Proof of Theorem itgaddlem2
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | itgadd.5 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | 
| 2 |  | max0sub 13238 | . . . . . . . . . 10
⊢ (𝐵 ∈ ℝ → (if(0
≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0)) = 𝐵) | 
| 3 | 1, 2 | syl 17 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0)) = 𝐵) | 
| 4 |  | itgadd.6 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ) | 
| 5 |  | max0sub 13238 | . . . . . . . . . 10
⊢ (𝐶 ∈ ℝ → (if(0
≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) = 𝐶) | 
| 6 | 4, 5 | syl 17 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) = 𝐶) | 
| 7 | 3, 6 | oveq12d 7449 | . . . . . . . 8
⊢ ((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)) | 
| 8 |  | 0re 11263 | . . . . . . . . . . 11
⊢ 0 ∈
ℝ | 
| 9 |  | ifcl 4571 | . . . . . . . . . . 11
⊢ ((𝐵 ∈ ℝ ∧ 0 ∈
ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ) | 
| 10 | 1, 8, 9 | sylancl 586 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ) | 
| 11 | 10 | recnd 11289 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℂ) | 
| 12 |  | ifcl 4571 | . . . . . . . . . . 11
⊢ ((𝐶 ∈ ℝ ∧ 0 ∈
ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ) | 
| 13 | 4, 8, 12 | sylancl 586 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ) | 
| 14 | 13 | recnd 11289 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℂ) | 
| 15 | 1 | renegcld 11690 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ) | 
| 16 |  | ifcl 4571 | . . . . . . . . . . 11
⊢ ((-𝐵 ∈ ℝ ∧ 0 ∈
ℝ) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ) | 
| 17 | 15, 8, 16 | sylancl 586 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ) | 
| 18 | 17 | recnd 11289 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℂ) | 
| 19 | 4 | renegcld 11690 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐶 ∈ ℝ) | 
| 20 |  | ifcl 4571 | . . . . . . . . . . 11
⊢ ((-𝐶 ∈ ℝ ∧ 0 ∈
ℝ) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ) | 
| 21 | 19, 8, 20 | sylancl 586 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ) | 
| 22 | 21 | recnd 11289 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℂ) | 
| 23 | 11, 14, 18, 22 | addsub4d 11667 | . . . . . . . 8
⊢ ((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)))) | 
| 24 | 1, 4 | readdcld 11290 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + 𝐶) ∈ ℝ) | 
| 25 |  | max0sub 13238 | . . . . . . . . 9
⊢ ((𝐵 + 𝐶) ∈ ℝ → (if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) − if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) = (𝐵 + 𝐶)) | 
| 26 | 24, 25 | syl 17 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) − if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) = (𝐵 + 𝐶)) | 
| 27 | 7, 23, 26 | 3eqtr4rd 2788 | . . . . . . 7
⊢ ((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)))) | 
| 28 | 24 | renegcld 11690 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(𝐵 + 𝐶) ∈ ℝ) | 
| 29 |  | ifcl 4571 | . . . . . . . . . 10
⊢ ((-(𝐵 + 𝐶) ∈ ℝ ∧ 0 ∈ ℝ)
→ if(0 ≤ -(𝐵 +
𝐶), -(𝐵 + 𝐶), 0) ∈ ℝ) | 
| 30 | 28, 8, 29 | sylancl 586 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) ∈ ℝ) | 
| 31 | 30 | recnd 11289 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) ∈ ℂ) | 
| 32 | 11, 14 | addcld 11280 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℂ) | 
| 33 |  | ifcl 4571 | . . . . . . . . . 10
⊢ (((𝐵 + 𝐶) ∈ ℝ ∧ 0 ∈ ℝ)
→ if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) ∈ ℝ) | 
| 34 | 24, 8, 33 | sylancl 586 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) ∈ ℝ) | 
| 35 | 34 | recnd 11289 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) ∈ ℂ) | 
| 36 | 18, 22 | addcld 11280 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) ∈ ℂ) | 
| 37 | 31, 32, 35, 36 | addsubeq4d 11671 | . . . . . . 7
⊢ ((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))))) | 
| 38 | 27, 37 | mpbird 257 | . . . . . 6
⊢ ((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)))) | 
| 39 | 38 | itgeq2dv 25817 | . . . . 5
⊢ (𝜑 → ∫𝐴(if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) + (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) d𝑥 = ∫𝐴(if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) + (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))) d𝑥) | 
| 40 |  | itgadd.1 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | 
| 41 |  | itgadd.2 | . . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈
