Step |
Hyp |
Ref |
Expression |
1 |
|
sqrtcvallem5 |
⊢ ( 𝐴 ∈ ℂ → ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ∈ ℝ ) |
2 |
1
|
recnd |
⊢ ( 𝐴 ∈ ℂ → ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ∈ ℂ ) |
3 |
|
ax-icn |
⊢ i ∈ ℂ |
4 |
3
|
a1i |
⊢ ( 𝐴 ∈ ℂ → i ∈ ℂ ) |
5 |
|
neg1rr |
⊢ - 1 ∈ ℝ |
6 |
|
1re |
⊢ 1 ∈ ℝ |
7 |
5 6
|
ifcli |
⊢ if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) ∈ ℝ |
8 |
7
|
a1i |
⊢ ( 𝐴 ∈ ℂ → if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) ∈ ℝ ) |
9 |
|
sqrtcvallem3 |
⊢ ( 𝐴 ∈ ℂ → ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ∈ ℝ ) |
10 |
8 9
|
remulcld |
⊢ ( 𝐴 ∈ ℂ → ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ∈ ℝ ) |
11 |
10
|
recnd |
⊢ ( 𝐴 ∈ ℂ → ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ∈ ℂ ) |
12 |
4 11
|
mulcld |
⊢ ( 𝐴 ∈ ℂ → ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ∈ ℂ ) |
13 |
2 12
|
addcld |
⊢ ( 𝐴 ∈ ℂ → ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ∈ ℂ ) |
14 |
|
id |
⊢ ( 𝐴 ∈ ℂ → 𝐴 ∈ ℂ ) |
15 |
|
binom2 |
⊢ ( ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ∈ ℂ ∧ ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ∈ ℂ ) → ( ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ↑ 2 ) = ( ( ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ↑ 2 ) + ( 2 · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) · ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ) + ( ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ↑ 2 ) ) ) |
16 |
2 12 15
|
syl2anc |
⊢ ( 𝐴 ∈ ℂ → ( ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ↑ 2 ) = ( ( ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ↑ 2 ) + ( 2 · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) · ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ) + ( ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ↑ 2 ) ) ) |
17 |
|
abscl |
⊢ ( 𝐴 ∈ ℂ → ( abs ‘ 𝐴 ) ∈ ℝ ) |
18 |
|
recl |
⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ∈ ℝ ) |
19 |
17 18
|
readdcld |
⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) ∈ ℝ ) |
20 |
19
|
rehalfcld |
⊢ ( 𝐴 ∈ ℂ → ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ∈ ℝ ) |
21 |
20
|
recnd |
⊢ ( 𝐴 ∈ ℂ → ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ∈ ℂ ) |
22 |
21
|
sqsqrtd |
⊢ ( 𝐴 ∈ ℂ → ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ↑ 2 ) = ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) |
23 |
4 11
|
sqmuld |
⊢ ( 𝐴 ∈ ℂ → ( ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ↑ 2 ) = ( ( i ↑ 2 ) · ( ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ↑ 2 ) ) ) |
24 |
|
i2 |
⊢ ( i ↑ 2 ) = - 1 |
25 |
24
|
a1i |
⊢ ( 𝐴 ∈ ℂ → ( i ↑ 2 ) = - 1 ) |
26 |
8
|
recnd |
⊢ ( 𝐴 ∈ ℂ → if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) ∈ ℂ ) |
27 |
9
|
recnd |
⊢ ( 𝐴 ∈ ℂ → ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ∈ ℂ ) |
28 |
26 27
|
sqmuld |
⊢ ( 𝐴 ∈ ℂ → ( ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ↑ 2 ) = ( ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) ↑ 2 ) · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ↑ 2 ) ) ) |
29 |
|
ovif |
⊢ ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) ↑ 2 ) = if ( ( ℑ ‘ 𝐴 ) < 0 , ( - 1 ↑ 2 ) , ( 1 ↑ 2 ) ) |
30 |
|
neg1sqe1 |
⊢ ( - 1 ↑ 2 ) = 1 |
31 |
|
sq1 |
⊢ ( 1 ↑ 2 ) = 1 |
32 |
|
ifeq12 |
⊢ ( ( ( - 1 ↑ 2 ) = 1 ∧ ( 1 ↑ 2 ) = 1 ) → if ( ( ℑ ‘ 𝐴 ) < 0 , ( - 1 ↑ 2 ) , ( 1 ↑ 2 ) ) = if ( ( ℑ ‘ 𝐴 ) < 0 , 1 , 1 ) ) |
33 |
30 31 32
|
mp2an |
⊢ if ( ( ℑ ‘ 𝐴 ) < 0 , ( - 1 ↑ 2 ) , ( 1 ↑ 2 ) ) = if ( ( ℑ ‘ 𝐴 ) < 0 , 1 , 1 ) |
34 |
|
ifid |
⊢ if ( ( ℑ ‘ 𝐴 ) < 0 , 1 , 1 ) = 1 |
35 |
29 33 34
|
3eqtri |
⊢ ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) ↑ 2 ) = 1 |
36 |
35
|
a1i |
⊢ ( 𝐴 ∈ ℂ → ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) ↑ 2 ) = 1 ) |
37 |
17 18
|
resubcld |
⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ∈ ℝ ) |
38 |
37
|
rehalfcld |
⊢ ( 𝐴 ∈ ℂ → ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ∈ ℝ ) |
39 |
38
|
recnd |
⊢ ( 𝐴 ∈ ℂ → ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ∈ ℂ ) |
40 |
39
|
sqsqrtd |
⊢ ( 𝐴 ∈ ℂ → ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ↑ 2 ) = ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) |
