| Step |
Hyp |
Ref |
Expression |
| 1 |
|
0re |
⊢ 0 ∈ ℝ |
| 2 |
|
atandm2 |
⊢ ( 𝐴 ∈ dom arctan ↔ ( 𝐴 ∈ ℂ ∧ ( 1 − ( i · 𝐴 ) ) ≠ 0 ∧ ( 1 + ( i · 𝐴 ) ) ≠ 0 ) ) |
| 3 |
2
|
simp1bi |
⊢ ( 𝐴 ∈ dom arctan → 𝐴 ∈ ℂ ) |
| 4 |
3
|
recld |
⊢ ( 𝐴 ∈ dom arctan → ( ℜ ‘ 𝐴 ) ∈ ℝ ) |
| 5 |
|
leloe |
⊢ ( ( 0 ∈ ℝ ∧ ( ℜ ‘ 𝐴 ) ∈ ℝ ) → ( 0 ≤ ( ℜ ‘ 𝐴 ) ↔ ( 0 < ( ℜ ‘ 𝐴 ) ∨ 0 = ( ℜ ‘ 𝐴 ) ) ) ) |
| 6 |
1 4 5
|
sylancr |
⊢ ( 𝐴 ∈ dom arctan → ( 0 ≤ ( ℜ ‘ 𝐴 ) ↔ ( 0 < ( ℜ ‘ 𝐴 ) ∨ 0 = ( ℜ ‘ 𝐴 ) ) ) ) |
| 7 |
6
|
biimpa |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 ≤ ( ℜ ‘ 𝐴 ) ) → ( 0 < ( ℜ ‘ 𝐴 ) ∨ 0 = ( ℜ ‘ 𝐴 ) ) ) |
| 8 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
| 9 |
|
ax-icn |
⊢ i ∈ ℂ |
| 10 |
|
mulcl |
⊢ ( ( i ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( i · 𝐴 ) ∈ ℂ ) |
| 11 |
9 3 10
|
sylancr |
⊢ ( 𝐴 ∈ dom arctan → ( i · 𝐴 ) ∈ ℂ ) |
| 12 |
|
addcl |
⊢ ( ( 1 ∈ ℂ ∧ ( i · 𝐴 ) ∈ ℂ ) → ( 1 + ( i · 𝐴 ) ) ∈ ℂ ) |
| 13 |
8 11 12
|
sylancr |
⊢ ( 𝐴 ∈ dom arctan → ( 1 + ( i · 𝐴 ) ) ∈ ℂ ) |
| 14 |
2
|
simp3bi |
⊢ ( 𝐴 ∈ dom arctan → ( 1 + ( i · 𝐴 ) ) ≠ 0 ) |
| 15 |
13 14
|
logcld |
⊢ ( 𝐴 ∈ dom arctan → ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ∈ ℂ ) |
| 16 |
|
subcl |
⊢ ( ( 1 ∈ ℂ ∧ ( i · 𝐴 ) ∈ ℂ ) → ( 1 − ( i · 𝐴 ) ) ∈ ℂ ) |
| 17 |
8 11 16
|
sylancr |
⊢ ( 𝐴 ∈ dom arctan → ( 1 − ( i · 𝐴 ) ) ∈ ℂ ) |
| 18 |
2
|
simp2bi |
⊢ ( 𝐴 ∈ dom arctan → ( 1 − ( i · 𝐴 ) ) ≠ 0 ) |
| 19 |
17 18
|
logcld |
⊢ ( 𝐴 ∈ dom arctan → ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ∈ ℂ ) |
| 20 |
15 19
|
addcld |
⊢ ( 𝐴 ∈ dom arctan → ( ( log ‘ ( 1 + ( i · 𝐴 ) ) ) + ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ∈ ℂ ) |
| 21 |
20
|
adantr |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( log ‘ ( 1 + ( i · 𝐴 ) ) ) + ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ∈ ℂ ) |
| 22 |
|
pire |
⊢ π ∈ ℝ |
| 23 |
22
|
renegcli |
⊢ - π ∈ ℝ |
| 24 |
23
|
a1i |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → - π ∈ ℝ ) |
| 25 |
19
|
adantr |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ∈ ℂ ) |
| 26 |
25
|
imcld |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ∈ ℝ ) |
| 27 |
15
|
adantr |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ∈ ℂ ) |
| 28 |
27
|
imcld |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) ∈ ℝ ) |
| 29 |
28 26
|
readdcld |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) + ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ) ∈ ℝ ) |
| 30 |
17
|
adantr |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( 1 − ( i · 𝐴 ) ) ∈ ℂ ) |
| 31 |
|
im1 |
⊢ ( ℑ ‘ 1 ) = 0 |
| 32 |
31
|
oveq1i |
⊢ ( ( ℑ ‘ 1 ) − ( ℑ ‘ ( i · 𝐴 ) ) ) = ( 0 − ( ℑ ‘ ( i · 𝐴 ) ) ) |
| 33 |
|
df-neg |
⊢ - ( ℑ ‘ ( i · 𝐴 ) ) = ( 0 − ( ℑ ‘ ( i · 𝐴 ) ) ) |
| 34 |
32 33
|
eqtr4i |
⊢ ( ( ℑ ‘ 1 ) − ( ℑ ‘ ( i · 𝐴 ) ) ) = - ( ℑ ‘ ( i · 𝐴 ) ) |
| 35 |
11
|
adantr |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( i · 𝐴 ) ∈ ℂ ) |
| 36 |
|
imsub |
⊢ ( ( 1 ∈ ℂ ∧ ( i · 𝐴 ) ∈ ℂ ) → ( ℑ ‘ ( 1 − ( i · 𝐴 ) ) ) = ( ( ℑ ‘ 1 ) − ( ℑ ‘ ( i · 𝐴 ) ) ) ) |
| 37 |
8 35 36
|
sylancr |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( 1 − ( i · 𝐴 ) ) ) = ( ( ℑ ‘ 1 ) − ( ℑ ‘ ( i · 𝐴 ) ) ) ) |
| 38 |
3
|
adantr |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → 𝐴 ∈ ℂ ) |
| 39 |
|
reim |
⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) = ( ℑ ‘ ( i · 𝐴 ) ) ) |
| 40 |
38 39
|
