Step |
Hyp |
Ref |
Expression |
1 |
|
ibladdnc.1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 ) |
2 |
|
ibladdnc.2 |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝐿1 ) |
3 |
|
ibladdnc.3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ 𝑉 ) |
4 |
|
ibladdnc.4 |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ∈ 𝐿1 ) |
5 |
|
ibladdnc.m |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 + 𝐶 ) ) ∈ MblFn ) |
6 |
|
itgaddnclem.1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
7 |
|
itgaddnclem.2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ ℝ ) |
8 |
|
max0sub |
⊢ ( 𝐵 ∈ ℝ → ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) − if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) = 𝐵 ) |
9 |
6 8
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) − if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) = 𝐵 ) |
10 |
|
max0sub |
⊢ ( 𝐶 ∈ ℝ → ( if ( 0 ≤ 𝐶 , 𝐶 , 0 ) − if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) = 𝐶 ) |
11 |
7 10
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( if ( 0 ≤ 𝐶 , 𝐶 , 0 ) − if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) = 𝐶 ) |
12 |
9 11
|
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 ) ) |
13 |
|
0re |
⊢ 0 ∈ ℝ |
14 |
|
ifcl |
⊢ ( ( 𝐵 ∈ ℝ ∧ 0 ∈ ℝ ) → if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ∈ ℝ ) |
15 |
6 13 14
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ∈ ℝ ) |
16 |
15
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ∈ ℂ ) |
17 |
|
ifcl |
⊢ ( ( 𝐶 ∈ ℝ ∧ 0 ∈ ℝ ) → if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ∈ ℝ ) |
18 |
7 13 17
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ∈ ℝ ) |
19 |
18
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ∈ ℂ ) |
20 |
6
|
renegcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → - 𝐵 ∈ ℝ ) |
21 |
|
ifcl |
⊢ ( ( - 𝐵 ∈ ℝ ∧ 0 ∈ ℝ ) → if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ∈ ℝ ) |
22 |
20 13 21
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ∈ ℝ ) |
23 |
22
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ∈ ℂ ) |
24 |
7
|
renegcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → - 𝐶 ∈ ℝ ) |
25 |
|
ifcl |
⊢ ( ( - 𝐶 ∈ ℝ ∧ 0 ∈ ℝ ) → if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ∈ ℝ ) |
26 |
24 13 25
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ∈ ℝ ) |
27 |
26
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ∈ ℂ ) |
28 |
16 19 23 27
|
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 ) ) ) ) |
29 |
6 7
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐵 + 𝐶 ) ∈ ℝ ) |
30 |
|
max0sub |
⊢ ( ( 𝐵 + 𝐶 ) ∈ ℝ → ( if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) − if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) ) = ( 𝐵 + 𝐶 ) ) |
31 |
29 30
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) − if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) ) = ( 𝐵 + 𝐶 ) ) |
32 |
12 28 31
|
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 ) ) ) ) |
33 |
29
|
renegcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → - ( 𝐵 + 𝐶 ) ∈ ℝ ) |
34 |
|
ifcl |
⊢ ( ( - ( 𝐵 + 𝐶 ) ∈ ℝ ∧ 0 ∈ ℝ ) → if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) ∈ ℝ ) |
35 |
33 13 34
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) ∈ ℝ ) |
36 |
35
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) ∈ ℂ ) |
37 |
15 18
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ∈ ℝ ) |
38 |
37
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ∈ ℂ ) |
39 |
|
ifcl |
⊢ ( ( ( 𝐵 + 𝐶 ) ∈ ℝ ∧ 0 ∈ ℝ ) → if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) ∈ ℝ ) |
40 |
29 13 39
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) ∈ ℝ ) |
41 |
40
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) ∈ ℂ ) |
42 |
22 26
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ∈ ℝ ) |
43 |
42
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ∈ ℂ ) |