41 |
36 40
|
oveq12d |
⊢ ( 𝐴 ∈ ℂ → ( ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) ↑ 2 ) · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ↑ 2 ) ) = ( 1 · ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) |
42 |
39
|
mulid2d |
⊢ ( 𝐴 ∈ ℂ → ( 1 · ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) = ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) |
43 |
28 41 42
|
3eqtrd |
⊢ ( 𝐴 ∈ ℂ → ( ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ↑ 2 ) = ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) |
44 |
25 43
|
oveq12d |
⊢ ( 𝐴 ∈ ℂ → ( ( i ↑ 2 ) · ( ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ↑ 2 ) ) = ( - 1 · ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) |
45 |
39
|
mulm1d |
⊢ ( 𝐴 ∈ ℂ → ( - 1 · ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) = - ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) |
46 |
23 44 45
|
3eqtrd |
⊢ ( 𝐴 ∈ ℂ → ( ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ↑ 2 ) = - ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) |
47 |
22 46
|
oveq12d |
⊢ ( 𝐴 ∈ ℂ → ( ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ↑ 2 ) + ( ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ↑ 2 ) ) = ( ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) + - ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) |
48 |
21 39
|
negsubd |
⊢ ( 𝐴 ∈ ℂ → ( ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) + - ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) = ( ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) − ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) |
49 |
17
|
recnd |
⊢ ( 𝐴 ∈ ℂ → ( abs ‘ 𝐴 ) ∈ ℂ ) |
50 |
18
|
recnd |
⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ∈ ℂ ) |
51 |
49 50 50
|
pnncand |
⊢ ( 𝐴 ∈ ℂ → ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) − ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ) = ( ( ℜ ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) ) |
52 |
50
|
2timesd |
⊢ ( 𝐴 ∈ ℂ → ( 2 · ( ℜ ‘ 𝐴 ) ) = ( ( ℜ ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) ) |
53 |
51 52
|
eqtr4d |
⊢ ( 𝐴 ∈ ℂ → ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) − ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ) = ( 2 · ( ℜ ‘ 𝐴 ) ) ) |
54 |
53
|
oveq1d |
⊢ ( 𝐴 ∈ ℂ → ( ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) − ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ) / 2 ) = ( ( 2 · ( ℜ ‘ 𝐴 ) ) / 2 ) ) |
55 |
19
|
recnd |
⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) ∈ ℂ ) |
56 |
37
|
recnd |
⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ∈ ℂ ) |
57 |
|
2cnd |
⊢ ( 𝐴 ∈ ℂ → 2 ∈ ℂ ) |
58 |
|
2ne0 |
⊢ 2 ≠ 0 |
59 |
58
|
a1i |
⊢ ( 𝐴 ∈ ℂ → 2 ≠ 0 ) |
60 |
55 56 57 59
|
divsubdird |
⊢ ( 𝐴 ∈ ℂ → ( ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) − ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ) / 2 ) = ( ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) − ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) |
61 |
50 57 59
|
divcan3d |
⊢ ( 𝐴 ∈ ℂ → ( ( 2 · ( ℜ ‘ 𝐴 ) ) / 2 ) = ( ℜ ‘ 𝐴 ) ) |
62 |
54 60 61
|
3eqtr3d |
⊢ ( 𝐴 ∈ ℂ → ( ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) − ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) = ( ℜ ‘ 𝐴 ) ) |
63 |
47 48 62
|
3eqtrd |
⊢ ( 𝐴 ∈ ℂ → ( ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ↑ 2 ) + ( ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ↑ 2 ) ) = ( ℜ ‘ 𝐴 ) ) |
64 |
57 2
|
mulcld |
⊢ ( 𝐴 ∈ ℂ → ( 2 · ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ∈ ℂ ) |
65 |
64 4 11
|
mul12d |
⊢ ( 𝐴 ∈ ℂ → ( ( 2 · ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) · ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) = ( i · ( ( 2 · ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) |
66 |
57 2 12
|
mulassd |
⊢ ( 𝐴 ∈ ℂ → ( ( 2 · ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) · ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) = ( 2 · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) · ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ) |
67 |
57 2 11
|
mulassd |