𝐿1) | 
| 42 |  | itgadd.3 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉) | 
| 43 |  | itgadd.4 | . . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈
𝐿1) | 
| 44 | 40, 41, 42, 43 | ibladd 25856 | . . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈
𝐿1) | 
| 45 | 24 | iblre 25829 | . . . . . . . 8
⊢ (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 232 | . . . . . . 7
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0)) ∈ 𝐿1 ∧
(𝑥 ∈ 𝐴 ↦ if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) ∈
𝐿1)) | 
| 47 | 46 | simprd 495 | . . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) ∈
𝐿1) | 
| 48 | 10, 13 | readdcld 11290 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ) | 
| 49 | 1 | iblre 25829 | . . . . . . . . 9
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔
((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1 ∧
(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈
𝐿1))) | 
| 50 | 41, 49 | mpbid 232 | . . . . . . . 8
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1 ∧
(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈
𝐿1)) | 
| 51 | 50 | simpld 494 | . . . . . . 7
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈
𝐿1) | 
| 52 | 4 | iblre 25829 | . . . . . . . . 9
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1 ↔
((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)) ∈ 𝐿1 ∧
(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐶, -𝐶, 0)) ∈
𝐿1))) | 
| 53 | 43, 52 | mpbid 232 | . . . . . . . 8
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)) ∈ 𝐿1 ∧
(𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐶, -𝐶, 0)) ∈
𝐿1)) | 
| 54 | 53 | simpld 494 | . . . . . . 7
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)) ∈
𝐿1) | 
| 55 | 10, 51, 13, 54 | ibladd 25856 | . . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ∈
𝐿1) | 
| 56 |  | max1 13227 | . . . . . . 7
⊢ ((0
∈ ℝ ∧ -(𝐵 +
𝐶) ∈ ℝ) → 0
≤ if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) | 
| 57 | 8, 28, 56 | sylancr 587 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) | 
| 58 |  | max1 13227 | . . . . . . . 8
⊢ ((0
∈ ℝ ∧ 𝐵
∈ ℝ) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0)) | 
| 59 | 8, 1, 58 | sylancr 587 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0)) | 
| 60 |  | max1 13227 | . . . . . . . 8
⊢ ((0
∈ ℝ ∧ 𝐶
∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0)) | 
| 61 | 8, 4, 60 | sylancr 587 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0)) | 
| 62 | 10, 13, 59, 61 | addge0d 11839 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) | 
| 63 | 30, 47, 48, 55, 30, 48, 57, 62 | itgaddlem1 25858 | . . . . 5
⊢ (𝜑 → ∫𝐴(if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) + (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) d𝑥 = (∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥 + ∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥)) | 
| 64 | 46 | simpld 494 | . . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0)) ∈
𝐿1) | 
| 65 | 17, 21 | readdcld 11290 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) ∈ ℝ) | 
| 66 | 50 | simprd 495 | . . . . . . 7
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈
𝐿1) | 
| 67 | 53 | simprd 495 | . . . . . . 7
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐶, -𝐶, 0)) ∈
𝐿1) | 
| 68 | 17, 66, 21, 67 | ibladd 25856 | . . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))) ∈
𝐿1) | 
| 69 |  | max1 13227 | . . . . . . 7
⊢ ((0
∈ ℝ ∧ (𝐵 +
𝐶) ∈ ℝ) → 0
≤ if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0)) | 
| 70 | 8, 24, 69 | sylancr 587 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0)) | 
| 71 |  | max1 13227 | . . . . . . . 8
⊢ ((0
∈ ℝ ∧ -𝐵
∈ ℝ) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0)) | 
| 72 | 8, 15, 71 | sylancr 587 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0)) | 
| 73 |  | max1 13227 | . . . . . . . 8
⊢ ((0
∈ ℝ ∧ -𝐶
∈ ℝ) → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0)) | 
| 74 | 8, 19, 73 | sylancr 587 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0)) | 
| 75 | 17, 21, 72, 74 | addge0d 11839 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))) | 
| 76 | 34, 64, 65, 68, 34, 65, 70, 75 | itgaddlem1 25858 | . . . . 5