44 |
36 38 41 43
|
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 ) ) ) ) ) |
45 |
32 44
|
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 ) ) ) ) |
46 |
45
|
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 ) |
47 |
6 2 7 4 5
|
ibladdnc |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 + 𝐶 ) ) ∈ 𝐿1 ) |
48 |
29
|
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 ) ) ) |
49 |
47 48
|
mpbid |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) ) ∈ 𝐿1 ∧ ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) ) ∈ 𝐿1 ) ) |
50 |
49
|
simprd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) ) ∈ 𝐿1 ) |
51 |
6
|
iblre |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝐿1 ↔ ( ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) ∈ 𝐿1 ∧ ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) ∈ 𝐿1 ) ) ) |
52 |
2 51
|
mpbid |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) ∈ 𝐿1 ∧ ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) ∈ 𝐿1 ) ) |
53 |
52
|
simpld |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) ∈ 𝐿1 ) |
54 |
7
|
iblre |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ∈ 𝐿1 ↔ ( ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ∈ 𝐿1 ∧ ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ∈ 𝐿1 ) ) ) |
55 |
4 54
|
mpbid |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ∈ 𝐿1 ∧ ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ∈ 𝐿1 ) ) |
56 |
55
|
simpld |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ∈ 𝐿1 ) |
57 |
|
iblmbf |
⊢ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝐿1 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ MblFn ) |
58 |
2 57
|
syl |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ MblFn ) |
59 |
|
iblmbf |
⊢ ( ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ∈ 𝐿1 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ∈ MblFn ) |
60 |
4 59
|
syl |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ∈ MblFn ) |
61 |
58 6 60 7 5
|
mbfposadd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) ∈ MblFn ) |
62 |
15 53 18 56 61
|
ibladdnc |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) ∈ 𝐿1 ) |
63 |
|
max1 |
⊢ ( ( 0 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → 0 ≤ if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) |
64 |
13 6 63
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) |
65 |
|
max1 |
⊢ ( ( 0 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 0 ≤ if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) |
66 |
13 7 65
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) |
67 |
15 18 64 66
|
addge0d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) |
68 |
67
|
iftrued |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) , ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) , 0 ) = ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) |
69 |
68
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) + if ( 0 ≤ ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) , ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) , 0 ) ) = ( if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) + ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) ) |
70 |
69
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) + if ( 0 ≤ ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) , ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) , 0 ) ) ) = ( 𝑥 ∈ 𝐴 ↦ ( if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) + ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) ) ) |
71 |
29 5
|
mbfneg |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ - ( 𝐵 + 𝐶 ) ) ∈ MblFn ) |
72 |
6
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
73 |
7
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ ℂ ) |
74 |
72 73
|
negdid |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → - ( 𝐵 + 𝐶 ) = ( - 𝐵 + - 𝐶 ) ) |
