Step |
Hyp |
Ref |
Expression |
1 |
|
itgadd.1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 ) |
2 |
|
itgadd.2 |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝐿1 ) |
3 |
|
itgadd.3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ 𝑉 ) |
4 |
|
itgadd.4 |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ∈ 𝐿1 ) |
5 |
|
itgadd.5 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
6 |
|
itgadd.6 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ ℝ ) |
7 |
|
max0sub |
⊢ ( 𝐵 ∈ ℝ → ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) − if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) = 𝐵 ) |
8 |
5 7
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) − if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) = 𝐵 ) |
9 |
|
max0sub |
⊢ ( 𝐶 ∈ ℝ → ( if ( 0 ≤ 𝐶 , 𝐶 , 0 ) − if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) = 𝐶 ) |
10 |
6 9
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( if ( 0 ≤ 𝐶 , 𝐶 , 0 ) − if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) = 𝐶 ) |
11 |
8 10
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) − if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) + ( if ( 0 ≤ 𝐶 , 𝐶 , 0 ) − if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) = ( 𝐵 + 𝐶 ) ) |
12 |
|
0re |
⊢ 0 ∈ ℝ |
13 |
|
ifcl |
⊢ ( ( 𝐵 ∈ ℝ ∧ 0 ∈ ℝ ) → if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ∈ ℝ ) |
14 |
5 12 13
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ∈ ℝ ) |
15 |
14
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ∈ ℂ ) |
16 |
|
ifcl |
⊢ ( ( 𝐶 ∈ ℝ ∧ 0 ∈ ℝ ) → if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ∈ ℝ ) |
17 |
6 12 16
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ∈ ℝ ) |
18 |
17
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ∈ ℂ ) |
19 |
5
|
renegcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → - 𝐵 ∈ ℝ ) |
20 |
|
ifcl |
⊢ ( ( - 𝐵 ∈ ℝ ∧ 0 ∈ ℝ ) → if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ∈ ℝ ) |
21 |
19 12 20
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ∈ ℝ ) |
22 |
21
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ∈ ℂ ) |
23 |
6
|
renegcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → - 𝐶 ∈ ℝ ) |
24 |
|
ifcl |
⊢ ( ( - 𝐶 ∈ ℝ ∧ 0 ∈ ℝ ) → if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ∈ ℝ ) |
25 |
23 12 24
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ∈ ℝ ) |
26 |
25
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ∈ ℂ ) |
27 |
15 18 22 26
|
addsub4d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) − ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) = ( ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) − if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) + ( if ( 0 ≤ 𝐶 , 𝐶 , 0 ) − if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) ) |
28 |
5 6
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐵 + 𝐶 ) ∈ ℝ ) |
29 |
|
max0sub |
⊢ ( ( 𝐵 + 𝐶 ) ∈ ℝ → ( if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) − if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) ) = ( 𝐵 + 𝐶 ) ) |
30 |
28 29
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) − if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) ) = ( 𝐵 + 𝐶 ) ) |
31 |
11 27 30
|
3eqtr4rd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) − if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) ) = ( ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) − ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) ) |
32 |
28
|
renegcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → - ( 𝐵 + 𝐶 ) ∈ ℝ ) |
33 |
|
ifcl |
⊢ ( ( - ( 𝐵 + 𝐶 ) ∈ ℝ ∧ 0 ∈ ℝ ) → if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) ∈ ℝ ) |
34 |
32 12 33
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) ∈ ℝ ) |
35 |
34
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) ∈ ℂ ) |
36 |
15 18
|
addcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ∈ ℂ ) |
37 |
|
ifcl |
⊢ ( ( ( 𝐵 + 𝐶 ) ∈ ℝ ∧ 0 ∈ ℝ ) → if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) ∈ ℝ ) |
38 |
28 12 37
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) ∈ ℝ ) |
39 |
38
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) ∈ ℂ ) |
40 |
22 26
|
addcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ∈ ℂ ) |
41 |
35 36 39 40
|
addsubeq4d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) + ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) = ( if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) + ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) ↔ ( if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) − if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) ) = ( ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) − ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) ) ) |
42 |
31 41
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) + ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) = ( if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) + ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) ) |
43 |
42
|
itgeq2dv |
⊢ ( 𝜑 → ∫ 𝐴 ( if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) + ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) d 𝑥 = ∫ 𝐴 ( if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) + ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) d 𝑥 ) |
44 |
1 2 3 4
|
ibladd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 + 𝐶 ) ) ∈ 𝐿1 ) |
45 |
28
|
iblre |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 + 𝐶 ) ) ∈ 𝐿1 ↔ ( ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) ) ∈ 𝐿1 ∧ ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) ) ∈ 𝐿1 ) ) ) |
46 |
44 45
|
mpbid |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) ) ∈ 𝐿1 ∧ ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) ) ∈ 𝐿1 ) ) |
47 |
46
|
simprd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) ) ∈ 𝐿1 ) |
48 |
14 17
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ∈ ℝ ) |
49 |
5
|
iblre |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝐿1 ↔ ( ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) ∈ 𝐿1 ∧ ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) ∈ 𝐿1 ) ) ) |
50 |
2 49
|
mpbid |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) ∈ 𝐿1 ∧ ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) ∈ 𝐿1 ) ) |
51 |
50
|
simpld |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) ∈ 𝐿1 ) |
52 |
6
|
iblre |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ∈ 𝐿1 ↔ ( ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ∈ 𝐿1 ∧ ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ∈ 𝐿1 ) ) ) |
53 |
4 52
|
mpbid |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ∈ 𝐿1 ∧ ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ∈ 𝐿1 ) ) |
54 |
53
|
simpld |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ∈ 𝐿1 ) |
55 |
14 51 17 54
|
ibladd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) ∈ 𝐿1 ) |
56 |
|
max1 |
⊢ ( ( 0 ∈ ℝ ∧ - ( 𝐵 + 𝐶 ) ∈ ℝ ) → 0 ≤ if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) ) |
57 |
12 32 56