syl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℜ ‘ 𝐴 ) = ( ℑ ‘ ( i · 𝐴 ) ) ) |
| 41 |
40
|
negeqd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → - ( ℜ ‘ 𝐴 ) = - ( ℑ ‘ ( i · 𝐴 ) ) ) |
| 42 |
34 37 41
|
3eqtr4a |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( 1 − ( i · 𝐴 ) ) ) = - ( ℜ ‘ 𝐴 ) ) |
| 43 |
4
|
lt0neg2d |
⊢ ( 𝐴 ∈ dom arctan → ( 0 < ( ℜ ‘ 𝐴 ) ↔ - ( ℜ ‘ 𝐴 ) < 0 ) ) |
| 44 |
43
|
biimpa |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → - ( ℜ ‘ 𝐴 ) < 0 ) |
| 45 |
42 44
|
eqbrtrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( 1 − ( i · 𝐴 ) ) ) < 0 ) |
| 46 |
|
argimlt0 |
⊢ ( ( ( 1 − ( i · 𝐴 ) ) ∈ ℂ ∧ ( ℑ ‘ ( 1 − ( i · 𝐴 ) ) ) < 0 ) → ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ∈ ( - π (,) 0 ) ) |
| 47 |
30 45 46
|
syl2anc |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ∈ ( - π (,) 0 ) ) |
| 48 |
|
eliooord |
⊢ ( ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ∈ ( - π (,) 0 ) → ( - π < ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ∧ ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) < 0 ) ) |
| 49 |
47 48
|
syl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( - π < ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ∧ ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) < 0 ) ) |
| 50 |
49
|
simpld |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → - π < ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ) |
| 51 |
13
|
adantr |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( 1 + ( i · 𝐴 ) ) ∈ ℂ ) |
| 52 |
|
simpr |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → 0 < ( ℜ ‘ 𝐴 ) ) |
| 53 |
|
imadd |
⊢ ( ( 1 ∈ ℂ ∧ ( i · 𝐴 ) ∈ ℂ ) → ( ℑ ‘ ( 1 + ( i · 𝐴 ) ) ) = ( ( ℑ ‘ 1 ) + ( ℑ ‘ ( i · 𝐴 ) ) ) ) |
| 54 |
8 35 53
|
sylancr |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( 1 + ( i · 𝐴 ) ) ) = ( ( ℑ ‘ 1 ) + ( ℑ ‘ ( i · 𝐴 ) ) ) ) |
| 55 |
40
|
oveq2d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℑ ‘ 1 ) + ( ℜ ‘ 𝐴 ) ) = ( ( ℑ ‘ 1 ) + ( ℑ ‘ ( i · 𝐴 ) ) ) ) |
| 56 |
31
|
oveq1i |
⊢ ( ( ℑ ‘ 1 ) + ( ℜ ‘ 𝐴 ) ) = ( 0 + ( ℜ ‘ 𝐴 ) ) |
| 57 |
38
|
recld |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℜ ‘ 𝐴 ) ∈ ℝ ) |
| 58 |
57
|
recnd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℜ ‘ 𝐴 ) ∈ ℂ ) |
| 59 |
58
|
addid2d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( 0 + ( ℜ ‘ 𝐴 ) ) = ( ℜ ‘ 𝐴 ) ) |
| 60 |
56 59
|
eqtrid |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℑ ‘ 1 ) + ( ℜ ‘ 𝐴 ) ) = ( ℜ ‘ 𝐴 ) ) |
| 61 |
54 55 60
|
3eqtr2d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( 1 + ( i · 𝐴 ) ) ) = ( ℜ ‘ 𝐴 ) ) |
| 62 |
52 61
|
breqtrrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → 0 < ( ℑ ‘ ( 1 + ( i · 𝐴 ) ) ) ) |
| 63 |
|
argimgt0 |
⊢ ( ( ( 1 + ( i · 𝐴 ) ) ∈ ℂ ∧ 0 < ( ℑ ‘ ( 1 + ( i · 𝐴 ) ) ) ) → ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) ∈ ( 0 (,) π ) ) |
| 64 |
51 62 63
|
syl2anc |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) ∈ ( 0 (,) π ) ) |
| 65 |
|
eliooord |
⊢ ( ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) ∈ ( 0 (,) π ) → ( 0 < ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) ∧ ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) < π ) ) |
| 66 |
64 65
|
syl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( 0 < ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) ∧ ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) < π ) ) |
