99	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘(exp‘𝑦)) = (exp‘(ℜ‘𝑦)))
101	97	recld 15148	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℜ‘𝑦) ∈ ℝ)
102	101	fvresd 6848	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((exp ↾ ℝ)‘(ℜ‘𝑦)) = (exp‘(ℜ‘𝑦)))
103	100, 102	eqtr4d 2777	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘(exp‘𝑦)) = ((exp ↾ ℝ)‘(ℜ‘𝑦)))
104	103	fveq2d 6832	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) = (◡(exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))))
105		f1ocnvfv1 7221	. . . . . . . . 9 ⊢ (((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ ∧ (ℜ‘𝑦) ∈ ℝ) → (◡(exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))) = (ℜ‘𝑦))
106	23, 101, 105	sylancr 593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡(exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))) = (ℜ‘𝑦))
107	104, 106	eqtrd 2774	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) = (ℜ‘𝑦))
108	97	imcld 15149	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℑ‘𝑦) ∈ ℝ)
109	108	recnd 11165	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℑ‘𝑦) ∈ ℂ)
110		mulcl 11114	. . . . . . . . . . . . . 14 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝑦) ∈ ℂ) → (i · (ℑ‘𝑦)) ∈ ℂ)
111	30, 109, 110	sylancr 593	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (i · (ℑ‘𝑦)) ∈ ℂ)
112		efcl 16039	. . . . . . . . . . . . 13 ⊢ ((i · (ℑ‘𝑦)) ∈ ℂ → (exp‘(i · (ℑ‘𝑦))) ∈ ℂ)
113	111, 112	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘(i · (ℑ‘𝑦))) ∈ ℂ)
114	101	recnd 11165	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℜ‘𝑦) ∈ ℂ)
115		efcl 16039	. . . . . . . . . . . . 13 ⊢ ((ℜ‘𝑦) ∈ ℂ → (exp‘(ℜ‘𝑦)) ∈ ℂ)
116	114, 115	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘(ℜ‘𝑦)) ∈ ℂ)
117		efne0 16055	. . . . . . . . . . . . 13 ⊢ ((ℜ‘𝑦) ∈ ℂ → (exp‘(ℜ‘𝑦)) ≠ 0)
118	114, 117	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘(ℜ‘𝑦)) ≠ 0)
119	113, 116, 118	divcan3d 11928	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))) / (exp‘(ℜ‘𝑦))) = (exp‘(i · (ℑ‘𝑦))))
120	98	fveq2d 6832	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘𝑦) = (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))))
121		efadd 16051	. . . . . . . . . . . . . 14 ⊢ (((ℜ‘𝑦) ∈ ℂ ∧ (i · (ℑ‘𝑦)) ∈ ℂ) → (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
122	114, 111, 121	syl2anc 590	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
123	120, 122	eqtrd 2774	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘𝑦) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
124	123, 100	oveq12d 7375	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((exp‘𝑦) / (abs‘(exp‘𝑦))) = (((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))) / (exp‘(ℜ‘𝑦))))
125		elpreima 7000	. . . . . . . . . . . . . . . 16 ⊢ (ℑ Fn ℂ → (𝑦 ∈ (◡ℑ “ 𝐷) ↔ (𝑦 ∈ ℂ ∧ (ℑ‘𝑦) ∈ 𝐷)))
126	3, 66, 125	mp2b 10	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (◡ℑ “ 𝐷) ↔ (𝑦 ∈ ℂ ∧ (ℑ‘𝑦) ∈ 𝐷))
127	126	simprbi 498	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (◡ℑ “ 𝐷) → (ℑ‘𝑦) ∈ 𝐷)
128	127, 2	eleq2s 2857	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑆 → (ℑ‘𝑦) ∈ 𝐷)
129	128	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℑ‘𝑦) ∈ 𝐷)
130		oveq2 7365	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (ℑ‘𝑦) → (i · 𝑤) = (i · (ℑ‘𝑦)))
131	130	fveq2d 6832	. . . . . . . . . . . . 13 ⊢ (𝑤 = (ℑ‘𝑦) → (exp‘(i · 𝑤)) = (exp‘(i · (ℑ‘𝑦))))
132		fvex 6841	. . . . . . . . . . . . 13 ⊢ (exp‘(i · (ℑ‘𝑦))) ∈ V
133	131, 33, 132	fvmpt 6936	. . . . . . . . . . . 12 ⊢ ((ℑ‘𝑦) ∈ 𝐷 → (𝐹‘(ℑ‘𝑦)) = (exp‘(i · (ℑ‘𝑦))))
134	129, 133	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝐹‘(ℑ‘𝑦)) = (exp‘(i · (ℑ‘𝑦))))
135	119, 124, 134	3eqtr4d 2784	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((exp‘𝑦) / (abs‘(exp‘𝑦))) = (𝐹‘(ℑ‘𝑦)))
136	135	fveq2d 6832	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))) = (◡𝐹‘(𝐹‘(ℑ‘𝑦))))
137		f1ocnvfv1 7221	. . . . . . . . . 10 ⊢ ((𝐹:𝐷–1-1-onto→(◡abs “ {1}) ∧ (ℑ‘𝑦) ∈ 𝐷) → (◡𝐹‘(𝐹‘(ℑ‘𝑦))) = (ℑ‘𝑦))
138	38, 128, 137	syl2an 602	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡𝐹‘(𝐹‘(ℑ‘𝑦))) = (ℑ‘𝑦))
139	136, 138	eqtrd 2774	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))) = (ℑ‘𝑦))
140	139	oveq2d 7373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦))))) = (i · (ℑ‘𝑦)))
141	107, 140	oveq12d 7375	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))) = ((ℜ‘𝑦) + (i · (ℑ‘𝑦))))
142	98, 141	eqtr4d 2777	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 = ((◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))))
143		fveq2 6828	. . . . . . . 8 ⊢ (𝑥 = (exp‘𝑦) → (abs‘𝑥) = (abs‘(exp‘𝑦)))
144	143	fveq2d 6832	. . . . . . 7 ⊢ (𝑥 = (exp‘𝑦) → (◡(exp ↾ ℝ)‘(abs‘𝑥)) = (◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))))
145		id 22	. . . . . . . . . 10 ⊢ (𝑥 = (exp‘𝑦) → 𝑥 = (exp‘𝑦))
146	145, 143	oveq12d 7375	. . . . . . . . 9 ⊢ (𝑥 = (exp‘𝑦) → (𝑥 / (abs‘𝑥)) = ((exp‘𝑦) / (abs‘(exp‘𝑦))))
147	146	fveq2d 6832	. . . . . . . 8 ⊢ (𝑥 = (exp‘𝑦) → (◡𝐹‘(𝑥 / (abs‘𝑥))) = (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))
148	147	oveq2d 7373	. . . . . . 7 ⊢ (𝑥 = (exp‘𝑦) → (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))) = (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦))))))
149	144, 148	oveq12d 7375	. . . . . 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)))))))
150	149	eqeq2d 2750	. . . . 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))))))))
151	142, 150	syl5ibrcom 248	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝑥 = (exp‘𝑦) → 𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
152	151	adantrr 723	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ (ℂ ∖ {0}))) → (𝑥 = (exp‘𝑦) → 𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
153	96, 152	impbid 213	. 2 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ (ℂ ∖ {0}))) → (𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ↔ 𝑥 = (exp‘𝑦)))
154	13, 17, 70, 153	f1o2d 7611	1 ⊢ (𝜑 → (exp ↾ 𝑆):𝑆–1-1-onto→(ℂ ∖ {0}))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∃wrex 3063   ∖ cdif 3880   ⊆ wss 3883  {csn 4556   class class class wbr 5073   ↦ cmpt 5154  ^◡ccnv 5618  dom cdm 5619   ↾ cres 5621   “ cima 5622   Fn wfn 6481  ⟶wf 6482  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7357  ℂcc 11028  ℝcr 11029  0cc0 11030  1c1 11031  ici 11032   + caddc 11033   · cmul 11035   < clt 11171   − cmin 11369  -cneg 11370   / cdiv 11799  2c2 12228  ℤcz 12516  ℝ⁺crp 12934  [,]cicc 13293  ℜcre 15051  ℑcim 15052  abscabs 15188  expce 16018  sincsin 16020  πcpi 16023

