Step |
Hyp |
Ref |
Expression |
1 |
|
iblabs.1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 ) |
2 |
|
iblabs.2 |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝐿1 ) |
3 |
|
absf |
⊢ abs : ℂ ⟶ ℝ |
4 |
3
|
a1i |
⊢ ( 𝜑 → abs : ℂ ⟶ ℝ ) |
5 |
|
iblmbf |
⊢ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝐿1 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ MblFn ) |
6 |
2 5
|
syl |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ MblFn ) |
7 |
6 1
|
mbfmptcl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
8 |
4 7
|
cofmpt |
⊢ ( 𝜑 → ( abs ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) = ( 𝑥 ∈ 𝐴 ↦ ( abs ‘ 𝐵 ) ) ) |
9 |
7
|
fmpttd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℂ ) |
10 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
11 |
|
ssid |
⊢ ℂ ⊆ ℂ |
12 |
|
cncfss |
⊢ ( ( ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( ℂ –cn→ ℝ ) ⊆ ( ℂ –cn→ ℂ ) ) |
13 |
10 11 12
|
mp2an |
⊢ ( ℂ –cn→ ℝ ) ⊆ ( ℂ –cn→ ℂ ) |
14 |
|
abscncf |
⊢ abs ∈ ( ℂ –cn→ ℝ ) |
15 |
13 14
|
sselii |
⊢ abs ∈ ( ℂ –cn→ ℂ ) |
16 |
15
|
a1i |
⊢ ( 𝜑 → abs ∈ ( ℂ –cn→ ℂ ) ) |
17 |
|
cncombf |
⊢ ( ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ MblFn ∧ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℂ ∧ abs ∈ ( ℂ –cn→ ℂ ) ) → ( abs ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ∈ MblFn ) |
18 |
6 9 16 17
|
syl3anc |
⊢ ( 𝜑 → ( abs ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ∈ MblFn ) |
19 |
8 18
|
eqeltrrd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( abs ‘ 𝐵 ) ) ∈ MblFn ) |
20 |
7
|
abscld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( abs ‘ 𝐵 ) ∈ ℝ ) |
21 |
20
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( abs ‘ 𝐵 ) ∈ ℝ* ) |
22 |
7
|
absge0d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ ( abs ‘ 𝐵 ) ) |
23 |
|
elxrge0 |
⊢ ( ( abs ‘ 𝐵 ) ∈ ( 0 [,] +∞ ) ↔ ( ( abs ‘ 𝐵 ) ∈ ℝ* ∧ 0 ≤ ( abs ‘ 𝐵 ) ) ) |
24 |
21 22 23
|
sylanbrc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( abs ‘ 𝐵 ) ∈ ( 0 [,] +∞ ) ) |
25 |
|
0e0iccpnf |
⊢ 0 ∈ ( 0 [,] +∞ ) |
26 |
25
|
a1i |
⊢ ( ( 𝜑 ∧ ¬ 𝑥 ∈ 𝐴 ) → 0 ∈ ( 0 [,] +∞ ) ) |
27 |
24 26
|
ifclda |
⊢ ( 𝜑 → if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) ∈ ( 0 [,] +∞ ) ) |
28 |
27
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) ∈ ( 0 [,] +∞ ) ) |
29 |
28
|
fmpttd |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) ) : ℝ ⟶ ( 0 [,] +∞ ) ) |
30 |
|
reex |
⊢ ℝ ∈ V |
31 |
30
|
a1i |
⊢ ( 𝜑 → ℝ ∈ V ) |
32 |
7
|
recld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ℜ ‘ 𝐵 ) ∈ ℝ ) |
33 |
32
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ℜ ‘ 𝐵 ) ∈ ℂ ) |
34 |
33
|
abscld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( abs ‘ ( ℜ ‘ 𝐵 ) ) ∈ ℝ ) |
35 |
33
|
absge0d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ ( abs ‘ ( ℜ ‘ 𝐵 ) ) ) |
36 |
|
elrege0 |
⊢ ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) ∈ ( 0 [,) +∞ ) ↔ ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) ∈ ℝ ∧ 0 ≤ ( abs ‘ ( ℜ ‘ 𝐵 ) ) ) ) |
37 |
34 35 36
|
sylanbrc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( abs ‘ ( ℜ ‘ 𝐵 ) ) ∈ ( 0 [,) +∞ ) ) |
38 |
|
0e0icopnf |
⊢ 0 ∈ ( 0 [,) +∞ ) |
39 |
38
|
a1i |
⊢ ( ( 𝜑 ∧ ¬ 𝑥 ∈ 𝐴 ) → 0 ∈ ( 0 [,) +∞ ) ) |
40 |
37 39
|
ifclda |
⊢ ( 𝜑 → if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) ∈ ( 0 [,) +∞ ) ) |
41 |
40
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) ∈ ( 0 [,) +∞ ) ) |
42 |
7
|
imcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ℑ ‘ 𝐵 ) ∈ ℝ ) |
43 |
42
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ℑ ‘ 𝐵 ) ∈ ℂ ) |
44 |
43
|
abscld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( abs ‘ ( ℑ ‘ 𝐵 ) ) ∈ ℝ ) |
45 |
43
|
absge0d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) |
46 |
|
elrege0 |
⊢ ( ( abs ‘ ( ℑ ‘ 𝐵 ) ) ∈ ( 0 [,) +∞ ) ↔ ( ( abs ‘ ( ℑ ‘ 𝐵 ) ) ∈ ℝ ∧ 0 ≤ ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) ) |
47 |
44 45 46
|
