Step |
Hyp |
Ref |
Expression |
1 |
|
efopn.j |
⊢ 𝐽 = ( TopOpen ‘ ℂfld ) |
2 |
|
logf1o |
⊢ log : ( ℂ ∖ { 0 } ) –1-1-onto→ ran log |
3 |
|
f1orn |
⊢ ( log : ( ℂ ∖ { 0 } ) –1-1-onto→ ran log ↔ ( log Fn ( ℂ ∖ { 0 } ) ∧ Fun ◡ log ) ) |
4 |
3
|
simprbi |
⊢ ( log : ( ℂ ∖ { 0 } ) –1-1-onto→ ran log → Fun ◡ log ) |
5 |
|
funcnvres |
⊢ ( Fun ◡ log → ◡ ( log ↾ ( ℂ ∖ ( -∞ (,] 0 ) ) ) = ( ◡ log ↾ ( log “ ( ℂ ∖ ( -∞ (,] 0 ) ) ) ) ) |
6 |
2 4 5
|
mp2b |
⊢ ◡ ( log ↾ ( ℂ ∖ ( -∞ (,] 0 ) ) ) = ( ◡ log ↾ ( log “ ( ℂ ∖ ( -∞ (,] 0 ) ) ) ) |
7 |
|
df-log |
⊢ log = ◡ ( exp ↾ ( ◡ ℑ “ ( - π (,] π ) ) ) |
8 |
7
|
cnveqi |
⊢ ◡ log = ◡ ◡ ( exp ↾ ( ◡ ℑ “ ( - π (,] π ) ) ) |
9 |
|
relres |
⊢ Rel ( exp ↾ ( ◡ ℑ “ ( - π (,] π ) ) ) |
10 |
|
dfrel2 |
⊢ ( Rel ( exp ↾ ( ◡ ℑ “ ( - π (,] π ) ) ) ↔ ◡ ◡ ( exp ↾ ( ◡ ℑ “ ( - π (,] π ) ) ) = ( exp ↾ ( ◡ ℑ “ ( - π (,] π ) ) ) ) |
11 |
9 10
|
mpbi |
⊢ ◡ ◡ ( exp ↾ ( ◡ ℑ “ ( - π (,] π ) ) ) = ( exp ↾ ( ◡ ℑ “ ( - π (,] π ) ) ) |
12 |
8 11
|
eqtri |
⊢ ◡ log = ( exp ↾ ( ◡ ℑ “ ( - π (,] π ) ) ) |
13 |
12
|
reseq1i |
⊢ ( ◡ log ↾ ( log “ ( ℂ ∖ ( -∞ (,] 0 ) ) ) ) = ( ( exp ↾ ( ◡ ℑ “ ( - π (,] π ) ) ) ↾ ( log “ ( ℂ ∖ ( -∞ (,] 0 ) ) ) ) |
14 |
|
imassrn |
⊢ ( log “ ( ℂ ∖ ( -∞ (,] 0 ) ) ) ⊆ ran log |
15 |
|
logrn |
⊢ ran log = ( ◡ ℑ “ ( - π (,] π ) ) |
16 |
14 15
|
sseqtri |
⊢ ( log “ ( ℂ ∖ ( -∞ (,] 0 ) ) ) ⊆ ( ◡ ℑ “ ( - π (,] π ) ) |
17 |
|
resabs1 |
⊢ ( ( log “ ( ℂ ∖ ( -∞ (,] 0 ) ) ) ⊆ ( ◡ ℑ “ ( - π (,] π ) ) → ( ( exp ↾ ( ◡ ℑ “ ( - π (,] π ) ) ) ↾ ( log “ ( ℂ ∖ ( -∞ (,] 0 ) ) ) ) = ( exp ↾ ( log “ ( ℂ ∖ ( -∞ (,] 0 ) ) ) ) ) |
18 |
16 17
|
ax-mp |
⊢ ( ( exp ↾ ( ◡ ℑ “ ( - π (,] π ) ) ) ↾ ( log “ ( ℂ ∖ ( -∞ (,] 0 ) ) ) ) = ( exp ↾ ( log “ ( ℂ ∖ ( -∞ (,] 0 ) ) ) ) |
19 |
6 13 18
|
3eqtri |
⊢ ◡ ( log ↾ ( ℂ ∖ ( -∞ (,] 0 ) ) ) = ( exp ↾ ( log “ ( ℂ ∖ ( -∞ (,] 0 ) ) ) ) |
20 |
19
