Theorem eff1olem 24294

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 5485	. . . 4 ⊢ (◡ℑ “ 𝐷) ⊆ dom ℑ
2		eff1olem.2	. . . 4 ⊢ 𝑆 = (◡ℑ “ 𝐷)
3		imf 13853	. . . . . 6 ⊢ ℑ:ℂ⟶ℝ
4	3	fdmi 6052	. . . . 5 ⊢ dom ℑ = ℂ
5	4	eqcomi 2631	. . . 4 ⊢ ℂ = dom ℑ
6	1, 2, 5	3sstr4i 3644	. . 3 ⊢ 𝑆 ⊆ ℂ
7		eff2 14829	. . . . . . 7 ⊢ exp:ℂ⟶(ℂ ∖ {0})
8	7	a1i 11	. . . . . 6 ⊢ (𝑆 ⊆ ℂ → exp:ℂ⟶(ℂ ∖ {0}))
9	8	feqmptd 6249	. . . . 5 ⊢ (𝑆 ⊆ ℂ → exp = (𝑦 ∈ ℂ ↦ (exp‘𝑦)))
10	9	reseq1d 5395	. . . 4 ⊢ (𝑆 ⊆ ℂ → (exp ↾ 𝑆) = ((𝑦 ∈ ℂ ↦ (exp‘𝑦)) ↾ 𝑆))
11		resmpt 5449	. . . 4 ⊢ (𝑆 ⊆ ℂ → ((𝑦 ∈ ℂ ↦ (exp‘𝑦)) ↾ 𝑆) = (𝑦 ∈ 𝑆 ↦ (exp‘𝑦)))
12	10, 11	eqtrd 2656	. . 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 3599	. . . 4 ⊢ (𝑦 ∈ 𝑆 → 𝑦 ∈ ℂ)
15	7	ffvelrni 6358	. . . 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		simpr 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ∈ (ℂ ∖ {0}))
19		eldifsn 4317	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
20	18, 19	sylib 208	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
21	20	simpld 475	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
22	20	simprd 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 ≠ 0)
23	21, 22	absrpcld 14187	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℝ+)
24		reeff1o 24201	. . . . . . . . 9 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ+
25		f1ocnv 6149	. . . . . . . . 9 ⊢ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ → ◡(exp ↾ ℝ):ℝ+–1-1-onto→ℝ)
26		f1of 6137	. . . . . . . . 9 ⊢ (◡(exp ↾ ℝ):ℝ+–1-1-onto→ℝ → ◡(exp ↾ ℝ):ℝ+⟶ℝ)
27	24, 25, 26	mp2b 10	. . . . . . . 8 ⊢ ◡(exp ↾ ℝ):ℝ+⟶ℝ
28	27	ffvelrni 6358	. . . . . . 7 ⊢ ((abs‘𝑥) ∈ ℝ+ → (◡(exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ)
29	23, 28	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡(exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ)
30	29	recnd 10068	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡(exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℂ)
31		ax-icn 9995	. . . . . 6 ⊢ i ∈ ℂ
32		eff1olem.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ⊆ ℝ)
33	32	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝐷 ⊆ ℝ)
34		eff1olem.1	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝑤 ∈ 𝐷 ↦ (exp‘(i · 𝑤)))
35		eqid 2622	. . . . . . . . . . . 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 2622	. . . . . . . . . . . 12 ⊢ (sin ↾ (-(π / 2)[,](π / 2))) = (sin ↾ (-(π / 2)[,](π / 2)))
39	34, 35, 32, 36, 37, 38	efif1olem4 24291	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐷–1-1-onto→(◡abs “ {1}))
40		f1ocnv 6149	. . . . . . . . . . 11 ⊢ (𝐹:𝐷–1-1-onto→(◡abs “ {1}) → ◡𝐹:(◡abs “ {1})–1-1-onto→𝐷)
41		f1of 6137	. . . . . . . . . . 11 ⊢ (◡𝐹:(◡abs “ {1})–1-1-onto→𝐷 → ◡𝐹:(◡abs “ {1})⟶𝐷)
42	39, 40, 41	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ◡𝐹:(◡abs “ {1})⟶𝐷)
43	42	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ◡𝐹:(◡abs “ {1})⟶𝐷)
44	21	abscld 14175	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℝ)
45	44	recnd 10068	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ∈ ℂ)
46	21, 22	absne0d 14186	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘𝑥) ≠ 0)
47	21, 45, 46	divcld 10801	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥 / (abs‘𝑥)) ∈ ℂ)