| 67 |
66
|
simpld |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → 0 < ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) ) |
| 68 |
28 26
|
ltaddpos2d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( 0 < ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) ↔ ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) < ( ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) + ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ) ) ) |
| 69 |
67 68
|
mpbid |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) < ( ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) + ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ) ) |
| 70 |
24 26 29 50 69
|
lttrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → - π < ( ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) + ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ) ) |
| 71 |
27 25
|
imaddd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( ( log ‘ ( 1 + ( i · 𝐴 ) ) ) + ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ) = ( ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) + ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ) ) |
| 72 |
70 71
|
breqtrrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → - π < ( ℑ ‘ ( ( log ‘ ( 1 + ( i · 𝐴 ) ) ) + ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ) ) |
| 73 |
22
|
a1i |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → π ∈ ℝ ) |
| 74 |
|
0red |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → 0 ∈ ℝ ) |
| 75 |
49
|
simprd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) < 0 ) |
| 76 |
26 74 28 75
|
ltadd2dd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) + ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ) < ( ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) + 0 ) ) |
| 77 |
28
|
recnd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) ∈ ℂ ) |
| 78 |
77
|
addid1d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) + 0 ) = ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) ) |
| 79 |
76 78
|
breqtrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) + ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ) < ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) ) |
| 80 |
66
|
simprd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) < π ) |
| 81 |
29 28 73 79 80
|
lttrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) + ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ) < π ) |
| 82 |
29 73 81
|
ltled |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( ℑ ‘ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) + ( ℑ ‘ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ) ≤ π ) |
| 83 |
71 82
|
eqbrtrd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( ( log ‘ ( 1 + ( i · 𝐴 ) ) ) + ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ) ≤ π ) |
| 84 |
|
ellogrn |
⊢ ( ( ( log ‘ ( 1 + ( i · 𝐴 ) ) ) + ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ∈ ran log ↔ ( ( ( log ‘ ( 1 + ( i · 𝐴 ) ) ) + ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ∈ ℂ ∧ - π < ( ℑ ‘ ( ( log ‘ ( 1 + ( i · 𝐴 ) ) ) + ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ) ∧ ( ℑ ‘ ( ( log ‘ ( 1 + ( i · 𝐴 ) ) ) + ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ) ≤ π ) ) |
| 85 |
21 72 83 84
|