75 |
74
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( - ( 𝐵 + 𝐶 ) + ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) = ( ( - 𝐵 + - 𝐶 ) + ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) ) |
76 |
20
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → - 𝐵 ∈ ℂ ) |
77 |
24
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → - 𝐶 ∈ ℂ ) |
78 |
76 77 16 19
|
add4d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( - 𝐵 + - 𝐶 ) + ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) = ( ( - 𝐵 + if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) + ( - 𝐶 + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) ) |
79 |
|
negeq |
⊢ ( 𝐵 = 0 → - 𝐵 = - 0 ) |
80 |
|
neg0 |
⊢ - 0 = 0 |
81 |
79 80
|
eqtrdi |
⊢ ( 𝐵 = 0 → - 𝐵 = 0 ) |
82 |
|
0le0 |
⊢ 0 ≤ 0 |
83 |
82 81
|
breqtrrid |
⊢ ( 𝐵 = 0 → 0 ≤ - 𝐵 ) |
84 |
83
|
iftrued |
⊢ ( 𝐵 = 0 → if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) = - 𝐵 ) |
85 |
|
id |
⊢ ( 𝐵 = 0 → 𝐵 = 0 ) |
86 |
82 85
|
breqtrrid |
⊢ ( 𝐵 = 0 → 0 ≤ 𝐵 ) |
87 |
86
|
iftrued |
⊢ ( 𝐵 = 0 → if ( 0 ≤ 𝐵 , 𝐵 , 0 ) = 𝐵 ) |
88 |
87 85
|
eqtrd |
⊢ ( 𝐵 = 0 → if ( 0 ≤ 𝐵 , 𝐵 , 0 ) = 0 ) |
89 |
81 88
|
oveq12d |
⊢ ( 𝐵 = 0 → ( - 𝐵 + if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) = ( 0 + 0 ) ) |
90 |
|
00id |
⊢ ( 0 + 0 ) = 0 |
91 |
89 90
|
eqtrdi |
⊢ ( 𝐵 = 0 → ( - 𝐵 + if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) = 0 ) |
92 |
81 84 91
|
3eqtr4rd |
⊢ ( 𝐵 = 0 → ( - 𝐵 + if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) = if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) |
93 |
92
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐵 = 0 ) → ( - 𝐵 + if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) = if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) |
94 |
|
ovif2 |
⊢ ( - 𝐵 + if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) = if ( 0 ≤ 𝐵 , ( - 𝐵 + 𝐵 ) , ( - 𝐵 + 0 ) ) |
95 |
72
|
negne0bd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐵 ≠ 0 ↔ - 𝐵 ≠ 0 ) ) |
96 |
95
|
biimpa |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐵 ≠ 0 ) → - 𝐵 ≠ 0 ) |
97 |
6
|
le0neg2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 0 ≤ 𝐵 ↔ - 𝐵 ≤ 0 ) ) |
98 |
|
leloe |
⊢ ( ( - 𝐵 ∈ ℝ ∧ 0 ∈ ℝ ) → ( - 𝐵 ≤ 0 ↔ ( - 𝐵 < 0 ∨ - 𝐵 = 0 ) ) ) |
99 |
20 13 98
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( - 𝐵 ≤ 0 ↔ ( - 𝐵 < 0 ∨ - 𝐵 = 0 ) ) ) |
100 |
97 99
|
bitrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 0 ≤ 𝐵 ↔ ( - 𝐵 < 0 ∨ - 𝐵 = 0 ) ) ) |
101 |
|
df-ne |
⊢ ( - 𝐵 ≠ 0 ↔ ¬ - 𝐵 = 0 ) |
102 |
|
biorf |
⊢ ( ¬ - 𝐵 = 0 → ( - 𝐵 < 0 ↔ ( - 𝐵 = 0 ∨ - 𝐵 < 0 ) ) ) |
103 |
101 102
|
sylbi |
⊢ ( - 𝐵 ≠ 0 → ( - 𝐵 < 0 ↔ ( - 𝐵 = 0 ∨ - 𝐵 < 0 ) ) ) |
104 |
|
orcom |
⊢ ( ( - 𝐵 = 0 ∨ - 𝐵 < 0 ) ↔ ( - 𝐵 < 0 ∨ - 𝐵 = 0 ) ) |
105 |
103 104
|
bitr2di |
⊢ ( - 𝐵 ≠ 0 → ( ( - 𝐵 < 0 ∨ - 𝐵 = 0 ) ↔ - 𝐵 < 0 ) ) |
106 |
100 105
|
sylan9bb |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ - 𝐵 ≠ 0 ) → ( 0 ≤ 𝐵 ↔ - 𝐵 < 0 ) ) |
107 |
96 106
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐵 ≠ 0 ) → ( 0 ≤ 𝐵 ↔ - 𝐵 < 0 ) ) |
108 |
|
ltnle |
⊢ ( ( - 𝐵 ∈ ℝ ∧ 0 ∈ ℝ ) → ( - 𝐵 < 0 ↔ ¬ 0 ≤ - 𝐵 ) ) |
109 |
20 13 108
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( - 𝐵 < 0 ↔ ¬ 0 ≤ - 𝐵 ) ) |
110 |
109
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐵 ≠ 0 ) → ( - 𝐵 < 0 ↔ ¬ 0 ≤ - 𝐵 ) ) |
111 |
107 110
|
bitrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐵 ≠ 0 ) → ( 0 ≤ 𝐵 ↔ ¬ 0 ≤ - 𝐵 ) ) |
112 |
76 72
|
addcomd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( - 𝐵 + 𝐵 ) = ( 𝐵 + - 𝐵 ) ) |
113 |
72
|
negidd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐵 + - 𝐵 ) = 0 ) |
114 |
112 113
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( - 𝐵 + 𝐵 ) = 0 ) |
115 |
114
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐵 ≠ 0 ) → ( - 𝐵 + 𝐵 ) = 0 ) |