⊢ (𝜑 → ∫𝐴(if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) + (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))) d𝑥 = (∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 + ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥)) | 
| 77 | 39, 63, 76 | 3eqtr3d 2785 | . . . 4
⊢ (𝜑 → (∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥 + ∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥) = (∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 + ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥)) | 
| 78 | 30, 47 | itgcl 25819 | . . . . 5
⊢ (𝜑 → ∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥 ∈ ℂ) | 
| 79 | 10, 51, 13, 54, 10, 13, 59, 61 | itgaddlem1 25858 | . . . . . 6
⊢ (𝜑 → ∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥 = (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥)) | 
| 80 | 10, 51 | itgcl 25819 | . . . . . . 7
⊢ (𝜑 → ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 ∈ ℂ) | 
| 81 | 13, 54 | itgcl 25819 | . . . . . . 7
⊢ (𝜑 → ∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥 ∈ ℂ) | 
| 82 | 80, 81 | addcld 11280 | . . . . . 6
⊢ (𝜑 → (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥) ∈ ℂ) | 
| 83 | 79, 82 | eqeltrd 2841 | . . . . 5
⊢ (𝜑 → ∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥 ∈ ℂ) | 
| 84 | 34, 64 | itgcl 25819 | . . . . 5
⊢ (𝜑 → ∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 ∈ ℂ) | 
| 85 | 17, 66, 21, 67, 17, 21, 72, 74 | itgaddlem1 25858 | . . . . . 6
⊢ (𝜑 → ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥 = (∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥)) | 
| 86 | 17, 66 | itgcl 25819 | . . . . . . 7
⊢ (𝜑 → ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 ∈ ℂ) | 
| 87 | 21, 67 | itgcl 25819 | . . . . . . 7
⊢ (𝜑 → ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥 ∈ ℂ) | 
| 88 | 86, 87 | addcld 11280 | . . . . . 6
⊢ (𝜑 → (∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥) ∈ ℂ) | 
| 89 | 85, 88 | eqeltrd 2841 | . . . . 5
⊢ (𝜑 → ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥 ∈ ℂ) | 
| 90 | 78, 83, 84, 89 | addsubeq4d 11671 | . . . 4
⊢ (𝜑 → ((∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥 + ∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥) = (∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 + ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥) ↔ (∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 − ∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥) = (∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥 − ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥))) | 
| 91 | 77, 90 | mpbid 232 | . . 3
⊢ (𝜑 → (∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 − ∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥) = (∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥 − ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥)) | 
| 92 | 79, 85 | oveq12d 7449 | . . 3
⊢ (𝜑 → (∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥 − ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥) = ((∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥) − (∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥))) | 
| 93 | 80, 81, 86, 87 | addsub4d 11667 | . . 3
⊢ (𝜑 → ((∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥) − (∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥)) = ((∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥) + (∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥))) | 
| 94 | 91, 92, 93 | 3eqtrd 2781 | . 2
⊢ (𝜑 → (∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 − ∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥) = ((∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥) + (∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥))) | 
| 95 | 24, 44 | itgreval 25832 | . 2
⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 − ∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥)) | 
| 96 | 1, 41 | itgreval 25832 | . . 3
⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥)) | 
| 97 | 4, 43 | itgreval 25832 | . . 3
⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = (∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥)) | 
| 98 | 96, 97 | oveq12d 7449 | . 2
⊢ (𝜑 → (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥) = ((∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥) + (∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥))) | 
| 99 | 94, 95, 98 | 3eqtr4d 2787 | 1
⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥)) |