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-inf2 9554  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107  ax-pre-sup 11108  ax-addf 11109

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-tp 4561  df-op 4563  df-uni 4840  df-int 4879  df-iun 4924  df-iin 4925  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7621  df-om 7808  df-1st 7932  df-2nd 7933  df-supp 8102  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-2o 8397  df-er 8634  df-map 8766  df-pm 8767  df-ixp 8837  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-fsupp 9266  df-fi 9315  df-sup 9346  df-inf 9347  df-oi 9416  df-card 9855  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-div 11800  df-nn 12167  df-2 12236  df-3 12237  df-4 12238  df-5 12239  df-6 12240  df-7 12241  df-8 12242  df-9 12243  df-n0 12430  df-z 12517  df-dec 12637  df-uz 12781  df-q 12891  df-rp 12935  df-xneg 13055  df-xadd 13056  df-xmul 13057  df-ioo 13294  df-ioc 13295  df-ico 13296  df-icc 13297  df-fz 13454  df-fzo 13601  df-fl 13743  df-mod 13821  df-seq 13956  df-exp 14016  df-fac 14228  df-bc 14257  df-hash 14285  df-shft 15021  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-limsup 15425  df-clim 15442  df-rlim 15443  df-sum 15641  df-ef 16024  df-sin 16026  df-cos 16027  df-pi 16029  df-struct 17109  df-sets 17126  df-slot 17144  df-ndx 17156  df-base 17172  df-ress 17193  df-plusg 17225  df-mulr 17226  df-starv 17227  df-sca 17228  df-vsca 17229  df-ip 17230  df-tset 17231  df-ple 17232  df-ds 17234  df-unif 17235  df-hom 17236  df-cco 17237  df-rest 17377  df-topn 17378  df-0g 17396  df-gsum 17397  df-topgen 17398  df-pt 17399  df-prds 17402  df-xrs 17458  df-qtop 17463  df-imas 17464  df-xps 17466  df-mre 17540  df-mrc 17541  df-acs 17543  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-submnd 18744  df-mulg 19036  df-cntz 19284  df-cmn 19749  df-psmet 21340  df-xmet 21341  df-met 21342  df-bl 21343  df-mopn 21344  df-fbas 21345  df-fg 21346  df-cnfld 21349  df-top 22878  df-topon 22895  df-topsp 22917  df-bases 22930  df-cld 23003  df-ntr 23004  df-cls 23005  df-nei 23082  df-lp 23120  df-perf 23121  df-cn 23211  df-cnp 23212  df-haus 23299  df-tx 23546  df-hmeo 23739  df-fil 23830  df-fm 23922  df-flim 23923  df-flf 23924  df-xms 24304  df-ms 24305  df-tms 24306  df-cncf 24864  df-limc 25852  df-dv 25853

This theorem is referenced by:  eff1o  26532