116 |
76
|
addid1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( - 𝐵 + 0 ) = - 𝐵 ) |
117 |
116
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐵 ≠ 0 ) → ( - 𝐵 + 0 ) = - 𝐵 ) |
118 |
111 115 117
|
ifbieq12d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐵 ≠ 0 ) → if ( 0 ≤ 𝐵 , ( - 𝐵 + 𝐵 ) , ( - 𝐵 + 0 ) ) = if ( ¬ 0 ≤ - 𝐵 , 0 , - 𝐵 ) ) |
119 |
|
ifnot |
⊢ if ( ¬ 0 ≤ - 𝐵 , 0 , - 𝐵 ) = if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) |
120 |
118 119
|
eqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐵 ≠ 0 ) → if ( 0 ≤ 𝐵 , ( - 𝐵 + 𝐵 ) , ( - 𝐵 + 0 ) ) = if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) |
121 |
94 120
|
syl5eq |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐵 ≠ 0 ) → ( - 𝐵 + if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) = if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) |
122 |
93 121
|
pm2.61dane |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( - 𝐵 + if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) = if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) |
123 |
|
negeq |
⊢ ( 𝐶 = 0 → - 𝐶 = - 0 ) |
124 |
123 80
|
eqtrdi |
⊢ ( 𝐶 = 0 → - 𝐶 = 0 ) |
125 |
82 124
|
breqtrrid |
⊢ ( 𝐶 = 0 → 0 ≤ - 𝐶 ) |
126 |
125
|
iftrued |
⊢ ( 𝐶 = 0 → if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) = - 𝐶 ) |
127 |
|
id |
⊢ ( 𝐶 = 0 → 𝐶 = 0 ) |
128 |
82 127
|
breqtrrid |
⊢ ( 𝐶 = 0 → 0 ≤ 𝐶 ) |
129 |
128
|
iftrued |
⊢ ( 𝐶 = 0 → if ( 0 ≤ 𝐶 , 𝐶 , 0 ) = 𝐶 ) |
130 |
129 127
|
eqtrd |
⊢ ( 𝐶 = 0 → if ( 0 ≤ 𝐶 , 𝐶 , 0 ) = 0 ) |
131 |
124 130
|
oveq12d |
⊢ ( 𝐶 = 0 → ( - 𝐶 + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) = ( 0 + 0 ) ) |
132 |
131 90
|
eqtrdi |
⊢ ( 𝐶 = 0 → ( - 𝐶 + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) = 0 ) |
133 |
124 126 132
|
3eqtr4rd |
⊢ ( 𝐶 = 0 → ( - 𝐶 + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) = if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) |
134 |
133
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐶 = 0 ) → ( - 𝐶 + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) = if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) |
135 |
|
ovif2 |
⊢ ( - 𝐶 + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) = if ( 0 ≤ 𝐶 , ( - 𝐶 + 𝐶 ) , ( - 𝐶 + 0 ) ) |
136 |
73
|
negne0bd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐶 ≠ 0 ↔ - 𝐶 ≠ 0 ) ) |
137 |
136
|
biimpa |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐶 ≠ 0 ) → - 𝐶 ≠ 0 ) |
138 |
7
|
le0neg2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 0 ≤ 𝐶 ↔ - 𝐶 ≤ 0 ) ) |
139 |
|
leloe |
⊢ ( ( - 𝐶 ∈ ℝ ∧ 0 ∈ ℝ ) → ( - 𝐶 ≤ 0 ↔ ( - 𝐶 < 0 ∨ - 𝐶 = 0 ) ) ) |
140 |
24 13 139
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( - 𝐶 ≤ 0 ↔ ( - 𝐶 < 0 ∨ - 𝐶 = 0 ) ) ) |
141 |
138 140
|
bitrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 0 ≤ 𝐶 ↔ ( - 𝐶 < 0 ∨ - 𝐶 = 0 ) ) ) |
142 |
|
df-ne |
⊢ ( - 𝐶 ≠ 0 ↔ ¬ - 𝐶 = 0 ) |
143 |
|
biorf |
⊢ ( ¬ - 𝐶 = 0 → ( - 𝐶 < 0 ↔ ( - 𝐶 = 0 ∨ - 𝐶 < 0 ) ) ) |
144 |
142 143
|
sylbi |
⊢ ( - 𝐶 ≠ 0 → ( - 𝐶 < 0 ↔ ( - 𝐶 = 0 ∨ - 𝐶 < 0 ) ) ) |
145 |
|
orcom |
⊢ ( ( - 𝐶 = 0 ∨ - 𝐶 < 0 ) ↔ ( - 𝐶 < 0 ∨ - 𝐶 = 0 ) ) |
146 |
144 145
|
bitr2di |
⊢ ( - 𝐶 ≠ 0 → ( ( - 𝐶 < 0 ∨ - 𝐶 = 0 ) ↔ - 𝐶 < 0 ) ) |
147 |
141 146
|
sylan9bb |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ - 𝐶 ≠ 0 ) → ( 0 ≤ 𝐶 ↔ - 𝐶 < 0 ) ) |
148 |
137 147
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐶 ≠ 0 ) → ( 0 ≤ 𝐶 ↔ - 𝐶 < 0 ) ) |
149 |
|
ltnle |
⊢ ( ( - 𝐶 ∈ ℝ ∧ 0 ∈ ℝ ) → ( - 𝐶 < 0 ↔ ¬ 0 ≤ - 𝐶 ) ) |
150 |
24 13 149
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( - 𝐶 < 0 ↔ ¬ 0 ≤ - 𝐶 ) ) |
151 |
150
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐶 ≠ 0 ) → ( - 𝐶 < 0 ↔ ¬ 0 ≤ - 𝐶 ) ) |
152 |
148 151
|
bitrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐶 ≠ 0 ) → ( 0 ≤ 𝐶 ↔ ¬ 0 ≤ - 𝐶 ) ) |
153 |
77 73
|