48	21, 45, 46	absdivd 14194	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘(𝑥 / (abs‘𝑥))) = ((abs‘𝑥) / (abs‘(abs‘𝑥))))
49		absidm 14063	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (abs‘(abs‘𝑥)) = (abs‘𝑥))
50	21, 49	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘(abs‘𝑥)) = (abs‘𝑥))
51	50	oveq2d 6666	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) / (abs‘(abs‘𝑥))) = ((abs‘𝑥) / (abs‘𝑥)))
52	45, 46	dividd 10799	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) / (abs‘𝑥)) = 1)
53	48, 51, 52	3eqtrd 2660	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (abs‘(𝑥 / (abs‘𝑥))) = 1)
54		absf 14077	. . . . . . . . . . 11 ⊢ abs:ℂ⟶ℝ
55		ffn 6045	. . . . . . . . . . 11 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
56		fniniseg 6338	. . . . . . . . . . 11 ⊢ (abs Fn ℂ → ((𝑥 / (abs‘𝑥)) ∈ (◡abs “ {1}) ↔ ((𝑥 / (abs‘𝑥)) ∈ ℂ ∧ (abs‘(𝑥 / (abs‘𝑥))) = 1)))
57	54, 55, 56	mp2b 10	. . . . . . . . . 10 ⊢ ((𝑥 / (abs‘𝑥)) ∈ (◡abs “ {1}) ↔ ((𝑥 / (abs‘𝑥)) ∈ ℂ ∧ (abs‘(𝑥 / (abs‘𝑥))) = 1))
58	47, 53, 57	sylanbrc 698	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝑥 / (abs‘𝑥)) ∈ (◡abs “ {1}))
59	43, 58	ffvelrnd 6360	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ 𝐷)
60	33, 59	sseldd 3604	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℝ)
61	60	recnd 10068	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℂ)
62		mulcl 10020	. . . . . 6 ⊢ ((i ∈ ℂ ∧ (◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ ℂ) → (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))) ∈ ℂ)
63	31, 61, 62	sylancr 695	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))) ∈ ℂ)
64	30, 63	addcld 10059	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ ℂ)
65	29, 60	crimd 13972	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (ℑ‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))) = (◡𝐹‘(𝑥 / (abs‘𝑥))))
66	65, 59	eqeltrd 2701	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (ℑ‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))) ∈ 𝐷)
67		ffn 6045	. . . . 5 ⊢ (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
68		elpreima 6337	. . . . 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)))
69	3, 67, 68	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))
70	64, 66, 69	sylanbrc 698	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ (◡ℑ “ 𝐷))
71	70, 2	syl6eleqr 2712	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ 𝑆)
72		efadd 14824	. . . . . . 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)))))))
73	30, 63, 72	syl2anc 693	. . . . . 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)))))))
74		fvres 6207	. . . . . . . . 9 ⊢ ((◡(exp ↾ ℝ)‘(abs‘𝑥)) ∈ ℝ → ((exp ↾ ℝ)‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) = (exp‘(◡(exp ↾ ℝ)‘(abs‘𝑥))))
75	29, 74	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) = (exp‘(◡(exp ↾ ℝ)‘(abs‘𝑥))))
76		f1ocnvfv2 6533	. . . . . . . . 9 ⊢ (((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ ∧ (abs‘𝑥) ∈ ℝ+) → ((exp ↾ ℝ)‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
77	24, 23, 76	sylancr 695	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((exp ↾ ℝ)‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