⊢ ( 𝐴 ∈ ℂ → ( ( 2 · ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) = ( 2 · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) |
68 |
2 26 27
|
mul12d |
⊢ ( 𝐴 ∈ ℂ → ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) = ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) |
69 |
|
sqrtcvallem4 |
⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) |
70 |
|
halfnneg2 |
⊢ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) ∈ ℝ → ( 0 ≤ ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) ↔ 0 ≤ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) |
71 |
19 70
|
syl |
⊢ ( 𝐴 ∈ ℂ → ( 0 ≤ ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) ↔ 0 ≤ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) |
72 |
69 71
|
mpbird |
⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) ) |
73 |
|
2rp |
⊢ 2 ∈ ℝ+ |
74 |
73
|
a1i |
⊢ ( 𝐴 ∈ ℂ → 2 ∈ ℝ+ ) |
75 |
19 72 74
|
sqrtdivd |
⊢ ( 𝐴 ∈ ℂ → ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) = ( ( √ ‘ ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) ) / ( √ ‘ 2 ) ) ) |
76 |
|
sqrtcvallem2 |
⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) |
77 |
|
halfnneg2 |
⊢ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ∈ ℝ → ( 0 ≤ ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ↔ 0 ≤ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) |
78 |
37 77
|
syl |
⊢ ( 𝐴 ∈ ℂ → ( 0 ≤ ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ↔ 0 ≤ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) |
79 |
76 78
|
mpbird |
⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ) |
80 |
37 79 74
|
sqrtdivd |
⊢ ( 𝐴 ∈ ℂ → ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) = ( ( √ ‘ ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ) / ( √ ‘ 2 ) ) ) |
81 |
75 80
|
oveq12d |
⊢ ( 𝐴 ∈ ℂ → ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) = ( ( ( √ ‘ ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) ) / ( √ ‘ 2 ) ) · ( ( √ ‘ ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ) / ( √ ‘ 2 ) ) ) ) |
82 |
19 72
|
resqrtcld |
⊢ ( 𝐴 ∈ ℂ → ( √ ‘ ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) ) ∈ ℝ ) |
83 |
82
|
recnd |
⊢ ( 𝐴 ∈ ℂ → ( √ ‘ ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) ) ∈ ℂ ) |
84 |
|
2re |
⊢ 2 ∈ ℝ |
85 |
84
|
a1i |
⊢ ( 𝐴 ∈ ℂ → 2 ∈ ℝ ) |
86 |
|
0le2 |
⊢ 0 ≤ 2 |
87 |
86
|
a1i |
⊢ ( 𝐴 ∈ ℂ → 0 ≤ 2 ) |
88 |
85 87
|
resqrtcld |
⊢ ( 𝐴 ∈ ℂ → ( √ ‘ 2 ) ∈ ℝ ) |
89 |
88
|
recnd |
⊢ ( 𝐴 ∈ ℂ → ( √ ‘ 2 ) ∈ ℂ ) |
90 |
37 79
|
resqrtcld |
⊢ ( 𝐴 ∈ ℂ → ( √ ‘ ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ) ∈ ℝ ) |
91 |
90
|
recnd |
⊢ ( 𝐴 ∈ ℂ → ( √ ‘ ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ) ∈ ℂ ) |
92 |
|
sqrt00 |
⊢ ( ( 2 ∈ ℝ ∧ 0 ≤ 2 ) → ( ( √ ‘ 2 ) = 0 ↔ 2 = 0 ) ) |
93 |
84 86 92
|
mp2an |
⊢ ( ( √ ‘ 2 ) = 0 ↔ 2 = 0 ) |
94 |
93
|
necon3bii |
⊢ ( ( √ ‘ 2 ) ≠ 0 ↔ 2 ≠ 0 ) |
95 |
58 94
|
mpbir |
⊢ ( √ ‘ 2 ) ≠ 0 |
96 |
95
|
a1i |
⊢ ( 𝐴 ∈ ℂ → ( √ ‘ 2 ) ≠ 0 ) |
97 |
83 89 91 89 96 96
|
divmuldivd |
⊢ ( 𝐴 ∈ ℂ → ( ( ( √ ‘ ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) ) / ( √ ‘ 2 ) ) · ( ( √ ‘ ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ) / ( √ ‘ 2 ) ) ) = ( ( ( √ ‘ ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) ) · ( √ ‘ ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ) ) / ( ( √ ‘ 2 ) · ( √ ‘ 2 ) ) ) ) |
98 |
18
|
resqcld |
⊢ ( 𝐴 ∈ ℂ → ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ∈ ℝ ) |
99 |
98
|
recnd |
⊢ ( 𝐴 ∈ ℂ → ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ∈ ℂ ) |
100 |
|
imcl |
⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) ∈ ℝ ) |
101 |
100
|
resqcld |
⊢ ( 𝐴 ∈ ℂ → ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ∈ ℝ ) |
102 |
101
|
recnd |
⊢ ( 𝐴 ∈ ℂ → ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ∈ ℂ ) |
103 |
|
absvalsq2 |
⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |
104 |
99 102 103
|
mvrladdd |
⊢ ( 𝐴 ∈ ℂ → ( ( ( abs ‘ 𝐴 ) ↑ 2 ) − ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) = ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) |
105 |
|
subsq |
