| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | itgabs.1 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | 
| 2 |  | itgabs.2 | . . . . . . . . . . . 12
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈
𝐿1) | 
| 3 | 1, 2 | itgcl 25819 | . . . . . . . . . . 11
⊢ (𝜑 → ∫𝐴𝐵 d𝑥 ∈ ℂ) | 
| 4 | 3 | cjcld 15235 | . . . . . . . . . 10
⊢ (𝜑 → (∗‘∫𝐴𝐵 d𝑥) ∈ ℂ) | 
| 5 |  | iblmbf 25802 | . . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) | 
| 6 | 2, 5 | syl 17 | . . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) | 
| 7 | 6, 1 | mbfmptcl 25671 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) | 
| 8 | 7 | ralrimiva 3146 | . . . . . . . . . . . 12
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ ℂ) | 
| 9 |  | nfv 1914 | . . . . . . . . . . . . 13
⊢
Ⅎ𝑦 𝐵 ∈ ℂ | 
| 10 |  | nfcsb1v 3923 | . . . . . . . . . . . . . 14
⊢
Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | 
| 11 | 10 | nfel1 2922 | . . . . . . . . . . . . 13
⊢
Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ | 
| 12 |  | csbeq1a 3913 | . . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | 
| 13 | 12 | eleq1d 2826 | . . . . . . . . . . . . 13
⊢ (𝑥 = 𝑦 → (𝐵 ∈ ℂ ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ)) | 
| 14 | 9, 11, 13 | cbvralw 3306 | . . . . . . . . . . . 12
⊢
(∀𝑥 ∈
𝐴 𝐵 ∈ ℂ ↔ ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ) | 
| 15 | 8, 14 | sylib 218 | . . . . . . . . . . 11
⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ) | 
| 16 | 15 | r19.21bi 3251 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ) | 
| 17 |  | nfcv 2905 | . . . . . . . . . . . 12
⊢
Ⅎ𝑦𝐵 | 
| 18 | 17, 10, 12 | cbvmpt 5253 | . . . . . . . . . . 11
⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) | 
| 19 | 18, 2 | eqeltrrid 2846 | . . . . . . . . . 10
⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) ∈
𝐿1) | 
| 20 | 4, 16, 19 | iblmulc2 25866 | . . . . . . . . 9
⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) ∈
𝐿1) | 
| 21 | 4 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (∗‘∫𝐴𝐵 d𝑥) ∈ ℂ) | 
| 22 | 21, 16 | mulcld 11281 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵) ∈ ℂ) | 
| 23 | 22 | iblcn 25834 | . . . . . . . . 9
⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦ ((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) ∈ 𝐿1 ↔
((𝑦 ∈ 𝐴 ↦
(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵))) ∈ 𝐿1 ∧
(𝑦 ∈ 𝐴 ↦
(ℑ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵))) ∈
𝐿1))) | 
| 24 | 20, 23 | mpbid 232 | . . . . . . . 8
⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦
(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵))) ∈ 𝐿1 ∧
(𝑦 ∈ 𝐴 ↦
(ℑ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵))) ∈
𝐿1)) | 
| 25 | 24 | simpld 494 | . . . . . . 7
⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦
(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵))) ∈
𝐿1) | 
| 26 |  | ovexd 7466 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵) ∈ V) | 
| 27 | 26, 20 | iblabs 25864 | . . . . . . 7
⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦
(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵))) ∈
𝐿1) | 
| 28 | 22 | recld 15233 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) →