78	75, 77	eqtr3d 2658	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (exp‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) = (abs‘𝑥))
79		oveq2 6658	. . . . . . . . . . 11 ⊢ (𝑧 = (◡𝐹‘(𝑥 / (abs‘𝑥))) → (i · 𝑧) = (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))
80	79	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝑧 = (◡𝐹‘(𝑥 / (abs‘𝑥))) → (exp‘(i · 𝑧)) = (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))))
81		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → (i · 𝑤) = (i · 𝑧))
82	81	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (exp‘(i · 𝑤)) = (exp‘(i · 𝑧)))
83	82	cbvmptv 4750	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝐷 ↦ (exp‘(i · 𝑤))) = (𝑧 ∈ 𝐷 ↦ (exp‘(i · 𝑧)))
84	34, 83	eqtri 2644	. . . . . . . . . 10 ⊢ 𝐹 = (𝑧 ∈ 𝐷 ↦ (exp‘(i · 𝑧)))
85		fvex 6201	. . . . . . . . . 10 ⊢ (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ∈ V
86	80, 84, 85	fvmpt 6282	. . . . . . . . 9 ⊢ ((◡𝐹‘(𝑥 / (abs‘𝑥))) ∈ 𝐷 → (𝐹‘(◡𝐹‘(𝑥 / (abs‘𝑥)))) = (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))))
87	59, 86	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(◡𝐹‘(𝑥 / (abs‘𝑥)))) = (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))))
88	39	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝐹:𝐷–1-1-onto→(◡abs “ {1}))
89		f1ocnvfv2 6533	. . . . . . . . 9 ⊢ ((𝐹:𝐷–1-1-onto→(◡abs “ {1}) ∧ (𝑥 / (abs‘𝑥)) ∈ (◡abs “ {1})) → (𝐹‘(◡𝐹‘(𝑥 / (abs‘𝑥)))) = (𝑥 / (abs‘𝑥)))
90	88, 58, 89	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (𝐹‘(◡𝐹‘(𝑥 / (abs‘𝑥)))) = (𝑥 / (abs‘𝑥)))
91	87, 90	eqtr3d 2658	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) = (𝑥 / (abs‘𝑥)))
92	78, 91	oveq12d 6668	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((exp‘(◡(exp ↾ ℝ)‘(abs‘𝑥))) · (exp‘(i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))) = ((abs‘𝑥) · (𝑥 / (abs‘𝑥))))
93	21, 45, 46	divcan2d 10803	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((abs‘𝑥) · (𝑥 / (abs‘𝑥))) = 𝑥)
94	73, 92, 93	3eqtrrd 2661	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℂ ∖ {0})) → 𝑥 = (exp‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
95	94	adantrl 752	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ (ℂ ∖ {0}))) → 𝑥 = (exp‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
96		fveq2 6191	. . . . 5 ⊢ (𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) → (exp‘𝑦) = (exp‘((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
97	96	eqeq2d 2632	. . . 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))))))))
98	95, 97	syl5ibrcom 237	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ (ℂ ∖ {0}))) → (𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) → 𝑥 = (exp‘𝑦)))
99	14	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ℂ)
100	99	replimd 13937	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 = ((ℜ‘𝑦) + (i · (ℑ‘𝑦))))
101		absef 14927	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ → (abs‘(exp‘𝑦)) = (exp‘(ℜ‘𝑦)))
102	99, 101	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘(exp‘𝑦)) = (exp‘(ℜ‘𝑦)))
103	99	recld 13934	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℜ‘𝑦) ∈ ℝ)
104		fvres 6207	. . . . . . . . . . 11 ⊢ ((ℜ‘𝑦) ∈ ℝ → ((exp ↾ ℝ)‘(ℜ‘𝑦)) = (exp‘(ℜ‘𝑦)))
105	103, 104	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((exp ↾ ℝ)‘(ℜ‘𝑦)) = (exp‘(ℜ‘𝑦)))