|
imaeq1i |
⊢ ( ◡ ( log ↾ ( ℂ ∖ ( -∞ (,] 0 ) ) ) “ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) = ( ( exp ↾ ( log “ ( ℂ ∖ ( -∞ (,] 0 ) ) ) ) “ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) |
21 |
|
cnxmet |
⊢ ( abs ∘ − ) ∈ ( ∞Met ‘ ℂ ) |
22 |
|
0cnd |
⊢ ( ( 𝑅 ∈ ℝ+ ∧ 𝑅 < π ) → 0 ∈ ℂ ) |
23 |
|
rpxr |
⊢ ( 𝑅 ∈ ℝ+ → 𝑅 ∈ ℝ* ) |
24 |
23
|
adantr |
⊢ ( ( 𝑅 ∈ ℝ+ ∧ 𝑅 < π ) → 𝑅 ∈ ℝ* ) |
25 |
|
blssm |
⊢ ( ( ( abs ∘ − ) ∈ ( ∞Met ‘ ℂ ) ∧ 0 ∈ ℂ ∧ 𝑅 ∈ ℝ* ) → ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ⊆ ℂ ) |
26 |
21 22 24 25
|
mp3an2i |
⊢ ( ( 𝑅 ∈ ℝ+ ∧ 𝑅 < π ) → ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ⊆ ℂ ) |
27 |
26
|
sselda |
⊢ ( ( ( 𝑅 ∈ ℝ+ ∧ 𝑅 < π ) ∧ 𝑥 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) → 𝑥 ∈ ℂ ) |
28 |
27
|
imcld |
⊢ ( ( ( 𝑅 ∈ ℝ+ ∧ 𝑅 < π ) ∧ 𝑥 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) → ( ℑ ‘ 𝑥 ) ∈ ℝ ) |
29 |
|
efopnlem1 |
⊢ ( ( ( 𝑅 ∈ ℝ+ ∧ 𝑅 < π ) ∧ 𝑥 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) → ( abs ‘ ( ℑ ‘ 𝑥 ) ) < π ) |
30 |
|
pire |
⊢ π ∈ ℝ |
31 |
|
abslt |
⊢ ( ( ( ℑ ‘ 𝑥 ) ∈ ℝ ∧ π ∈ ℝ ) → ( ( abs ‘ ( ℑ ‘ 𝑥 ) ) < π ↔ ( - π < ( ℑ ‘ 𝑥 ) ∧ ( ℑ ‘ 𝑥 ) < π ) ) ) |
32 |
28 30 31
|
sylancl |
⊢ ( ( ( 𝑅 ∈ ℝ+ ∧ 𝑅 < π ) ∧ 𝑥 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) → ( ( abs ‘ ( ℑ ‘ 𝑥 ) ) < π ↔ ( - π < ( ℑ ‘ 𝑥 ) ∧ ( ℑ ‘ 𝑥 ) < π ) ) ) |
33 |
29 32
|
mpbid |
⊢ ( ( ( 𝑅 ∈ ℝ+ ∧ 𝑅 < π ) ∧ 𝑥 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) → ( - π < ( ℑ ‘ 𝑥 ) ∧ ( ℑ ‘ 𝑥 ) < π ) ) |
34 |
33
|
simpld |
⊢ ( ( ( 𝑅 ∈ ℝ+ ∧ 𝑅 < π ) ∧ 𝑥 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) → - π < ( ℑ ‘ 𝑥 ) ) |
35 |
33
|
simprd |
⊢ ( ( ( 𝑅 ∈ ℝ+ ∧ 𝑅 < π ) ∧ 𝑥 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) → ( ℑ ‘ 𝑥 ) < π ) |
36 |
30
|
renegcli |
⊢ - π ∈ ℝ |
37 |
36
|
rexri |
⊢ - π ∈ ℝ* |
38 |