addcomd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( - 𝐶 + 𝐶 ) = ( 𝐶 + - 𝐶 ) ) |
154 |
73
|
negidd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐶 + - 𝐶 ) = 0 ) |
155 |
153 154
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( - 𝐶 + 𝐶 ) = 0 ) |
156 |
155
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐶 ≠ 0 ) → ( - 𝐶 + 𝐶 ) = 0 ) |
157 |
77
|
addid1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( - 𝐶 + 0 ) = - 𝐶 ) |
158 |
157
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐶 ≠ 0 ) → ( - 𝐶 + 0 ) = - 𝐶 ) |
159 |
152 156 158
|
ifbieq12d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐶 ≠ 0 ) → if ( 0 ≤ 𝐶 , ( - 𝐶 + 𝐶 ) , ( - 𝐶 + 0 ) ) = if ( ¬ 0 ≤ - 𝐶 , 0 , - 𝐶 ) ) |
160 |
|
ifnot |
⊢ if ( ¬ 0 ≤ - 𝐶 , 0 , - 𝐶 ) = if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) |
161 |
159 160
|
eqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐶 ≠ 0 ) → if ( 0 ≤ 𝐶 , ( - 𝐶 + 𝐶 ) , ( - 𝐶 + 0 ) ) = if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) |
162 |
135 161
|
syl5eq |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐶 ≠ 0 ) → ( - 𝐶 + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) = if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) |
163 |
134 162
|
pm2.61dane |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( - 𝐶 + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) = if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) |
164 |
122 163
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( - 𝐵 + if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) + ( - 𝐶 + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) = ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) |
165 |
75 78 164
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( - ( 𝐵 + 𝐶 ) + ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) = ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) |
166 |
165
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( - ( 𝐵 + 𝐶 ) + ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) ) = ( 𝑥 ∈ 𝐴 ↦ ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) ) |
167 |
6 58
|
mbfneg |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) ∈ MblFn ) |
168 |
7 60
|
mbfneg |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ - 𝐶 ) ∈ MblFn ) |
169 |
74
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ - ( 𝐵 + 𝐶 ) ) = ( 𝑥 ∈ 𝐴 ↦ ( - 𝐵 + - 𝐶 ) ) ) |
170 |
169 71
|
eqeltrrd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( - 𝐵 + - 𝐶 ) ) ∈ MblFn ) |
171 |
167 20 168 24 170
|
mbfposadd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) ∈ MblFn ) |
172 |
166 171
|
eqeltrd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( - ( 𝐵 + 𝐶 ) + ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) ) ∈ MblFn ) |
173 |
71 33 61 37 172
|
mbfposadd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) + if ( 0 ≤ ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) , ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) , 0 ) ) ) ∈ MblFn ) |
174 |
70 173
|
eqeltrrd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) + ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) ) ∈ MblFn ) |
175 |
|
max1 |
⊢ ( ( 0 ∈ ℝ ∧ - ( 𝐵 + 𝐶 ) ∈ ℝ ) → 0 ≤ if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) ) |
176 |
13 33 175
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) ) |
177 |
35 50 37 62 174 35 37 176 67
|
itgaddnclem1 |
⊢ ( 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 ) ) |
178 |
49
|
simpld |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) ) ∈ 𝐿1 ) |
179 |
52
|
simprd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) ∈ 𝐿1 ) |
180 |
55
|
simprd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ∈ 𝐿1 ) |
181 |
22 179 26 180 171
|
ibladdnc |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) ∈ 𝐿1 ) |
182 |
|
max1 |
⊢ ( ( 0 ∈ ℝ ∧ - 𝐵 ∈ ℝ ) → 0 ≤ if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) |