106	102, 105	eqtr4d 2659	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (abs‘(exp‘𝑦)) = ((exp ↾ ℝ)‘(ℜ‘𝑦)))
107	106	fveq2d 6195	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) = (◡(exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))))
108		f1ocnvfv1 6532	. . . . . . . . 9 ⊢ (((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ ∧ (ℜ‘𝑦) ∈ ℝ) → (◡(exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))) = (ℜ‘𝑦))
109	24, 103, 108	sylancr 695	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡(exp ↾ ℝ)‘((exp ↾ ℝ)‘(ℜ‘𝑦))) = (ℜ‘𝑦))
110	107, 109	eqtrd 2656	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) = (ℜ‘𝑦))
111	99	imcld 13935	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℑ‘𝑦) ∈ ℝ)
112	111	recnd 10068	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℑ‘𝑦) ∈ ℂ)
113		mulcl 10020	. . . . . . . . . . . . . 14 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝑦) ∈ ℂ) → (i · (ℑ‘𝑦)) ∈ ℂ)
114	31, 112, 113	sylancr 695	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (i · (ℑ‘𝑦)) ∈ ℂ)
115		efcl 14813	. . . . . . . . . . . . 13 ⊢ ((i · (ℑ‘𝑦)) ∈ ℂ → (exp‘(i · (ℑ‘𝑦))) ∈ ℂ)
116	114, 115	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘(i · (ℑ‘𝑦))) ∈ ℂ)
117	103	recnd 10068	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℜ‘𝑦) ∈ ℂ)
118		efcl 14813	. . . . . . . . . . . . 13 ⊢ ((ℜ‘𝑦) ∈ ℂ → (exp‘(ℜ‘𝑦)) ∈ ℂ)
119	117, 118	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘(ℜ‘𝑦)) ∈ ℂ)
120		efne0 14827	. . . . . . . . . . . . 13 ⊢ ((ℜ‘𝑦) ∈ ℂ → (exp‘(ℜ‘𝑦)) ≠ 0)
121	117, 120	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘(ℜ‘𝑦)) ≠ 0)
122	116, 119, 121	divcan3d 10806	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))) / (exp‘(ℜ‘𝑦))) = (exp‘(i · (ℑ‘𝑦))))
123	100	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘𝑦) = (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))))
124		efadd 14824	. . . . . . . . . . . . . 14 ⊢ (((ℜ‘𝑦) ∈ ℂ ∧ (i · (ℑ‘𝑦)) ∈ ℂ) → (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
125	117, 114, 124	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘((ℜ‘𝑦) + (i · (ℑ‘𝑦)))) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
126	123, 125	eqtrd 2656	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (exp‘𝑦) = ((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))))
127	126, 102	oveq12d 6668	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((exp‘𝑦) / (abs‘(exp‘𝑦))) = (((exp‘(ℜ‘𝑦)) · (exp‘(i · (ℑ‘𝑦)))) / (exp‘(ℜ‘𝑦))))
128		elpreima 6337	. . . . . . . . . . . . . . . 16 ⊢ (ℑ Fn ℂ → (𝑦 ∈ (◡ℑ “ 𝐷) ↔ (𝑦 ∈ ℂ ∧ (ℑ‘𝑦) ∈ 𝐷)))
129	3, 67, 128	mp2b 10	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (◡ℑ “ 𝐷) ↔ (𝑦 ∈ ℂ ∧ (ℑ‘𝑦) ∈ 𝐷))
130	129	simprbi 480	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (◡ℑ “ 𝐷) → (ℑ‘𝑦) ∈ 𝐷)
131	130, 2	eleq2s 2719	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑆 → (ℑ‘𝑦) ∈ 𝐷)
132	131	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (ℑ‘𝑦) ∈ 𝐷)
133		oveq2 6658	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (ℑ‘𝑦) → (i · 𝑤) = (i · (ℑ‘𝑦)))
134	133	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝑤 = (ℑ‘𝑦) → (exp‘(i · 𝑤)) = (exp‘(i · (ℑ‘𝑦))))
135		fvex 6201	. . . . . . . . . . . . 13 ⊢ (exp‘(i · (ℑ‘𝑦))) ∈ V