sylanbrc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( abs ‘ ( ℑ ‘ 𝐵 ) ) ∈ ( 0 [,) +∞ ) ) |
48 |
47 39
|
ifclda |
⊢ ( 𝜑 → if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ∈ ( 0 [,) +∞ ) ) |
49 |
48
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ∈ ( 0 [,) +∞ ) ) |
50 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) ) = ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) ) ) |
51 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ) = ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ) ) |
52 |
31 41 49 50 51
|
offval2 |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) ) ∘f + ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ) ) = ( 𝑥 ∈ ℝ ↦ ( if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) + if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ) ) ) |
53 |
|
iftrue |
⊢ ( 𝑥 ∈ 𝐴 → if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) = ( abs ‘ ( ℜ ‘ 𝐵 ) ) ) |
54 |
|
iftrue |
⊢ ( 𝑥 ∈ 𝐴 → if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) = ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) |
55 |
53 54
|
oveq12d |
⊢ ( 𝑥 ∈ 𝐴 → ( if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) + if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ) = ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) ) |
56 |
|
iftrue |
⊢ ( 𝑥 ∈ 𝐴 → if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) = ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) ) |
57 |
55 56
|
eqtr4d |
⊢ ( 𝑥 ∈ 𝐴 → ( if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) + if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ) = if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) |
58 |
|
00id |
⊢ ( 0 + 0 ) = 0 |
59 |
|
iffalse |
⊢ ( ¬ 𝑥 ∈ 𝐴 → if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) = 0 ) |
60 |
|
iffalse |
⊢ ( ¬ 𝑥 ∈ 𝐴 → if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) = 0 ) |
61 |
59 60
|
oveq12d |
⊢ ( ¬ 𝑥 ∈ 𝐴 → ( if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) + if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ) = ( 0 + 0 ) ) |
62 |
|
iffalse |
⊢ ( ¬ 𝑥 ∈ 𝐴 → if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) = 0 ) |
63 |
58 61 62
|
3eqtr4a |
⊢ ( ¬ 𝑥 ∈ 𝐴 → ( if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) + if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ) = if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) |
64 |
57 63
|
pm2.61i |
⊢ ( if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) + if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ) = if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) |
65 |
64
|
mpteq2i |
⊢ ( 𝑥 ∈ ℝ ↦ ( if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) + if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ) ) = ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) |
66 |
52 65
|
eqtr2di |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) = ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) ) ∘f + ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ) ) ) |
67 |
66
|
fveq2d |
⊢ ( 𝜑 → ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) ) = ( ∫2 ‘ ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) ) ∘f + ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ) ) ) ) |
68 |
|
eqid |
⊢ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) ) = ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) ) |
69 |
7
|
iblcn |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝐿1 ↔ ( ( 𝑥 ∈ 𝐴 ↦ ( ℜ ‘ 𝐵 ) ) ∈ 𝐿1 ∧ ( 𝑥 ∈ 𝐴 ↦ ( ℑ ‘ 𝐵 ) ) ∈ 𝐿1 ) ) ) |
70 |
2 69
|
mpbid |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ ( ℜ ‘ 𝐵 ) ) ∈ 𝐿1 ∧ ( 𝑥 ∈ 𝐴 ↦ ( ℑ ‘ 𝐵 ) ) ∈ 𝐿1 ) ) |
71 |
70
|
simpld |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( ℜ ‘ 𝐵 ) ) ∈ 𝐿1 ) |
72 |
1 2 68 71 32
|
iblabslem |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) ) ∈ MblFn ∧ ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) ) ) ∈ ℝ ) ) |
73 |
72
|
simpld |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) ) ∈ MblFn ) |
74 |
41
|