syl3anbrc |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 < ( ℜ ‘ 𝐴 ) ) → ( ( log ‘ ( 1 + ( i · 𝐴 ) ) ) + ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ∈ ran log ) |
| 86 |
|
0red |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) → 0 ∈ ℝ ) |
| 87 |
11
|
adantr |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) → ( i · 𝐴 ) ∈ ℂ ) |
| 88 |
|
simpr |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) → 0 = ( ℜ ‘ 𝐴 ) ) |
| 89 |
3
|
adantr |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) → 𝐴 ∈ ℂ ) |
| 90 |
89 39
|
syl |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) → ( ℜ ‘ 𝐴 ) = ( ℑ ‘ ( i · 𝐴 ) ) ) |
| 91 |
88 90
|
eqtr2d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) → ( ℑ ‘ ( i · 𝐴 ) ) = 0 ) |
| 92 |
87 91
|
reim0bd |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) → ( i · 𝐴 ) ∈ ℝ ) |
| 93 |
15 19
|
addcomd |
⊢ ( 𝐴 ∈ dom arctan → ( ( log ‘ ( 1 + ( i · 𝐴 ) ) ) + ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) = ( ( log ‘ ( 1 − ( i · 𝐴 ) ) ) + ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) ) |
| 94 |
93
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ 0 ≤ ( i · 𝐴 ) ) → ( ( log ‘ ( 1 + ( i · 𝐴 ) ) ) + ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) = ( ( log ‘ ( 1 − ( i · 𝐴 ) ) ) + ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) ) |
| 95 |
|
logrncl |
⊢ ( ( ( 1 − ( i · 𝐴 ) ) ∈ ℂ ∧ ( 1 − ( i · 𝐴 ) ) ≠ 0 ) → ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ∈ ran log ) |
| 96 |
17 18 95
|
syl2anc |
⊢ ( 𝐴 ∈ dom arctan → ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ∈ ran log ) |
| 97 |
96
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ 0 ≤ ( i · 𝐴 ) ) → ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ∈ ran log ) |
| 98 |
|
1re |
⊢ 1 ∈ ℝ |
| 99 |
92
|
adantr |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ 0 ≤ ( i · 𝐴 ) ) → ( i · 𝐴 ) ∈ ℝ ) |
| 100 |
|
readdcl |
⊢ ( ( 1 ∈ ℝ ∧ ( i · 𝐴 ) ∈ ℝ ) → ( 1 + ( i · 𝐴 ) ) ∈ ℝ ) |
| 101 |
98 99 100
|
sylancr |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ 0 ≤ ( i · 𝐴 ) ) → ( 1 + ( i · 𝐴 ) ) ∈ ℝ ) |
| 102 |
|
0red |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ 0 ≤ ( i · 𝐴 ) ) → 0 ∈ ℝ ) |
| 103 |
|
1red |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ 0 ≤ ( i · 𝐴 ) ) → 1 ∈ ℝ ) |
| 104 |
|
0lt1 |
⊢ 0 < 1 |
| 105 |
104
|
a1i |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ 0 ≤ ( i · 𝐴 ) ) → 0 < 1 ) |
| 106 |
|
addge01 |
⊢ ( ( 1 ∈ ℝ ∧ ( i · 𝐴 ) ∈ ℝ ) → ( 0 ≤ ( i · 𝐴 ) ↔ 1 ≤ ( 1 + ( i · 𝐴 ) ) ) ) |
| 107 |
98 92 106
|
sylancr |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) → ( 0 ≤ ( i · 𝐴 ) ↔ 1 ≤ ( 1 + ( i · 𝐴 ) ) ) ) |
| 108 |
107
|
biimpa |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ 0 ≤ ( i · 𝐴 ) ) → 1 ≤ ( 1 + ( i · 𝐴 ) ) ) |
| 109 |
102 103 101 105 108
|
ltletrd |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ 0 ≤ ( i · 𝐴 ) ) → 0 < ( 1 + ( i · 𝐴 ) ) ) |
| 110 |
101 109
|
elrpd |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ 0 ≤ ( i · 𝐴 ) ) → ( 1 + ( i · 𝐴 ) ) ∈ ℝ+ ) |
| 111 |
110
|
relogcld |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ 0 ≤ ( i · 𝐴 ) ) → ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ∈ ℝ ) |
| 112 |
|
logrnaddcl |
⊢ ( ( ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ∈ ran log ∧ ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ∈ ℝ ) → ( ( log ‘ ( 1 − ( i · 𝐴 ) ) ) + ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) ∈ ran log ) |