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) ) |
58 |
|
max1 |
⊢ ( ( 0 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → 0 ≤ if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) |
59 |
12 5 58
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) |
60 |
|
max1 |
⊢ ( ( 0 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 0 ≤ if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) |
61 |
12 6 60
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) |
62 |
14 17 59 61
|
addge0d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) |
63 |
34 47 48 55 34 48 57 62
|
itgaddlem1 |
⊢ ( 𝜑 → ∫ 𝐴 ( if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) + ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) d 𝑥 = ( ∫ 𝐴 if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) d 𝑥 + ∫ 𝐴 ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) d 𝑥 ) ) |
64 |
46
|
simpld |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) ) ∈ 𝐿1 ) |
65 |
21 25
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ∈ ℝ ) |
66 |
50
|
simprd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) ∈ 𝐿1 ) |
67 |
53
|
simprd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ∈ 𝐿1 ) |
68 |
21 66 25 67
|
ibladd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) ∈ 𝐿1 ) |
69 |
|
max1 |
⊢ ( ( 0 ∈ ℝ ∧ ( 𝐵 + 𝐶 ) ∈ ℝ ) → 0 ≤ if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) ) |
70 |
12 28 69
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) ) |
71 |
|
max1 |
⊢ ( ( 0 ∈ ℝ ∧ - 𝐵 ∈ ℝ ) → 0 ≤ if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) |
72 |
12 19 71
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) |
73 |
|
max1 |
⊢ ( ( 0 ∈ ℝ ∧ - 𝐶 ∈ ℝ ) → 0 ≤ if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) |
74 |
12 23 73
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) |
75 |
21 25 72 74
|
addge0d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) |
76 |
38 64 65 68 38 65 70 75
|
itgaddlem1 |
⊢ ( 𝜑 → ∫ 𝐴 ( if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) + ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) d 𝑥 = ( ∫ 𝐴 if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) d 𝑥 + ∫ 𝐴 ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) d 𝑥 ) ) |
77 |
43 63 76
|
3eqtr3d |
⊢ ( 𝜑 → ( ∫ 𝐴 if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) d 𝑥 + ∫ 𝐴 ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) d 𝑥 ) = ( ∫ 𝐴 if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) d 𝑥 + ∫ 𝐴 ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) d 𝑥 ) ) |
78 |
34 47
|
itgcl |
⊢ ( 𝜑 → ∫ 𝐴 if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) d 𝑥 ∈ ℂ ) |
79 |
14 51 17 54 14 17 59 61
|
itgaddlem1 |
⊢ ( 𝜑 → ∫ 𝐴 ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) d 𝑥 = ( ∫ 𝐴 if ( 0 ≤ 𝐵 , 𝐵 , 0 ) d 𝑥 + ∫ 𝐴 if ( 0 ≤ 𝐶 , 𝐶 , 0 ) d 𝑥 ) ) |
80 |
14 51
|
itgcl |
⊢ ( 𝜑 → ∫ 𝐴 if ( 0 ≤ 𝐵 , 𝐵 , 0 ) d 𝑥 ∈ ℂ ) |
81 |
17 54
|
itgcl |
⊢ ( 𝜑 → ∫ 𝐴 if ( 0 ≤ 𝐶 , 𝐶 , 0 ) d 𝑥 ∈ ℂ ) |
82 |
80 81
|
addcld |
⊢ ( 𝜑 → ( ∫ 𝐴 if ( 0 ≤ 𝐵 , 𝐵 , 0 ) d 𝑥 + ∫ 𝐴 if ( 0 ≤ 𝐶 , 𝐶 , 0 ) d 𝑥 ) ∈ ℂ ) |
83 |
79 82
|
eqeltrd |
⊢ ( 𝜑 → ∫ 𝐴 ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) d 𝑥 ∈ ℂ ) |
84 |
38 64
|
itgcl |
⊢ ( 𝜑 → ∫ 𝐴 if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) d 𝑥 ∈ ℂ ) |
85 |
21 66 25 67 21 25 72 74
|
itgaddlem1 |
⊢ ( 𝜑 → ∫ 𝐴 ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) d 𝑥 = ( ∫ 𝐴 if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) d 𝑥 + ∫ 𝐴 if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) d 𝑥 ) ) |