183 |
13 20 182
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) |
184 |
|
max1 |
⊢ ( ( 0 ∈ ℝ ∧ - 𝐶 ∈ ℝ ) → 0 ≤ if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) |
185 |
13 24 184
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) |
186 |
22 26 183 185
|
addge0d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) |
187 |
186
|
iftrued |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( 0 ≤ ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) , ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) , 0 ) = ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) |
188 |
187
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) + if ( 0 ≤ ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) , ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) , 0 ) ) = ( if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) + ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) ) |
189 |
188
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) + if ( 0 ≤ ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) , ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) , 0 ) ) ) = ( 𝑥 ∈ 𝐴 ↦ ( if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) + ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) ) ) |
190 |
72 73 23 27
|
add4d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐵 + 𝐶 ) + ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) = ( ( 𝐵 + if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) + ( 𝐶 + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) ) |
191 |
84 81
|
eqtrd |
⊢ ( 𝐵 = 0 → if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) = 0 ) |
192 |
85 191
|
oveq12d |
⊢ ( 𝐵 = 0 → ( 𝐵 + if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) = ( 0 + 0 ) ) |
193 |
192 90
|
eqtrdi |
⊢ ( 𝐵 = 0 → ( 𝐵 + if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) = 0 ) |
194 |
85 87 193
|
3eqtr4rd |
⊢ ( 𝐵 = 0 → ( 𝐵 + if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) = if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) |
195 |
194
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐵 = 0 ) → ( 𝐵 + if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) = if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) |
196 |
|
ovif2 |
⊢ ( 𝐵 + if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) = if ( 0 ≤ - 𝐵 , ( 𝐵 + - 𝐵 ) , ( 𝐵 + 0 ) ) |
197 |
6
|
le0neg1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐵 ≤ 0 ↔ 0 ≤ - 𝐵 ) ) |
198 |
|
leloe |
⊢ ( ( 𝐵 ∈ ℝ ∧ 0 ∈ ℝ ) → ( 𝐵 ≤ 0 ↔ ( 𝐵 < 0 ∨ 𝐵 = 0 ) ) ) |
199 |
6 13 198
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐵 ≤ 0 ↔ ( 𝐵 < 0 ∨ 𝐵 = 0 ) ) ) |
200 |
197 199
|
bitr3d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 0 ≤ - 𝐵 ↔ ( 𝐵 < 0 ∨ 𝐵 = 0 ) ) ) |
201 |
|
df-ne |
⊢ ( 𝐵 ≠ 0 ↔ ¬ 𝐵 = 0 ) |
202 |
|
biorf |
⊢ ( ¬ 𝐵 = 0 → ( 𝐵 < 0 ↔ ( 𝐵 = 0 ∨ 𝐵 < 0 ) ) ) |
203 |
201 202
|
sylbi |
⊢ ( 𝐵 ≠ 0 → ( 𝐵 < 0 ↔ ( 𝐵 = 0 ∨ 𝐵 < 0 ) ) ) |
204 |
|
orcom |
⊢ ( ( 𝐵 = 0 ∨ 𝐵 < 0 ) ↔ ( 𝐵 < 0 ∨ 𝐵 = 0 ) ) |
205 |
203 204
|
bitr2di |
⊢ ( 𝐵 ≠ 0 → ( ( 𝐵 < 0 ∨ 𝐵 = 0 ) ↔ 𝐵 < 0 ) ) |
206 |
200 205
|
sylan9bb |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐵 ≠ 0 ) → ( 0 ≤ - 𝐵 ↔ 𝐵 < 0 ) ) |
207 |
|
ltnle |
⊢ ( ( 𝐵 ∈ ℝ ∧ 0 ∈ ℝ ) → ( 𝐵 < 0 ↔ ¬ 0 ≤ 𝐵 ) ) |
208 |
6 13 207
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐵 < 0 ↔ ¬ 0 ≤ 𝐵 ) ) |
209 |
208
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐵 ≠ 0 ) → ( 𝐵 < 0 ↔ ¬ 0 ≤ 𝐵 ) ) |
210 |
206 209
|
bitrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐵 ≠ 0 ) → ( 0 ≤ - 𝐵 ↔ ¬ 0 ≤ 𝐵 ) ) |
211 |
113
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐵 ≠ 0 ) → ( 𝐵 + - 𝐵 ) = 0 ) |
212 |
72
|
addid1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐵 + 0 ) = 𝐵 ) |
213 |
212
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐵 ≠ 0 ) → ( 𝐵 + 0 ) = 𝐵 ) |