30
|
rexri |
⊢ π ∈ ℝ* |
39 |
|
elioo2 |
⊢ ( ( - π ∈ ℝ* ∧ π ∈ ℝ* ) → ( ( ℑ ‘ 𝑥 ) ∈ ( - π (,) π ) ↔ ( ( ℑ ‘ 𝑥 ) ∈ ℝ ∧ - π < ( ℑ ‘ 𝑥 ) ∧ ( ℑ ‘ 𝑥 ) < π ) ) ) |
40 |
37 38 39
|
mp2an |
⊢ ( ( ℑ ‘ 𝑥 ) ∈ ( - π (,) π ) ↔ ( ( ℑ ‘ 𝑥 ) ∈ ℝ ∧ - π < ( ℑ ‘ 𝑥 ) ∧ ( ℑ ‘ 𝑥 ) < π ) ) |
41 |
28 34 35 40
|
syl3anbrc |
⊢ ( ( ( 𝑅 ∈ ℝ+ ∧ 𝑅 < π ) ∧ 𝑥 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) → ( ℑ ‘ 𝑥 ) ∈ ( - π (,) π ) ) |
42 |
|
imf |
⊢ ℑ : ℂ ⟶ ℝ |
43 |
|
ffn |
⊢ ( ℑ : ℂ ⟶ ℝ → ℑ Fn ℂ ) |
44 |
|
elpreima |
⊢ ( ℑ Fn ℂ → ( 𝑥 ∈ ( ◡ ℑ “ ( - π (,) π ) ) ↔ ( 𝑥 ∈ ℂ ∧ ( ℑ ‘ 𝑥 ) ∈ ( - π (,) π ) ) ) ) |
45 |
42 43 44
|
mp2b |
⊢ ( 𝑥 ∈ ( ◡ ℑ “ ( - π (,) π ) ) ↔ ( 𝑥 ∈ ℂ ∧ ( ℑ ‘ 𝑥 ) ∈ ( - π (,) π ) ) ) |
46 |
27 41 45
|
sylanbrc |
⊢ ( ( ( 𝑅 ∈ ℝ+ ∧ 𝑅 < π ) ∧ 𝑥 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) → 𝑥 ∈ ( ◡ ℑ “ ( - π (,) π ) ) ) |
47 |
46
|
ex |
⊢ ( ( 𝑅 ∈ ℝ+ ∧ 𝑅 < π ) → ( 𝑥 ∈ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) → 𝑥 ∈ ( ◡ ℑ “ ( - π (,) π ) ) ) ) |
48 |
47
|
ssrdv |
⊢ ( ( 𝑅 ∈ ℝ+ ∧ 𝑅 < π ) → ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ⊆ ( ◡ ℑ “ ( - π (,) π ) ) ) |
49 |
|
df-ima |
⊢ ( log “ ( ℂ ∖ ( -∞ (,] 0 ) ) ) = ran ( log ↾ ( ℂ ∖ ( -∞ (,] 0 ) ) ) |
50 |
|
eqid |
⊢ ( ℂ ∖ ( -∞ (,] 0 ) ) = ( ℂ ∖ ( -∞ (,] 0 ) ) |
51 |
50
|
logf1o2 |
⊢ ( log ↾ ( ℂ ∖ ( -∞ (,] 0 ) ) ) : ( ℂ ∖ ( -∞ (,] 0 ) ) –1-1-onto→ ( ◡ ℑ “ ( - π (,) π ) ) |
52 |
|
f1ofo |
⊢ ( ( log ↾ ( ℂ ∖ ( -∞ (,] 0 ) ) ) : ( ℂ ∖ ( -∞ (,] 0 ) ) –1-1-onto→ ( ◡ ℑ “ ( - π (,) π ) ) → ( log ↾ ( ℂ ∖ ( -∞ (,] 0 ) ) ) : ( ℂ ∖ ( -∞ (,] 0 ) ) –onto→ ( ◡ ℑ “ ( - π (,) π ) ) ) |
53 |
|
forn |
⊢ ( ( log ↾ ( ℂ ∖ ( -∞ (,] 0 ) ) ) : ( ℂ ∖ ( -∞ (,] 0 ) ) –onto→ ( ◡ ℑ “ ( - π (,) π ) ) → ran ( log ↾ ( ℂ ∖ ( -∞ (,] 0 ) ) ) = ( ◡ ℑ “ ( - π (,) π ) ) ) |
54 |
51 52 53
|
mp2b |
⊢ ran ( log ↾ ( ℂ ∖ ( -∞ (,] 0 ) ) ) = ( ◡ ℑ “ ( - π (,) π ) ) |