(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) ∈ ℝ) | 
| 29 | 22 | abscld 15475 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) →
(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) ∈ ℝ) | 
| 30 | 22 | releabsd 15490 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) →
(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) ≤
(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵))) | 
| 31 | 25, 27, 28, 29, 30 | itgle 25845 | . . . . . 6
⊢ (𝜑 → ∫𝐴(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦 ≤ ∫𝐴(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦) | 
| 32 | 3 | abscld 15475 | . . . . . . . . 9
⊢ (𝜑 → (abs‘∫𝐴𝐵 d𝑥) ∈ ℝ) | 
| 33 | 32 | recnd 11289 | . . . . . . . 8
⊢ (𝜑 → (abs‘∫𝐴𝐵 d𝑥) ∈ ℂ) | 
| 34 | 33 | sqvald 14183 | . . . . . . 7
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥)↑2) = ((abs‘∫𝐴𝐵 d𝑥) · (abs‘∫𝐴𝐵 d𝑥))) | 
| 35 | 3 | absvalsqd 15481 | . . . . . . . . . 10
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥)↑2) = (∫𝐴𝐵 d𝑥 · (∗‘∫𝐴𝐵 d𝑥))) | 
| 36 | 3, 4 | mulcomd 11282 | . . . . . . . . . 10
⊢ (𝜑 → (∫𝐴𝐵 d𝑥 · (∗‘∫𝐴𝐵 d𝑥)) = ((∗‘∫𝐴𝐵 d𝑥) · ∫𝐴𝐵 d𝑥)) | 
| 37 | 12, 17, 10 | cbvitg 25811 | . . . . . . . . . . . 12
⊢
∫𝐴𝐵 d𝑥 = ∫𝐴⦋𝑦 / 𝑥⦌𝐵 d𝑦 | 
| 38 | 37 | oveq2i 7442 | . . . . . . . . . . 11
⊢
((∗‘∫𝐴𝐵 d𝑥) · ∫𝐴𝐵 d𝑥) = ((∗‘∫𝐴𝐵 d𝑥) · ∫𝐴⦋𝑦 / 𝑥⦌𝐵 d𝑦) | 
| 39 | 4, 16, 19 | itgmulc2 25869 | . . . . . . . . . . 11
⊢ (𝜑 →
((∗‘∫𝐴𝐵 d𝑥) · ∫𝐴⦋𝑦 / 𝑥⦌𝐵 d𝑦) = ∫𝐴((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵) d𝑦) | 
| 40 | 38, 39 | eqtrid 2789 | . . . . . . . . . 10
⊢ (𝜑 →
((∗‘∫𝐴𝐵 d𝑥) · ∫𝐴𝐵 d𝑥) = ∫𝐴((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵) d𝑦) | 
| 41 | 35, 36, 40 | 3eqtrd 2781 | . . . . . . . . 9
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥)↑2) = ∫𝐴((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵) d𝑦) | 
| 42 | 41 | fveq2d 6910 | . . . . . . . 8
⊢ (𝜑 →
(ℜ‘((abs‘∫𝐴𝐵 d𝑥)↑2)) = (ℜ‘∫𝐴((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵) d𝑦)) | 
| 43 | 32 | resqcld 14165 | . . . . . . . . 9
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥)↑2) ∈ ℝ) | 
| 44 | 43 | rered 15263 | . . . . . . . 8
⊢ (𝜑 →
(ℜ‘((abs‘∫𝐴𝐵 d𝑥)↑2)) = ((abs‘∫𝐴𝐵 d𝑥)↑2)) | 
| 45 | 26, 20 | itgre 25836 | . . . . . . . 8
⊢ (𝜑 → (ℜ‘∫𝐴((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵) d𝑦) = ∫𝐴(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦) | 
| 46 | 42, 44, 45 | 3eqtr3d 2785 | . . . . . . 7
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥)↑2) = ∫𝐴(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦) | 
| 47 | 34, 46 | eqtr3d 2779 | . . . . . 6
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥) · (abs‘∫𝐴𝐵 d𝑥)) = ∫𝐴(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦) | 
| 48 | 12 | fveq2d 6910 | . . . . . . . . 9
⊢ (𝑥 = 𝑦 → (abs‘𝐵) = (abs‘⦋𝑦 / 𝑥⦌𝐵)) | 
| 49 |  | nfcv 2905 | . . . . . . . . 9
⊢
Ⅎ𝑦(abs‘𝐵) | 
| 50 |  | nfcv 2905 | . . . . . . . . . 10
⊢
Ⅎ𝑥abs | 
| 51 | 50, 10 | nffv 6916 | . . . . . . . . 9
⊢
Ⅎ𝑥(abs‘⦋𝑦 / 𝑥⦌𝐵) | 
| 52 | 48, 49, 51 | cbvitg 25811 | . . . . . . . 8
⊢
∫𝐴(abs‘𝐵) d𝑥 = ∫𝐴(abs‘⦋𝑦 / 𝑥⦌𝐵) d𝑦 | 
| 53 | 52 | oveq2i 7442 | . . . . . . 7
⊢
((abs‘∫𝐴𝐵 d𝑥) · ∫𝐴(abs‘𝐵) d𝑥) = ((abs‘∫𝐴𝐵 d𝑥) · ∫𝐴(abs‘⦋𝑦 / 𝑥⦌𝐵) d𝑦) | 