214 |
210 211 213
|
ifbieq12d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐵 ≠ 0 ) → if ( 0 ≤ - 𝐵 , ( 𝐵 + - 𝐵 ) , ( 𝐵 + 0 ) ) = if ( ¬ 0 ≤ 𝐵 , 0 , 𝐵 ) ) |
215 |
|
ifnot |
⊢ if ( ¬ 0 ≤ 𝐵 , 0 , 𝐵 ) = if ( 0 ≤ 𝐵 , 𝐵 , 0 ) |
216 |
214 215
|
eqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐵 ≠ 0 ) → if ( 0 ≤ - 𝐵 , ( 𝐵 + - 𝐵 ) , ( 𝐵 + 0 ) ) = if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) |
217 |
196 216
|
syl5eq |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐵 ≠ 0 ) → ( 𝐵 + if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) = if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) |
218 |
195 217
|
pm2.61dane |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐵 + if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) = if ( 0 ≤ 𝐵 , 𝐵 , 0 ) ) |
219 |
126 124
|
eqtrd |
⊢ ( 𝐶 = 0 → if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) = 0 ) |
220 |
127 219
|
oveq12d |
⊢ ( 𝐶 = 0 → ( 𝐶 + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) = ( 0 + 0 ) ) |
221 |
220 90
|
eqtrdi |
⊢ ( 𝐶 = 0 → ( 𝐶 + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) = 0 ) |
222 |
127 129 221
|
3eqtr4rd |
⊢ ( 𝐶 = 0 → ( 𝐶 + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) = if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) |
223 |
222
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐶 = 0 ) → ( 𝐶 + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) = if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) |
224 |
|
ovif2 |
⊢ ( 𝐶 + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) = if ( 0 ≤ - 𝐶 , ( 𝐶 + - 𝐶 ) , ( 𝐶 + 0 ) ) |
225 |
7
|
le0neg1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐶 ≤ 0 ↔ 0 ≤ - 𝐶 ) ) |
226 |
|
leloe |
⊢ ( ( 𝐶 ∈ ℝ ∧ 0 ∈ ℝ ) → ( 𝐶 ≤ 0 ↔ ( 𝐶 < 0 ∨ 𝐶 = 0 ) ) ) |
227 |
7 13 226
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐶 ≤ 0 ↔ ( 𝐶 < 0 ∨ 𝐶 = 0 ) ) ) |
228 |
225 227
|
bitr3d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 0 ≤ - 𝐶 ↔ ( 𝐶 < 0 ∨ 𝐶 = 0 ) ) ) |
229 |
|
df-ne |
⊢ ( 𝐶 ≠ 0 ↔ ¬ 𝐶 = 0 ) |
230 |
|
biorf |
⊢ ( ¬ 𝐶 = 0 → ( 𝐶 < 0 ↔ ( 𝐶 = 0 ∨ 𝐶 < 0 ) ) ) |
231 |
229 230
|
sylbi |
⊢ ( 𝐶 ≠ 0 → ( 𝐶 < 0 ↔ ( 𝐶 = 0 ∨ 𝐶 < 0 ) ) ) |
232 |
|
orcom |
⊢ ( ( 𝐶 = 0 ∨ 𝐶 < 0 ) ↔ ( 𝐶 < 0 ∨ 𝐶 = 0 ) ) |
233 |
231 232
|
bitr2di |
⊢ ( 𝐶 ≠ 0 → ( ( 𝐶 < 0 ∨ 𝐶 = 0 ) ↔ 𝐶 < 0 ) ) |
234 |
228 233
|
sylan9bb |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐶 ≠ 0 ) → ( 0 ≤ - 𝐶 ↔ 𝐶 < 0 ) ) |
235 |
|
ltnle |
⊢ ( ( 𝐶 ∈ ℝ ∧ 0 ∈ ℝ ) → ( 𝐶 < 0 ↔ ¬ 0 ≤ 𝐶 ) ) |
236 |
7 13 235
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐶 < 0 ↔ ¬ 0 ≤ 𝐶 ) ) |
237 |
236
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐶 ≠ 0 ) → ( 𝐶 < 0 ↔ ¬ 0 ≤ 𝐶 ) ) |
238 |
234 237
|
bitrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐶 ≠ 0 ) → ( 0 ≤ - 𝐶 ↔ ¬ 0 ≤ 𝐶 ) ) |
239 |
154
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐶 ≠ 0 ) → ( 𝐶 + - 𝐶 ) = 0 ) |
240 |
73
|
addid1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐶 + 0 ) = 𝐶 ) |
241 |
240
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐶 ≠ 0 ) → ( 𝐶 + 0 ) = 𝐶 ) |
242 |
238 239 241
|
ifbieq12d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐶 ≠ 0 ) → if ( 0 ≤ - 𝐶 , ( 𝐶 + - 𝐶 ) , ( 𝐶 + 0 ) ) = if ( ¬ 0 ≤ 𝐶 , 0 , 𝐶 ) ) |
243 |
|
ifnot |
⊢ if ( ¬ 0 ≤ 𝐶 , 0 , 𝐶 ) = if ( 0 ≤ 𝐶 , 𝐶 , 0 ) |
244 |
242 243
|
eqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐶 ≠ 0 ) → if ( 0 ≤ - 𝐶 , ( 𝐶 + - 𝐶 ) , ( 𝐶 + 0 ) ) = if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) |
245 |
224 244
|
syl5eq |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐶 ≠ 0 ) → ( 𝐶 + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) = if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) |
246 |
223 245
|
pm2.61dane |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐶 + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) = if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) |