⊢ ( ( ( abs ‘ 𝐴 ) ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) ∈ ℂ ) → ( ( ( abs ‘ 𝐴 ) ↑ 2 ) − ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) = ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) · ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ) ) |
106 |
49 50 105
|
syl2anc |
⊢ ( 𝐴 ∈ ℂ → ( ( ( abs ‘ 𝐴 ) ↑ 2 ) − ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) = ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) · ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ) ) |
107 |
104 106
|
eqtr3d |
⊢ ( 𝐴 ∈ ℂ → ( ( ℑ ‘ 𝐴 ) ↑ 2 ) = ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) · ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ) ) |
108 |
107
|
fveq2d |
⊢ ( 𝐴 ∈ ℂ → ( √ ‘ ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) = ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) · ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ) ) ) |
109 |
100
|
absred |
⊢ ( 𝐴 ∈ ℂ → ( abs ‘ ( ℑ ‘ 𝐴 ) ) = ( √ ‘ ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |
110 |
|
reabsifneg |
⊢ ( ( ℑ ‘ 𝐴 ) ∈ ℝ → ( abs ‘ ( ℑ ‘ 𝐴 ) ) = if ( ( ℑ ‘ 𝐴 ) < 0 , - ( ℑ ‘ 𝐴 ) , ( ℑ ‘ 𝐴 ) ) ) |
111 |
100 110
|
syl |
⊢ ( 𝐴 ∈ ℂ → ( abs ‘ ( ℑ ‘ 𝐴 ) ) = if ( ( ℑ ‘ 𝐴 ) < 0 , - ( ℑ ‘ 𝐴 ) , ( ℑ ‘ 𝐴 ) ) ) |
112 |
109 111
|
eqtr3d |
⊢ ( 𝐴 ∈ ℂ → ( √ ‘ ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) = if ( ( ℑ ‘ 𝐴 ) < 0 , - ( ℑ ‘ 𝐴 ) , ( ℑ ‘ 𝐴 ) ) ) |
113 |
19 72 37 79
|
sqrtmuld |
⊢ ( 𝐴 ∈ ℂ → ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) · ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ) ) = ( ( √ ‘ ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) ) · ( √ ‘ ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ) ) ) |
114 |
108 112 113
|
3eqtr3rd |
⊢ ( 𝐴 ∈ ℂ → ( ( √ ‘ ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) ) · ( √ ‘ ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ) ) = if ( ( ℑ ‘ 𝐴 ) < 0 , - ( ℑ ‘ 𝐴 ) , ( ℑ ‘ 𝐴 ) ) ) |
115 |
|
remsqsqrt |
⊢ ( ( 2 ∈ ℝ ∧ 0 ≤ 2 ) → ( ( √ ‘ 2 ) · ( √ ‘ 2 ) ) = 2 ) |
116 |
84 86 115
|
mp2an |
⊢ ( ( √ ‘ 2 ) · ( √ ‘ 2 ) ) = 2 |
117 |
116
|
a1i |
⊢ ( 𝐴 ∈ ℂ → ( ( √ ‘ 2 ) · ( √ ‘ 2 ) ) = 2 ) |
118 |
114 117
|
oveq12d |
⊢ ( 𝐴 ∈ ℂ → ( ( ( √ ‘ ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) ) · ( √ ‘ ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ) ) / ( ( √ ‘ 2 ) · ( √ ‘ 2 ) ) ) = ( if ( ( ℑ ‘ 𝐴 ) < 0 , - ( ℑ ‘ 𝐴 ) , ( ℑ ‘ 𝐴 ) ) / 2 ) ) |
119 |
81 97 118
|
3eqtrd |
⊢ ( 𝐴 ∈ ℂ → ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) = ( if ( ( ℑ ‘ 𝐴 ) < 0 , - ( ℑ ‘ 𝐴 ) , ( ℑ ‘ 𝐴 ) ) / 2 ) ) |
120 |
119
|
oveq2d |
⊢ ( 𝐴 ∈ ℂ → ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) = ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - ( ℑ ‘ 𝐴 ) , ( ℑ ‘ 𝐴 ) ) / 2 ) ) ) |
121 |
68 120
|
eqtrd |
⊢ ( 𝐴 ∈ ℂ → ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) = ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - ( ℑ ‘ 𝐴 ) , ( ℑ ‘ 𝐴 ) ) / 2 ) ) ) |
122 |
121
|
oveq2d |
⊢ ( 𝐴 ∈ ℂ → ( 2 · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) = ( 2 · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - ( ℑ ‘ 𝐴 ) , ( ℑ ‘ 𝐴 ) ) / 2 ) ) ) ) |
123 |
100
|
renegcld |
⊢ ( 𝐴 ∈ ℂ → - ( ℑ ‘ 𝐴 ) ∈ ℝ ) |
124 |
123 100
|
ifcld |
⊢ ( 𝐴 ∈ ℂ → if ( ( ℑ ‘ 𝐴 ) < 0 , - ( ℑ ‘ 𝐴 ) , ( ℑ ‘ 𝐴 ) ) ∈ ℝ ) |
125 |
124
|
recnd |
⊢ ( 𝐴 ∈ ℂ → if ( ( ℑ ‘ 𝐴 ) < 0 , - ( ℑ ‘ 𝐴 ) , ( ℑ ‘ 𝐴 ) ) ∈ ℂ ) |
126 |
26 125 57 59
|
divassd |
⊢ ( 𝐴 ∈ ℂ → ( ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · if ( ( ℑ ‘ 𝐴 ) < 0 , - ( ℑ ‘ 𝐴 ) , ( ℑ ‘ 𝐴 ) ) ) / 2 ) = ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - ( ℑ ‘ 𝐴 ) , ( ℑ ‘ 𝐴 ) ) / 2 ) ) ) |
127 |
|
ovif12 |
⊢ ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · if ( ( ℑ ‘ 𝐴 ) < 0 , - ( ℑ ‘ 𝐴 ) , ( ℑ ‘ 𝐴 ) ) ) = if ( ( ℑ ‘ 𝐴 ) < 0 , ( - 1 · - ( ℑ ‘ 𝐴 ) ) , ( 1 · ( ℑ ‘ 𝐴 ) ) ) |
128 |
5
|
a1i |
⊢ ( 𝐴 ∈ ℂ → - 1 ∈ ℝ ) |
129 |
128
|
recnd |
⊢ ( 𝐴 ∈ ℂ → - 1 ∈ ℂ ) |
130 |
100
|
recnd |
⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) ∈ ℂ ) |
131 |
129 129 130
|
mulassd |
⊢ ( 𝐴 ∈ ℂ → ( ( - 1 · - 1 ) · ( ℑ ‘ 𝐴 ) ) = ( - 1 · ( - 1 · ( ℑ ‘ 𝐴 ) ) ) ) |
132 |
|
neg1mulneg1e1 |
⊢ ( - 1 · - 1 ) = 1 |
133 |
132
|
a1i |
⊢ ( 𝐴 ∈ ℂ → ( - 1 · - 1 ) = 1 ) |
134 |
133
|
oveq1d |