86 |
21 66
|
itgcl |
⊢ ( 𝜑 → ∫ 𝐴 if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) d 𝑥 ∈ ℂ ) |
87 |
25 67
|
itgcl |
⊢ ( 𝜑 → ∫ 𝐴 if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) d 𝑥 ∈ ℂ ) |
88 |
86 87
|
addcld |
⊢ ( 𝜑 → ( ∫ 𝐴 if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) d 𝑥 + ∫ 𝐴 if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) d 𝑥 ) ∈ ℂ ) |
89 |
85 88
|
eqeltrd |
⊢ ( 𝜑 → ∫ 𝐴 ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) d 𝑥 ∈ ℂ ) |
90 |
78 83 84 89
|
addsubeq4d |
⊢ ( 𝜑 → ( ( ∫ 𝐴 if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) d 𝑥 + ∫ 𝐴 ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) d 𝑥 ) = ( ∫ 𝐴 if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) d 𝑥 + ∫ 𝐴 ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) d 𝑥 ) ↔ ( ∫ 𝐴 if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) d 𝑥 − ∫ 𝐴 if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) d 𝑥 ) = ( ∫ 𝐴 ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) d 𝑥 − ∫ 𝐴 ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) d 𝑥 ) ) ) |
91 |
77 90
|
mpbid |
⊢ ( 𝜑 → ( ∫ 𝐴 if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) d 𝑥 − ∫ 𝐴 if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) d 𝑥 ) = ( ∫ 𝐴 ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) d 𝑥 − ∫ 𝐴 ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) d 𝑥 ) ) |
92 |
79 85
|
oveq12d |
⊢ ( 𝜑 → ( ∫ 𝐴 ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) d 𝑥 − ∫ 𝐴 ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) d 𝑥 ) = ( ( ∫ 𝐴 if ( 0 ≤ 𝐵 , 𝐵 , 0 ) d 𝑥 + ∫ 𝐴 if ( 0 ≤ 𝐶 , 𝐶 , 0 ) d 𝑥 ) − ( ∫ 𝐴 if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) d 𝑥 + ∫ 𝐴 if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) d 𝑥 ) ) ) |
93 |
80 81 86 87
|
addsub4d |
⊢ ( 𝜑 → ( ( ∫ 𝐴 if ( 0 ≤ 𝐵 , 𝐵 , 0 ) d 𝑥 + ∫ 𝐴 if ( 0 ≤ 𝐶 , 𝐶 , 0 ) d 𝑥 ) − ( ∫ 𝐴 if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) d 𝑥 + ∫ 𝐴 if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) d 𝑥 ) ) = ( ( ∫ 𝐴 if ( 0 ≤ 𝐵 , 𝐵 , 0 ) d 𝑥 − ∫ 𝐴 if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) d 𝑥 ) + ( ∫ 𝐴 if ( 0 ≤ 𝐶 , 𝐶 , 0 ) d 𝑥 − ∫ 𝐴 if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) d 𝑥 ) ) ) |
94 |
91 92 93
|
3eqtrd |
⊢ ( 𝜑 → ( ∫ 𝐴 if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) d 𝑥 − ∫ 𝐴 if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) d 𝑥 ) = ( ( ∫ 𝐴 if ( 0 ≤ 𝐵 , 𝐵 , 0 ) d 𝑥 − ∫ 𝐴 if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) d 𝑥 ) + ( ∫ 𝐴 if ( 0 ≤ 𝐶 , 𝐶 , 0 ) d 𝑥 − ∫ 𝐴 if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) d 𝑥 ) ) ) |
95 |
28 44
|
itgreval |
⊢ ( 𝜑 → ∫ 𝐴 ( 𝐵 + 𝐶 ) d 𝑥 = ( ∫ 𝐴 if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) d 𝑥 − ∫ 𝐴 if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) d 𝑥 ) ) |
96 |
5 2
|
itgreval |
⊢ ( 𝜑 → ∫ 𝐴 𝐵 d 𝑥 = ( ∫ 𝐴 if ( 0 ≤ 𝐵 , 𝐵 , 0 ) d 𝑥 − ∫ 𝐴 if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) d 𝑥 ) ) |
97 |
6 4
|
itgreval |
⊢ ( 𝜑 → ∫ 𝐴 𝐶 d 𝑥 = ( ∫ 𝐴 if ( 0 ≤ 𝐶 , 𝐶 , 0 ) d 𝑥 − ∫ 𝐴 if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) d 𝑥 ) ) |
98 |
96 97
|
oveq12d |
⊢ ( 𝜑 → ( ∫ 𝐴 𝐵 d 𝑥 + ∫ 𝐴 𝐶 d 𝑥 ) = ( ( ∫ 𝐴 if ( 0 ≤ 𝐵 , 𝐵 , 0 ) d 𝑥 − ∫ 𝐴 if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) d 𝑥 ) + ( ∫ 𝐴 if ( 0 ≤ 𝐶 , 𝐶 , 0 ) d 𝑥 − ∫ 𝐴 if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) d 𝑥 ) ) ) |
99 |
94 95 98
|
3eqtr4d |
⊢ ( 𝜑 → ∫ 𝐴 ( 𝐵 + 𝐶 ) d 𝑥 = ( ∫ 𝐴 𝐵 d 𝑥 + ∫ 𝐴 𝐶 d 𝑥 ) ) |