247 |
218 246
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐵 + if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) ) + ( 𝐶 + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) = ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) |
248 |
190 247
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐵 + 𝐶 ) + ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) = ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) |
249 |
248
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( ( 𝐵 + 𝐶 ) + ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) ) = ( 𝑥 ∈ 𝐴 ↦ ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) ) ) |
250 |
249 61
|
eqeltrd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( ( 𝐵 + 𝐶 ) + ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) ) ∈ MblFn ) |
251 |
5 29 171 42 250
|
mbfposadd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) + if ( 0 ≤ ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) , ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) , 0 ) ) ) ∈ MblFn ) |
252 |
189 251
|
eqeltrrd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) + ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) ) ) ∈ MblFn ) |
253 |
|
max1 |
⊢ ( ( 0 ∈ ℝ ∧ ( 𝐵 + 𝐶 ) ∈ ℝ ) → 0 ≤ if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) ) |
254 |
13 29 253
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) ) |
255 |
40 178 42 181 252 40 42 254 186
|
itgaddnclem1 |
⊢ ( 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 ) ) |
256 |
46 177 255
|
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 ) ) |
257 |
35 50
|
itgcl |
⊢ ( 𝜑 → ∫ 𝐴 if ( 0 ≤ - ( 𝐵 + 𝐶 ) , - ( 𝐵 + 𝐶 ) , 0 ) d 𝑥 ∈ ℂ ) |
258 |
15 53 18 56 61 15 18 64 66
|
itgaddnclem1 |
⊢ ( 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 ) ) |
259 |
15 53
|
itgcl |
⊢ ( 𝜑 → ∫ 𝐴 if ( 0 ≤ 𝐵 , 𝐵 , 0 ) d 𝑥 ∈ ℂ ) |
260 |
18 56
|
itgcl |
⊢ ( 𝜑 → ∫ 𝐴 if ( 0 ≤ 𝐶 , 𝐶 , 0 ) d 𝑥 ∈ ℂ ) |
261 |
259 260
|
addcld |
⊢ ( 𝜑 → ( ∫ 𝐴 if ( 0 ≤ 𝐵 , 𝐵 , 0 ) d 𝑥 + ∫ 𝐴 if ( 0 ≤ 𝐶 , 𝐶 , 0 ) d 𝑥 ) ∈ ℂ ) |
262 |
258 261
|
eqeltrd |
⊢ ( 𝜑 → ∫ 𝐴 ( if ( 0 ≤ 𝐵 , 𝐵 , 0 ) + if ( 0 ≤ 𝐶 , 𝐶 , 0 ) ) d 𝑥 ∈ ℂ ) |
263 |
40 178
|
itgcl |
⊢ ( 𝜑 → ∫ 𝐴 if ( 0 ≤ ( 𝐵 + 𝐶 ) , ( 𝐵 + 𝐶 ) , 0 ) d 𝑥 ∈ ℂ ) |
264 |
22 179 26 180 171 22 26 183 185
|
itgaddnclem1 |
⊢ ( 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 ) ) |
265 |
22 179
|
itgcl |
⊢ ( 𝜑 → ∫ 𝐴 if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) d 𝑥 ∈ ℂ ) |
266 |
26 180
|
itgcl |
⊢ ( 𝜑 → ∫ 𝐴 if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) d 𝑥 ∈ ℂ ) |
267 |
265 266
|
addcld |
⊢ ( 𝜑 → ( ∫ 𝐴 if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) d 𝑥 + ∫ 𝐴 if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) d 𝑥 ) ∈ ℂ ) |
268 |
264 267
|
eqeltrd |
⊢ ( 𝜑 → ∫ 𝐴 ( if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) + if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) ) d 𝑥 ∈ ℂ ) |
269 |
257 262 263 268
|
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 ) ) ) |
270 |
256 269
|
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 ) ) |
271 |
258 264
|
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 ) ) ) |
272 |
259 260 265 266
|
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 ) ) ) |
273 |
270 271 272
|
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 ) ) ) |
274 |
29 47
|
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 ) ) |
275 |
6 2
|
itgreval |
⊢ ( 𝜑 → ∫ 𝐴 𝐵 d 𝑥 = ( ∫ 𝐴 if ( 0 ≤ 𝐵 , 𝐵 , 0 ) d 𝑥 − ∫ 𝐴 if ( 0 ≤ - 𝐵 , - 𝐵 , 0 ) d 𝑥 ) ) |
276 |
7 4
|
itgreval |
⊢ ( 𝜑 → ∫ 𝐴 𝐶 d 𝑥 = ( ∫ 𝐴 if ( 0 ≤ 𝐶 , 𝐶 , 0 ) d 𝑥 − ∫ 𝐴 if ( 0 ≤ - 𝐶 , - 𝐶 , 0 ) d 𝑥 ) ) |
277 |
275 276
|
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 ) ) ) |
278 |
273 274 277
|
3eqtr4d |
⊢ ( 𝜑 → ∫ 𝐴 ( 𝐵 + 𝐶 ) d 𝑥 = ( ∫ 𝐴 𝐵 d 𝑥 + ∫ 𝐴 𝐶 d 𝑥 ) ) |