⊢ ( 𝐴 ∈ ℂ → ( ( - 1 · - 1 ) · ( ℑ ‘ 𝐴 ) ) = ( 1 · ( ℑ ‘ 𝐴 ) ) ) |
135 |
130
|
mulid2d |
⊢ ( 𝐴 ∈ ℂ → ( 1 · ( ℑ ‘ 𝐴 ) ) = ( ℑ ‘ 𝐴 ) ) |
136 |
134 135
|
eqtrd |
⊢ ( 𝐴 ∈ ℂ → ( ( - 1 · - 1 ) · ( ℑ ‘ 𝐴 ) ) = ( ℑ ‘ 𝐴 ) ) |
137 |
130
|
mulm1d |
⊢ ( 𝐴 ∈ ℂ → ( - 1 · ( ℑ ‘ 𝐴 ) ) = - ( ℑ ‘ 𝐴 ) ) |
138 |
137
|
oveq2d |
⊢ ( 𝐴 ∈ ℂ → ( - 1 · ( - 1 · ( ℑ ‘ 𝐴 ) ) ) = ( - 1 · - ( ℑ ‘ 𝐴 ) ) ) |
139 |
131 136 138
|
3eqtr3rd |
⊢ ( 𝐴 ∈ ℂ → ( - 1 · - ( ℑ ‘ 𝐴 ) ) = ( ℑ ‘ 𝐴 ) ) |
140 |
139
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) < 0 ) → ( - 1 · - ( ℑ ‘ 𝐴 ) ) = ( ℑ ‘ 𝐴 ) ) |
141 |
135
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ¬ ( ℑ ‘ 𝐴 ) < 0 ) → ( 1 · ( ℑ ‘ 𝐴 ) ) = ( ℑ ‘ 𝐴 ) ) |
142 |
140 141
|
ifeqda |
⊢ ( 𝐴 ∈ ℂ → if ( ( ℑ ‘ 𝐴 ) < 0 , ( - 1 · - ( ℑ ‘ 𝐴 ) ) , ( 1 · ( ℑ ‘ 𝐴 ) ) ) = ( ℑ ‘ 𝐴 ) ) |
143 |
127 142
|
syl5eq |
⊢ ( 𝐴 ∈ ℂ → ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · if ( ( ℑ ‘ 𝐴 ) < 0 , - ( ℑ ‘ 𝐴 ) , ( ℑ ‘ 𝐴 ) ) ) = ( ℑ ‘ 𝐴 ) ) |
144 |
143
|
oveq1d |
⊢ ( 𝐴 ∈ ℂ → ( ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · if ( ( ℑ ‘ 𝐴 ) < 0 , - ( ℑ ‘ 𝐴 ) , ( ℑ ‘ 𝐴 ) ) ) / 2 ) = ( ( ℑ ‘ 𝐴 ) / 2 ) ) |
145 |
126 144
|
eqtr3d |
⊢ ( 𝐴 ∈ ℂ → ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - ( ℑ ‘ 𝐴 ) , ( ℑ ‘ 𝐴 ) ) / 2 ) ) = ( ( ℑ ‘ 𝐴 ) / 2 ) ) |
146 |
145
|
oveq2d |
⊢ ( 𝐴 ∈ ℂ → ( 2 · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - ( ℑ ‘ 𝐴 ) , ( ℑ ‘ 𝐴 ) ) / 2 ) ) ) = ( 2 · ( ( ℑ ‘ 𝐴 ) / 2 ) ) ) |
147 |
130 57 59
|
divcan2d |
⊢ ( 𝐴 ∈ ℂ → ( 2 · ( ( ℑ ‘ 𝐴 ) / 2 ) ) = ( ℑ ‘ 𝐴 ) ) |
148 |
146 147
|
eqtrd |
⊢ ( 𝐴 ∈ ℂ → ( 2 · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - ( ℑ ‘ 𝐴 ) , ( ℑ ‘ 𝐴 ) ) / 2 ) ) ) = ( ℑ ‘ 𝐴 ) ) |
149 |
67 122 148
|
3eqtrd |
⊢ ( 𝐴 ∈ ℂ → ( ( 2 · ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) = ( ℑ ‘ 𝐴 ) ) |
150 |
149
|
oveq2d |
⊢ ( 𝐴 ∈ ℂ → ( i · ( ( 2 · ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) = ( i · ( ℑ ‘ 𝐴 ) ) ) |
151 |
65 66 150
|
3eqtr3d |
⊢ ( 𝐴 ∈ ℂ → ( 2 · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) · ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) = ( i · ( ℑ ‘ 𝐴 ) ) ) |
152 |
63 151
|
oveq12d |
⊢ ( 𝐴 ∈ ℂ → ( ( ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ↑ 2 ) + ( ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ↑ 2 ) ) + ( 2 · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) · ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ) = ( ( ℜ ‘ 𝐴 ) + ( i · ( ℑ ‘ 𝐴 ) ) ) ) |
153 |
1
|
resqcld |
⊢ ( 𝐴 ∈ ℂ → ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ↑ 2 ) ∈ ℝ ) |
154 |
153
|
recnd |
⊢ ( 𝐴 ∈ ℂ → ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ↑ 2 ) ∈ ℂ ) |
155 |
2 12
|
mulcld |
⊢ ( 𝐴 ∈ ℂ → ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) · ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ∈ ℂ ) |
156 |
57 155
|
mulcld |
⊢ ( 𝐴 ∈ ℂ → ( 2 · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) · ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ∈ ℂ ) |
157 |
12
|
sqcld |
⊢ ( 𝐴 ∈ ℂ → ( ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ↑ 2 ) ∈ ℂ ) |
158 |
154 156 157
|
add32d |
⊢ ( 𝐴 ∈ ℂ → ( ( ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ↑ 2 ) + ( 2 · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) · ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ) + ( ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ↑ 2 ) ) = ( ( ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ↑ 2 ) + ( ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ↑ 2 ) ) + ( 2 · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) · ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ) ) |
159 |
|
replim |
⊢ ( 𝐴 ∈ ℂ → 𝐴 = ( ( ℜ ‘ 𝐴 ) + ( i · ( ℑ ‘ 𝐴 ) ) ) ) |
160 |
152 158 159
|
3eqtr4d |
⊢ ( 𝐴 ∈ ℂ → ( ( ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ↑ 2 ) + ( 2 · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) · ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ) + ( ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ↑ 2 ) ) = 𝐴 ) |
161 |
16 160
|
eqtrd |
⊢ ( 𝐴 ∈ ℂ → ( ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ↑ 2 ) = 𝐴 ) |
162 |
20 69
|
sqrtge0d |
⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) |