55 |
49 54
|
eqtri |
⊢ ( log “ ( ℂ ∖ ( -∞ (,] 0 ) ) ) = ( ◡ ℑ “ ( - π (,) π ) ) |
56 |
48 55
|
sseqtrrdi |
⊢ ( ( 𝑅 ∈ ℝ+ ∧ 𝑅 < π ) → ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ⊆ ( log “ ( ℂ ∖ ( -∞ (,] 0 ) ) ) ) |
57 |
|
resima2 |
⊢ ( ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ⊆ ( log “ ( ℂ ∖ ( -∞ (,] 0 ) ) ) → ( ( exp ↾ ( log “ ( ℂ ∖ ( -∞ (,] 0 ) ) ) ) “ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) = ( exp “ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) ) |
58 |
56 57
|
syl |
⊢ ( ( 𝑅 ∈ ℝ+ ∧ 𝑅 < π ) → ( ( exp ↾ ( log “ ( ℂ ∖ ( -∞ (,] 0 ) ) ) ) “ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) = ( exp “ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) ) |
59 |
20 58
|
syl5eq |
⊢ ( ( 𝑅 ∈ ℝ+ ∧ 𝑅 < π ) → ( ◡ ( log ↾ ( ℂ ∖ ( -∞ (,] 0 ) ) ) “ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) = ( exp “ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) ) |
60 |
50
|
logcn |
⊢ ( log ↾ ( ℂ ∖ ( -∞ (,] 0 ) ) ) ∈ ( ( ℂ ∖ ( -∞ (,] 0 ) ) –cn→ ℂ ) |
61 |
|
difss |
⊢ ( ℂ ∖ ( -∞ (,] 0 ) ) ⊆ ℂ |
62 |
|
ssid |
⊢ ℂ ⊆ ℂ |
63 |
|
eqid |
⊢ ( 𝐽 ↾t ( ℂ ∖ ( -∞ (,] 0 ) ) ) = ( 𝐽 ↾t ( ℂ ∖ ( -∞ (,] 0 ) ) ) |
64 |
1
|
cnfldtopon |
⊢ 𝐽 ∈ ( TopOn ‘ ℂ ) |
65 |
64
|
toponrestid |
⊢ 𝐽 = ( 𝐽 ↾t ℂ ) |
66 |
1 63 65
|
cncfcn |
⊢ ( ( ( ℂ ∖ ( -∞ (,] 0 ) ) ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( ( ℂ ∖ ( -∞ (,] 0 ) ) –cn→ ℂ ) = ( ( 𝐽 ↾t ( ℂ ∖ ( -∞ (,] 0 ) ) ) Cn 𝐽 ) ) |
67 |
61 62 66
|
mp2an |
⊢ ( ( ℂ ∖ ( -∞ (,] 0 ) ) –cn→ ℂ ) = ( ( 𝐽 ↾t ( ℂ ∖ ( -∞ (,] 0 ) ) ) Cn 𝐽 ) |
68 |
60 67
|
eleqtri |
⊢ ( log ↾ ( ℂ ∖ ( -∞ (,] 0 ) ) ) ∈ ( ( 𝐽 ↾t ( ℂ ∖ ( -∞ (,] 0 ) ) ) Cn 𝐽 ) |
69 |
1
|
cnfldtopn |
⊢ 𝐽 = ( MetOpen ‘ ( abs ∘ − ) ) |
70 |
69
|
blopn |
⊢ ( ( ( abs ∘ − ) ∈ ( ∞Met ‘ ℂ ) ∧ 0 ∈ ℂ ∧ 𝑅 ∈ ℝ* ) → ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ∈ 𝐽 ) |