136	134, 34, 135	fvmpt 6282	. . . . . . . . . . . 12 ⊢ ((ℑ‘𝑦) ∈ 𝐷 → (𝐹‘(ℑ‘𝑦)) = (exp‘(i · (ℑ‘𝑦))))
137	132, 136	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝐹‘(ℑ‘𝑦)) = (exp‘(i · (ℑ‘𝑦))))
138	122, 127, 137	3eqtr4d 2666	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((exp‘𝑦) / (abs‘(exp‘𝑦))) = (𝐹‘(ℑ‘𝑦)))
139	138	fveq2d 6195	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))) = (◡𝐹‘(𝐹‘(ℑ‘𝑦))))
140		f1ocnvfv1 6532	. . . . . . . . . 10 ⊢ ((𝐹:𝐷–1-1-onto→(◡abs “ {1}) ∧ (ℑ‘𝑦) ∈ 𝐷) → (◡𝐹‘(𝐹‘(ℑ‘𝑦))) = (ℑ‘𝑦))
141	39, 131, 140	syl2an 494	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡𝐹‘(𝐹‘(ℑ‘𝑦))) = (ℑ‘𝑦))
142	139, 141	eqtrd 2656	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))) = (ℑ‘𝑦))
143	142	oveq2d 6666	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦))))) = (i · (ℑ‘𝑦)))
144	110, 143	oveq12d 6668	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ((◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))) = ((ℜ‘𝑦) + (i · (ℑ‘𝑦))))
145	100, 144	eqtr4d 2659	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 = ((◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))) + (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))))
146		fveq2 6191	. . . . . . . 8 ⊢ (𝑥 = (exp‘𝑦) → (abs‘𝑥) = (abs‘(exp‘𝑦)))
147	146	fveq2d 6195	. . . . . . 7 ⊢ (𝑥 = (exp‘𝑦) → (◡(exp ↾ ℝ)‘(abs‘𝑥)) = (◡(exp ↾ ℝ)‘(abs‘(exp‘𝑦))))
148		id 22	. . . . . . . . . 10 ⊢ (𝑥 = (exp‘𝑦) → 𝑥 = (exp‘𝑦))
149	148, 146	oveq12d 6668	. . . . . . . . 9 ⊢ (𝑥 = (exp‘𝑦) → (𝑥 / (abs‘𝑥)) = ((exp‘𝑦) / (abs‘(exp‘𝑦))))
150	149	fveq2d 6195	. . . . . . . 8 ⊢ (𝑥 = (exp‘𝑦) → (◡𝐹‘(𝑥 / (abs‘𝑥))) = (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦)))))
151	150	oveq2d 6666	. . . . . . 7 ⊢ (𝑥 = (exp‘𝑦) → (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))) = (i · (◡𝐹‘((exp‘𝑦) / (abs‘(exp‘𝑦))))))
152	147, 151	oveq12d 6668	. . . . . 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)))))))
153	152	eqeq2d 2632	. . . . 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))))))))
154	145, 153	syl5ibrcom 237	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝑥 = (exp‘𝑦) → 𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
155	154	adantrr 753	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ (ℂ ∖ {0}))) → (𝑥 = (exp‘𝑦) → 𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥)))))))
156	98, 155	impbid 202	. 2 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ (ℂ ∖ {0}))) → (𝑦 = ((◡(exp ↾ ℝ)‘(abs‘𝑥)) + (i · (◡𝐹‘(𝑥 / (abs‘𝑥))))) ↔ 𝑥 = (exp‘𝑦)))
157	13, 17, 71, 156	f1o2d 6887	1 ⊢ (𝜑 → (exp ↾ 𝑆):𝑆–1-1-onto→(ℂ ∖ {0}))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913   ∖ cdif 3571   ⊆ wss 3574  {csn 4177   class class class wbr 4653   ↦ cmpt 4729  ^◡ccnv 5113  dom cdm 5114   ↾ cres 5116   “ cima 5117   Fn wfn 5883  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  ℝcr 9935  0cc0 9936  1c1 9937  ici 9938   + caddc 9939   · cmul 9941   < clt 10074   − cmin 10266  -cneg 10267   / cdiv 10684  2c2 11070  ℤcz 11377  ℝ⁺crp 11832  [,]cicc 12178  ℜcre 13837  ℑcim 13838  abscabs 13974  expce 14792  sincsin 14794  πcpi 14797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631

This theorem is referenced by:  eff1o  24295