163 |
1 10
|
crred |
⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) = ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) |
164 |
162 163
|
breqtrrd |
⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( ℜ ‘ ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ) |
165 |
|
reim |
⊢ ( ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ∈ ℂ → ( ℜ ‘ ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) = ( ℑ ‘ ( i · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ) ) |
166 |
13 165
|
syl |
⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) = ( ℑ ‘ ( i · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ) ) |
167 |
166 163
|
eqtr3d |
⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ ( i · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ) = ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) |
168 |
167
|
eqeq1d |
⊢ ( 𝐴 ∈ ℂ → ( ( ℑ ‘ ( i · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ) = 0 ↔ ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) = 0 ) ) |
169 |
|
cnsqrt00 |
⊢ ( ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ∈ ℂ → ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) = 0 ↔ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) = 0 ) ) |
170 |
21 169
|
syl |
⊢ ( 𝐴 ∈ ℂ → ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) = 0 ↔ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) = 0 ) ) |
171 |
|
half0 |
⊢ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) ∈ ℂ → ( ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) = 0 ↔ ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) = 0 ) ) |
172 |
55 171
|
syl |
⊢ ( 𝐴 ∈ ℂ → ( ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) = 0 ↔ ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) = 0 ) ) |
173 |
49 50
|
addcomd |
⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) = ( ( ℜ ‘ 𝐴 ) + ( abs ‘ 𝐴 ) ) ) |
174 |
173
|
eqeq1d |
⊢ ( 𝐴 ∈ ℂ → ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) = 0 ↔ ( ( ℜ ‘ 𝐴 ) + ( abs ‘ 𝐴 ) ) = 0 ) ) |
175 |
|
addeq0 |
⊢ ( ( ( ℜ ‘ 𝐴 ) ∈ ℂ ∧ ( abs ‘ 𝐴 ) ∈ ℂ ) → ( ( ( ℜ ‘ 𝐴 ) + ( abs ‘ 𝐴 ) ) = 0 ↔ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ) ) |
176 |
50 49 175
|
syl2anc |
⊢ ( 𝐴 ∈ ℂ → ( ( ( ℜ ‘ 𝐴 ) + ( abs ‘ 𝐴 ) ) = 0 ↔ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ) ) |
177 |
172 174 176
|
3bitrd |
⊢ ( 𝐴 ∈ ℂ → ( ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) = 0 ↔ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ) ) |
178 |
168 170 177
|
3bitrd |
⊢ ( 𝐴 ∈ ℂ → ( ( ℑ ‘ ( i · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ) = 0 ↔ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ) ) |
179 |
|
olc |
⊢ ( ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) → ( ( ℜ ‘ 𝐴 ) = ( abs ‘ 𝐴 ) ∨ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ) ) |
180 |
|
eqcom |
⊢ ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) = ( ( abs ‘ 𝐴 ) ↑ 2 ) ↔ ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) |
181 |
180
|
a1i |
⊢ ( 𝐴 ∈ ℂ → ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) = ( ( abs ‘ 𝐴 ) ↑ 2 ) ↔ ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) ) |
182 |
|
sqeqor |
⊢ ( ( ( ℜ ‘ 𝐴 ) ∈ ℂ ∧ ( abs ‘ 𝐴 ) ∈ ℂ ) → ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) = ( ( abs ‘ 𝐴 ) ↑ 2 ) ↔ ( ( ℜ ‘ 𝐴 ) = ( abs ‘ 𝐴 ) ∨ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ) ) ) |
183 |
50 49 182
|
syl2anc |
⊢ ( 𝐴 ∈ ℂ → ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) = ( ( abs ‘ 𝐴 ) ↑ 2 ) ↔ ( ( ℜ ‘ 𝐴 ) = ( abs ‘ 𝐴 ) ∨ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ) ) ) |
184 |
103
|
eqeq1d |
⊢ ( 𝐴 ∈ ℂ → ( ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ↔ ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) = ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ) ) |
185 |
|
addid0 |
⊢ ( ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ∈ ℂ ∧ ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ∈ ℂ ) → ( ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) = ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ↔ ( ( ℑ ‘ 𝐴 ) ↑ 2 ) = 0 ) ) |
186 |
99 102 185
|
syl2anc |