fmpttd |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) ) : ℝ ⟶ ( 0 [,) +∞ ) ) |
75 |
72
|
simprd |
⊢ ( 𝜑 → ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) ) ) ∈ ℝ ) |
76 |
|
eqid |
⊢ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ) = ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ) |
77 |
70
|
simprd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( ℑ ‘ 𝐵 ) ) ∈ 𝐿1 ) |
78 |
1 2 76 77 42
|
iblabslem |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ) ∈ MblFn ∧ ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ) ) ∈ ℝ ) ) |
79 |
78
|
simpld |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ) ∈ MblFn ) |
80 |
49
|
fmpttd |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ) : ℝ ⟶ ( 0 [,) +∞ ) ) |
81 |
78
|
simprd |
⊢ ( 𝜑 → ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ) ) ∈ ℝ ) |
82 |
73 74 75 79 80 81
|
itg2add |
⊢ ( 𝜑 → ( ∫2 ‘ ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) ) ∘f + ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ) ) ) = ( ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) ) ) + ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ) ) ) ) |
83 |
67 82
|
eqtrd |
⊢ ( 𝜑 → ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) ) = ( ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) ) ) + ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ) ) ) ) |
84 |
75 81
|
readdcld |
⊢ ( 𝜑 → ( ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℜ ‘ 𝐵 ) ) , 0 ) ) ) + ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ ( ℑ ‘ 𝐵 ) ) , 0 ) ) ) ) ∈ ℝ ) |
85 |
83 84
|
eqeltrd |
⊢ ( 𝜑 → ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) ) ∈ ℝ ) |
86 |
34 44
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) ∈ ℝ ) |
87 |
86
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) ∈ ℝ* ) |
88 |
34 44 35 45
|
addge0d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) ) |
89 |
|
elxrge0 |
⊢ ( ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) ∈ ( 0 [,] +∞ ) ↔ ( ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) ∈ ℝ* ∧ 0 ≤ ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) ) ) |
90 |
87 88 89
|
sylanbrc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) ∈ ( 0 [,] +∞ ) ) |
91 |
90 26
|
ifclda |
⊢ ( 𝜑 → if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ∈ ( 0 [,] +∞ ) ) |
92 |
91
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ∈ ( 0 [,] +∞ ) ) |
93 |
92
|
fmpttd |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) : ℝ ⟶ ( 0 [,] +∞ ) ) |
94 |
|
ax-icn |
⊢ i ∈ ℂ |
95 |
|
mulcl |
⊢ ( ( i ∈ ℂ ∧ ( ℑ ‘ 𝐵 ) ∈ ℂ ) → ( i · ( ℑ ‘ 𝐵 ) ) ∈ ℂ ) |
96 |
94 43 95
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( i · ( ℑ ‘ 𝐵 ) ) ∈ ℂ ) |
97 |
33 96
|
abstrid |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( abs ‘ ( ( ℜ ‘ 𝐵 ) + ( i · ( ℑ ‘ 𝐵 ) ) ) ) ≤ ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( i · ( ℑ ‘ 𝐵 ) ) ) ) ) |
98 |
7
|
replimd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = ( ( ℜ ‘ 𝐵 ) + ( i · ( ℑ ‘ 𝐵 ) ) ) ) |
99 |
98
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( abs ‘ 𝐵 ) = ( abs ‘ ( ( ℜ ‘ 𝐵 ) + ( i · ( ℑ ‘ 𝐵 ) ) ) ) ) |
100 |
|
absmul |
⊢ ( ( i ∈ ℂ ∧ ( ℑ ‘ 𝐵 ) ∈ ℂ ) → ( abs ‘ ( i · ( ℑ ‘ 𝐵 ) ) ) = ( ( abs ‘ i ) · ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) ) |
101 |
94 43 100
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( abs ‘ ( i · ( ℑ ‘ 𝐵 ) ) ) = ( ( abs ‘ i ) · ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) ) |
102 |
|
absi |
⊢ ( abs ‘ i ) = 1 |
103 |
102
|
oveq1i |
⊢ ( ( abs ‘ i ) · ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) = ( 1 · ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) |
104 |
44
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( abs ‘ ( ℑ ‘ 𝐵 ) ) ∈ ℂ ) |
105 |
104
|
mulid2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 1 · ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) = ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) |
106 |
103 105
|