71 |
21 22 24 70
|
mp3an2i |
⊢ ( ( 𝑅 ∈ ℝ+ ∧ 𝑅 < π ) → ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ∈ 𝐽 ) |
72 |
|
cnima |
⊢ ( ( ( log ↾ ( ℂ ∖ ( -∞ (,] 0 ) ) ) ∈ ( ( 𝐽 ↾t ( ℂ ∖ ( -∞ (,] 0 ) ) ) Cn 𝐽 ) ∧ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ∈ 𝐽 ) → ( ◡ ( log ↾ ( ℂ ∖ ( -∞ (,] 0 ) ) ) “ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) ∈ ( 𝐽 ↾t ( ℂ ∖ ( -∞ (,] 0 ) ) ) ) |
73 |
68 71 72
|
sylancr |
⊢ ( ( 𝑅 ∈ ℝ+ ∧ 𝑅 < π ) → ( ◡ ( log ↾ ( ℂ ∖ ( -∞ (,] 0 ) ) ) “ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) ∈ ( 𝐽 ↾t ( ℂ ∖ ( -∞ (,] 0 ) ) ) ) |
74 |
59 73
|
eqeltrrd |
⊢ ( ( 𝑅 ∈ ℝ+ ∧ 𝑅 < π ) → ( exp “ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) ∈ ( 𝐽 ↾t ( ℂ ∖ ( -∞ (,] 0 ) ) ) ) |
75 |
1
|
cnfldtop |
⊢ 𝐽 ∈ Top |
76 |
50
|
logdmopn |
⊢ ( ℂ ∖ ( -∞ (,] 0 ) ) ∈ ( TopOpen ‘ ℂfld ) |
77 |
76 1
|
eleqtrri |
⊢ ( ℂ ∖ ( -∞ (,] 0 ) ) ∈ 𝐽 |
78 |
|
restopn2 |
⊢ ( ( 𝐽 ∈ Top ∧ ( ℂ ∖ ( -∞ (,] 0 ) ) ∈ 𝐽 ) → ( ( exp “ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) ∈ ( 𝐽 ↾t ( ℂ ∖ ( -∞ (,] 0 ) ) ) ↔ ( ( exp “ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) ∈ 𝐽 ∧ ( exp “ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) ⊆ ( ℂ ∖ ( -∞ (,] 0 ) ) ) ) ) |
79 |
75 77 78
|
mp2an |
⊢ ( ( exp “ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) ∈ ( 𝐽 ↾t ( ℂ ∖ ( -∞ (,] 0 ) ) ) ↔ ( ( exp “ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) ∈ 𝐽 ∧ ( exp “ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) ⊆ ( ℂ ∖ ( -∞ (,] 0 ) ) ) ) |
80 |
74 79
|
sylib |
⊢ ( ( 𝑅 ∈ ℝ+ ∧ 𝑅 < π ) → ( ( exp “ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) ∈ 𝐽 ∧ ( exp “ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) ⊆ ( ℂ ∖ ( -∞ (,] 0 ) ) ) ) |
81 |
80
|
simpld |
⊢ ( ( 𝑅 ∈ ℝ+ ∧ 𝑅 < π ) → ( exp “ ( 0 ( ball ‘ ( abs ∘ − ) ) 𝑅 ) ) ∈ 𝐽 ) |