⊢ ( 𝐴 ∈ ℂ → ( ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) = ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ↔ ( ( ℑ ‘ 𝐴 ) ↑ 2 ) = 0 ) ) |
187 |
|
sqeq0 |
⊢ ( ( ℑ ‘ 𝐴 ) ∈ ℂ → ( ( ( ℑ ‘ 𝐴 ) ↑ 2 ) = 0 ↔ ( ℑ ‘ 𝐴 ) = 0 ) ) |
188 |
130 187
|
syl |
⊢ ( 𝐴 ∈ ℂ → ( ( ( ℑ ‘ 𝐴 ) ↑ 2 ) = 0 ↔ ( ℑ ‘ 𝐴 ) = 0 ) ) |
189 |
184 186 188
|
3bitrd |
⊢ ( 𝐴 ∈ ℂ → ( ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( ( ℜ ‘ 𝐴 ) ↑ 2 ) ↔ ( ℑ ‘ 𝐴 ) = 0 ) ) |
190 |
181 183 189
|
3bitr3d |
⊢ ( 𝐴 ∈ ℂ → ( ( ( ℜ ‘ 𝐴 ) = ( abs ‘ 𝐴 ) ∨ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ) ↔ ( ℑ ‘ 𝐴 ) = 0 ) ) |
191 |
179 190
|
syl5ib |
⊢ ( 𝐴 ∈ ℂ → ( ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) → ( ℑ ‘ 𝐴 ) = 0 ) ) |
192 |
191
|
ancld |
⊢ ( 𝐴 ∈ ℂ → ( ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) → ( ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) ) ) |
193 |
178 192
|
sylbid |
⊢ ( 𝐴 ∈ ℂ → ( ( ℑ ‘ ( i · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ) = 0 → ( ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) ) ) |
194 |
|
simp2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ) |
195 |
194
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) = ( ( abs ‘ 𝐴 ) + - ( abs ‘ 𝐴 ) ) ) |
196 |
|
simp1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → 𝐴 ∈ ℂ ) |
197 |
196 49
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( abs ‘ 𝐴 ) ∈ ℂ ) |
198 |
197
|
negidd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( ( abs ‘ 𝐴 ) + - ( abs ‘ 𝐴 ) ) = 0 ) |
199 |
195 198
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) = 0 ) |
200 |
199
|
oveq1d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) = ( 0 / 2 ) ) |
201 |
|
2cn |
⊢ 2 ∈ ℂ |
202 |
201 58
|
div0i |
⊢ ( 0 / 2 ) = 0 |
203 |
200 202
|
eqtrdi |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) = 0 ) |
204 |
203
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) = ( √ ‘ 0 ) ) |
205 |
|
sqrt0 |
⊢ ( √ ‘ 0 ) = 0 |
206 |
204 205
|
eqtrdi |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) = 0 ) |
207 |
|
simp3 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( ℑ ‘ 𝐴 ) = 0 ) |
208 |
|
0red |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → 0 ∈ ℝ ) |
209 |
208
|
ltnrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ¬ 0 < 0 ) |
210 |
207 209
|
eqnbrtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ¬ ( ℑ ‘ 𝐴 ) < 0 ) |
211 |
210
|
iffalsed |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) = 1 ) |
212 |
194
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) = ( ( abs ‘ 𝐴 ) − - ( abs ‘ 𝐴 ) ) ) |
213 |
49 49
|
subnegd |
⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) − - ( abs ‘ 𝐴 ) ) = ( ( abs ‘ 𝐴 ) + ( abs ‘ 𝐴 ) ) ) |
214 |
49
|
2timesd |
⊢ ( 𝐴 ∈ ℂ → ( 2 · ( abs ‘ 𝐴 ) ) = ( ( abs ‘ 𝐴 ) + ( abs ‘ 𝐴 ) ) ) |
215 |
213 214
|
eqtr4d |
⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) − - ( abs ‘ 𝐴 ) ) = ( 2 · ( abs ‘ 𝐴 ) ) ) |
216 |
196 215
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( ( abs ‘ 𝐴 ) − - ( abs ‘ 𝐴 ) ) = ( 2 · ( abs ‘ 𝐴 ) ) ) |
217 |
212 216
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) = ( 2 · ( abs ‘ 𝐴 ) ) ) |
218 |
217
|
oveq1d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) = ( ( 2 · ( abs ‘ 𝐴 ) ) / 2 ) ) |
219 |
49 57 59
|
divcan3d |
⊢ ( 𝐴 ∈ ℂ → ( ( 2 · ( abs ‘ 𝐴 ) ) / 2 ) = ( abs ‘ 𝐴 ) ) |
220 |
196 219
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( ( 2 · ( abs ‘ 𝐴 ) ) / 2 ) = ( abs ‘ 𝐴 ) ) |
221 |
218 220
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) = ( abs ‘ 𝐴 ) ) |
222 |
221
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) = ( √ ‘ ( abs ‘ 𝐴 ) ) ) |
223 |
211 222
|
oveq12d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) = ( 1 · ( √ ‘ ( abs ‘ 𝐴 ) ) ) ) |
224 |
|
absge0 |
⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( abs ‘ 𝐴 ) ) |
225 |
17 224
|
resqrtcld |
⊢ ( 𝐴 ∈ ℂ → ( √ ‘ ( abs ‘ 𝐴 ) ) ∈ ℝ ) |
226 |
225
|
recnd |
⊢ ( 𝐴 ∈ ℂ → ( √ ‘ ( abs ‘ 𝐴 ) ) ∈ ℂ ) |
227 |
226
|
mulid2d |
⊢ ( 𝐴 ∈ ℂ → ( 1 · ( √ ‘ ( abs ‘ 𝐴 ) ) ) = ( √ ‘ ( abs ‘ 𝐴 ) ) ) |
228 |
196 227
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( 1 · ( √ ‘ ( abs ‘ 𝐴 ) ) ) = ( √ ‘ ( abs ‘ 𝐴 ) ) ) |