| 113 |
97 111 112
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ 0 ≤ ( i · 𝐴 ) ) → ( ( log ‘ ( 1 − ( i · 𝐴 ) ) ) + ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ) ∈ ran log ) |
| 114 |
94 113
|
eqeltrd |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ 0 ≤ ( i · 𝐴 ) ) → ( ( log ‘ ( 1 + ( i · 𝐴 ) ) ) + ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ∈ ran log ) |
| 115 |
|
logrncl |
⊢ ( ( ( 1 + ( i · 𝐴 ) ) ∈ ℂ ∧ ( 1 + ( i · 𝐴 ) ) ≠ 0 ) → ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ∈ ran log ) |
| 116 |
13 14 115
|
syl2anc |
⊢ ( 𝐴 ∈ dom arctan → ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ∈ ran log ) |
| 117 |
116
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ ( i · 𝐴 ) ≤ 0 ) → ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ∈ ran log ) |
| 118 |
92
|
adantr |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ ( i · 𝐴 ) ≤ 0 ) → ( i · 𝐴 ) ∈ ℝ ) |
| 119 |
|
resubcl |
⊢ ( ( 1 ∈ ℝ ∧ ( i · 𝐴 ) ∈ ℝ ) → ( 1 − ( i · 𝐴 ) ) ∈ ℝ ) |
| 120 |
98 118 119
|
sylancr |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ ( i · 𝐴 ) ≤ 0 ) → ( 1 − ( i · 𝐴 ) ) ∈ ℝ ) |
| 121 |
|
0red |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ ( i · 𝐴 ) ≤ 0 ) → 0 ∈ ℝ ) |
| 122 |
|
1red |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ ( i · 𝐴 ) ≤ 0 ) → 1 ∈ ℝ ) |
| 123 |
104
|
a1i |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ ( i · 𝐴 ) ≤ 0 ) → 0 < 1 ) |
| 124 |
|
1m0e1 |
⊢ ( 1 − 0 ) = 1 |
| 125 |
|
1red |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) → 1 ∈ ℝ ) |
| 126 |
92 86 125
|
lesub2d |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) → ( ( i · 𝐴 ) ≤ 0 ↔ ( 1 − 0 ) ≤ ( 1 − ( i · 𝐴 ) ) ) ) |
| 127 |
126
|
biimpa |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ ( i · 𝐴 ) ≤ 0 ) → ( 1 − 0 ) ≤ ( 1 − ( i · 𝐴 ) ) ) |
| 128 |
124 127
|
eqbrtrrid |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ ( i · 𝐴 ) ≤ 0 ) → 1 ≤ ( 1 − ( i · 𝐴 ) ) ) |
| 129 |
121 122 120 123 128
|
ltletrd |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ ( i · 𝐴 ) ≤ 0 ) → 0 < ( 1 − ( i · 𝐴 ) ) ) |
| 130 |
120 129
|
elrpd |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ ( i · 𝐴 ) ≤ 0 ) → ( 1 − ( i · 𝐴 ) ) ∈ ℝ+ ) |
| 131 |
130
|
relogcld |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ ( i · 𝐴 ) ≤ 0 ) → ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ∈ ℝ ) |
| 132 |
|
logrnaddcl |
⊢ ( ( ( log ‘ ( 1 + ( i · 𝐴 ) ) ) ∈ ran log ∧ ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ∈ ℝ ) → ( ( log ‘ ( 1 + ( i · 𝐴 ) ) ) + ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ∈ ran log ) |
| 133 |
117 131 132
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) ∧ ( i · 𝐴 ) ≤ 0 ) → ( ( log ‘ ( 1 + ( i · 𝐴 ) ) ) + ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ∈ ran log ) |
| 134 |
86 92 114 133
|
lecasei |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 = ( ℜ ‘ 𝐴 ) ) → ( ( log ‘ ( 1 + ( i · 𝐴 ) ) ) + ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ∈ ran log ) |
| 135 |
85 134
|
jaodan |
⊢ ( ( 𝐴 ∈ dom arctan ∧ ( 0 < ( ℜ ‘ 𝐴 ) ∨ 0 = ( ℜ ‘ 𝐴 ) ) ) → ( ( log ‘ ( 1 + ( i · 𝐴 ) ) ) + ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ∈ ran log ) |
| 136 |
7 135
|
syldan |
⊢ ( ( 𝐴 ∈ dom arctan ∧ 0 ≤ ( ℜ ‘ 𝐴 ) ) → ( ( log ‘ ( 1 + ( i · 𝐴 ) ) ) + ( log ‘ ( 1 − ( i · 𝐴 ) ) ) ) ∈ ran log ) |