syl5eq |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( abs ‘ i ) · ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) = ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) |
107 |
101 106
|
eqtr2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( abs ‘ ( ℑ ‘ 𝐵 ) ) = ( abs ‘ ( i · ( ℑ ‘ 𝐵 ) ) ) ) |
108 |
107
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) = ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( i · ( ℑ ‘ 𝐵 ) ) ) ) ) |
109 |
97 99 108
|
3brtr4d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( abs ‘ 𝐵 ) ≤ ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) ) |
110 |
|
iftrue |
⊢ ( 𝑥 ∈ 𝐴 → if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) = ( abs ‘ 𝐵 ) ) |
111 |
110
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) = ( abs ‘ 𝐵 ) ) |
112 |
56
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) = ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) ) |
113 |
109 111 112
|
3brtr4d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) ≤ if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) |
114 |
113
|
ex |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) ≤ if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) ) |
115 |
|
0le0 |
⊢ 0 ≤ 0 |
116 |
115
|
a1i |
⊢ ( ¬ 𝑥 ∈ 𝐴 → 0 ≤ 0 ) |
117 |
|
iffalse |
⊢ ( ¬ 𝑥 ∈ 𝐴 → if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) = 0 ) |
118 |
116 117 62
|
3brtr4d |
⊢ ( ¬ 𝑥 ∈ 𝐴 → if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) ≤ if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) |
119 |
114 118
|
pm2.61d1 |
⊢ ( 𝜑 → if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) ≤ if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) |
120 |
119
|
ralrimivw |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) ≤ if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) |
121 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) ) = ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) ) ) |
122 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) = ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) ) |
123 |
31 28 92 121 122
|
ofrfval2 |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) ) ∘r ≤ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) ↔ ∀ 𝑥 ∈ ℝ if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) ≤ if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) ) |
124 |
120 123
|
mpbird |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) ) ∘r ≤ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) ) |
125 |
|
itg2le |
⊢ ( ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) ) : ℝ ⟶ ( 0 [,] +∞ ) ∧ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) : ℝ ⟶ ( 0 [,] +∞ ) ∧ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) ) ∘r ≤ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) ) → ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) ) ) ≤ ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) ) ) |
126 |
29 93 124 125
|
syl3anc |
⊢ ( 𝜑 → ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) ) ) ≤ ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) ) ) |
127 |
|
itg2lecl |
⊢ ( ( ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) ) : ℝ ⟶ ( 0 [,] +∞ ) ∧ ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) ) ∈ ℝ ∧ ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) ) ) ≤ ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( ( abs ‘ ( ℜ ‘ 𝐵 ) ) + ( abs ‘ ( ℑ ‘ 𝐵 ) ) ) , 0 ) ) ) ) → ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) ) ) ∈ ℝ ) |
128 |
29 85 126 127
|
syl3anc |
⊢ ( 𝜑 → ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) ) ) ∈ ℝ ) |
129 |
20 22
|
iblpos |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ ( abs ‘ 𝐵 ) ) ∈ 𝐿1 ↔ ( ( 𝑥 ∈ 𝐴 ↦ ( abs ‘ 𝐵 ) ) ∈ MblFn ∧ ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ 𝐴 , ( abs ‘ 𝐵 ) , 0 ) ) ) ∈ ℝ ) ) ) |
130 |
19 128 129
|
mpbir2and |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( abs ‘ 𝐵 ) ) ∈ 𝐿1 ) |