229 |
223 228
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) = ( √ ‘ ( abs ‘ 𝐴 ) ) ) |
230 |
229
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) = ( i · ( √ ‘ ( abs ‘ 𝐴 ) ) ) ) |
231 |
206 230
|
oveq12d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) = ( 0 + ( i · ( √ ‘ ( abs ‘ 𝐴 ) ) ) ) ) |
232 |
4 226
|
mulcld |
⊢ ( 𝐴 ∈ ℂ → ( i · ( √ ‘ ( abs ‘ 𝐴 ) ) ) ∈ ℂ ) |
233 |
196 232
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( i · ( √ ‘ ( abs ‘ 𝐴 ) ) ) ∈ ℂ ) |
234 |
233
|
addid2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( 0 + ( i · ( √ ‘ ( abs ‘ 𝐴 ) ) ) ) = ( i · ( √ ‘ ( abs ‘ 𝐴 ) ) ) ) |
235 |
231 234
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) = ( i · ( √ ‘ ( abs ‘ 𝐴 ) ) ) ) |
236 |
235
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( i · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) = ( i · ( i · ( √ ‘ ( abs ‘ 𝐴 ) ) ) ) ) |
237 |
|
ixi |
⊢ ( i · i ) = - 1 |
238 |
237
|
a1i |
⊢ ( 𝐴 ∈ ℂ → ( i · i ) = - 1 ) |
239 |
238
|
oveq1d |
⊢ ( 𝐴 ∈ ℂ → ( ( i · i ) · ( √ ‘ ( abs ‘ 𝐴 ) ) ) = ( - 1 · ( √ ‘ ( abs ‘ 𝐴 ) ) ) ) |
240 |
4 4 226
|
mulassd |
⊢ ( 𝐴 ∈ ℂ → ( ( i · i ) · ( √ ‘ ( abs ‘ 𝐴 ) ) ) = ( i · ( i · ( √ ‘ ( abs ‘ 𝐴 ) ) ) ) ) |
241 |
226
|
mulm1d |
⊢ ( 𝐴 ∈ ℂ → ( - 1 · ( √ ‘ ( abs ‘ 𝐴 ) ) ) = - ( √ ‘ ( abs ‘ 𝐴 ) ) ) |
242 |
239 240 241
|
3eqtr3d |
⊢ ( 𝐴 ∈ ℂ → ( i · ( i · ( √ ‘ ( abs ‘ 𝐴 ) ) ) ) = - ( √ ‘ ( abs ‘ 𝐴 ) ) ) |
243 |
196 242
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( i · ( i · ( √ ‘ ( abs ‘ 𝐴 ) ) ) ) = - ( √ ‘ ( abs ‘ 𝐴 ) ) ) |
244 |
236 243
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( i · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) = - ( √ ‘ ( abs ‘ 𝐴 ) ) ) |
245 |
244
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( ℜ ‘ ( i · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ) = ( ℜ ‘ - ( √ ‘ ( abs ‘ 𝐴 ) ) ) ) |
246 |
225
|
renegcld |
⊢ ( 𝐴 ∈ ℂ → - ( √ ‘ ( abs ‘ 𝐴 ) ) ∈ ℝ ) |
247 |
246
|
rered |
⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ - ( √ ‘ ( abs ‘ 𝐴 ) ) ) = - ( √ ‘ ( abs ‘ 𝐴 ) ) ) |
248 |
196 247
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( ℜ ‘ - ( √ ‘ ( abs ‘ 𝐴 ) ) ) = - ( √ ‘ ( abs ‘ 𝐴 ) ) ) |
249 |
245 248
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( ℜ ‘ ( i · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ) = - ( √ ‘ ( abs ‘ 𝐴 ) ) ) |
250 |
17 224
|
sqrtge0d |
⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( √ ‘ ( abs ‘ 𝐴 ) ) ) |
251 |
225
|
le0neg2d |
⊢ ( 𝐴 ∈ ℂ → ( 0 ≤ ( √ ‘ ( abs ‘ 𝐴 ) ) ↔ - ( √ ‘ ( abs ‘ 𝐴 ) ) ≤ 0 ) ) |
252 |
250 251
|
mpbid |
⊢ ( 𝐴 ∈ ℂ → - ( √ ‘ ( abs ‘ 𝐴 ) ) ≤ 0 ) |
253 |
196 252
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → - ( √ ‘ ( abs ‘ 𝐴 ) ) ≤ 0 ) |
254 |
249 253
|
eqbrtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( ℜ ‘ ( i · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ) ≤ 0 ) |
255 |
254
|
3expib |
⊢ ( 𝐴 ∈ ℂ → ( ( ( ℜ ‘ 𝐴 ) = - ( abs ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) = 0 ) → ( ℜ ‘ ( i · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ) ≤ 0 ) ) |
256 |
193 255
|
syld |
⊢ ( 𝐴 ∈ ℂ → ( ( ℑ ‘ ( i · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ) = 0 → ( ℜ ‘ ( i · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ) ≤ 0 ) ) |
257 |
4 13
|
mulcld |
⊢ ( 𝐴 ∈ ℂ → ( i · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ∈ ℂ ) |
258 |
257
|
sqrtcvallem1 |
⊢ ( 𝐴 ∈ ℂ → ( ( ( ℑ ‘ ( i · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ) = 0 → ( ℜ ‘ ( i · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ) ≤ 0 ) ↔ ¬ ( i · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ∈ ℝ+ ) ) |
259 |
256 258
|
mpbid |
⊢ ( 𝐴 ∈ ℂ → ¬ ( i · ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) ∈ ℝ+ ) |
260 |
13 14 161 164 259
|
eqsqrtd |
⊢ ( 𝐴 ∈ ℂ → ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) = ( √ ‘ 𝐴 ) ) |
261 |
260
|
eqcomd |
⊢ ( 𝐴 ∈ ℂ → ( √ ‘ 𝐴 ) = ( ( √ ‘ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) ) + ( i · ( if ( ( ℑ ‘ 𝐴 ) < 0 , - 1 , 1 ) · ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ) ) ) ) |