| 54 | 16 | abscld 15475 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (abs‘⦋𝑦 / 𝑥⦌𝐵) ∈ ℝ) | 
| 55 | 16, 19 | iblabs 25864 | . . . . . . . . 9
⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ (abs‘⦋𝑦 / 𝑥⦌𝐵)) ∈
𝐿1) | 
| 56 | 33, 54, 55 | itgmulc2 25869 | . . . . . . . 8
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥) · ∫𝐴(abs‘⦋𝑦 / 𝑥⦌𝐵) d𝑦) = ∫𝐴((abs‘∫𝐴𝐵 d𝑥) · (abs‘⦋𝑦 / 𝑥⦌𝐵)) d𝑦) | 
| 57 | 21, 16 | absmuld 15493 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) →
(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) = ((abs‘(∗‘∫𝐴𝐵 d𝑥)) · (abs‘⦋𝑦 / 𝑥⦌𝐵))) | 
| 58 | 3 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∫𝐴𝐵 d𝑥 ∈ ℂ) | 
| 59 | 58 | abscjd 15489 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) →
(abs‘(∗‘∫𝐴𝐵 d𝑥)) = (abs‘∫𝐴𝐵 d𝑥)) | 
| 60 | 59 | oveq1d 7446 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) →
((abs‘(∗‘∫𝐴𝐵 d𝑥)) · (abs‘⦋𝑦 / 𝑥⦌𝐵)) = ((abs‘∫𝐴𝐵 d𝑥) · (abs‘⦋𝑦 / 𝑥⦌𝐵))) | 
| 61 | 57, 60 | eqtrd 2777 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) →
(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) = ((abs‘∫𝐴𝐵 d𝑥) · (abs‘⦋𝑦 / 𝑥⦌𝐵))) | 
| 62 | 61 | itgeq2dv 25817 | . . . . . . . 8
⊢ (𝜑 → ∫𝐴(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦 = ∫𝐴((abs‘∫𝐴𝐵 d𝑥) · (abs‘⦋𝑦 / 𝑥⦌𝐵)) d𝑦) | 
| 63 | 56, 62 | eqtr4d 2780 | . . . . . . 7
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥) · ∫𝐴(abs‘⦋𝑦 / 𝑥⦌𝐵) d𝑦) = ∫𝐴(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦) | 
| 64 | 53, 63 | eqtrid 2789 | . . . . . 6
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥) · ∫𝐴(abs‘𝐵) d𝑥) = ∫𝐴(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦) | 
| 65 | 31, 47, 64 | 3brtr4d 5175 | . . . . 5
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥) · (abs‘∫𝐴𝐵 d𝑥)) ≤ ((abs‘∫𝐴𝐵 d𝑥) · ∫𝐴(abs‘𝐵) d𝑥)) | 
| 66 | 65 | adantr 480 | . . . 4
⊢ ((𝜑 ∧ 0 <
(abs‘∫𝐴𝐵 d𝑥)) → ((abs‘∫𝐴𝐵 d𝑥) · (abs‘∫𝐴𝐵 d𝑥)) ≤ ((abs‘∫𝐴𝐵 d𝑥) · ∫𝐴(abs‘𝐵) d𝑥)) | 
| 67 | 32 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ 0 <
(abs‘∫𝐴𝐵 d𝑥)) → (abs‘∫𝐴𝐵 d𝑥) ∈ ℝ) | 
| 68 | 7 | abscld 15475 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℝ) | 
| 69 | 1, 2 | iblabs 25864 | . . . . . . 7
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈
𝐿1) | 
| 70 | 68, 69 | itgrecl 25833 | . . . . . 6
⊢ (𝜑 → ∫𝐴(abs‘𝐵) d𝑥 ∈ ℝ) | 
| 71 | 70 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ 0 <
(abs‘∫𝐴𝐵 d𝑥)) → ∫𝐴(abs‘𝐵) d𝑥 ∈ ℝ) | 
| 72 |  | simpr 484 | . . . . 5
⊢ ((𝜑 ∧ 0 <
(abs‘∫𝐴𝐵 d𝑥)) → 0 < (abs‘∫𝐴𝐵 d𝑥)) | 
| 73 |  | lemul2 12120 | . . . . 5
⊢
(((abs‘∫𝐴𝐵 d𝑥) ∈ ℝ ∧ ∫𝐴(abs‘𝐵) d𝑥 ∈ ℝ ∧ ((abs‘∫𝐴𝐵 d𝑥) ∈ ℝ ∧ 0 <
(abs‘∫𝐴𝐵 d𝑥))) → ((abs‘∫𝐴𝐵 d𝑥) ≤ ∫𝐴(abs‘𝐵) d𝑥 ↔ ((abs‘∫𝐴𝐵 d𝑥) · (abs‘∫𝐴𝐵 d𝑥)) ≤ ((abs‘∫𝐴𝐵 d𝑥) · ∫𝐴(abs‘𝐵) d𝑥))) | 
| 74 | 67, 71, 67, 72, 73 | syl112anc 1376 | . . . 4
⊢ ((𝜑 ∧ 0 <
(abs‘∫𝐴𝐵 d𝑥)) → ((abs‘∫𝐴𝐵 d𝑥) ≤ ∫𝐴(abs‘𝐵) d𝑥 ↔ ((abs‘∫𝐴𝐵 d𝑥) · (abs‘∫𝐴𝐵 d𝑥)) ≤ ((abs‘∫𝐴𝐵 d𝑥) · ∫𝐴(abs‘𝐵) d𝑥))) | 
| 75 | 66, 74 | mpbird 257 | . . 3
⊢ ((𝜑 ∧ 0 <
(abs‘∫𝐴𝐵 d𝑥)) → (abs‘∫𝐴𝐵 d𝑥) ≤ ∫𝐴(abs‘𝐵) d𝑥) | 
| 76 | 75 | ex 412 | . 2
⊢ (𝜑 → (0 <
(abs‘∫𝐴𝐵 d𝑥) → (abs‘∫𝐴𝐵 d𝑥) ≤ ∫𝐴(abs‘𝐵) d𝑥)) | 
| 77 | 7 | absge0d 15483 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘𝐵)) | 
| 78 | 69, 68, 77 | itgge0 25846 | . . 3
⊢ (𝜑 → 0 ≤ ∫𝐴(abs‘𝐵) d𝑥) | 
| 79 |  | breq1 5146 | . . 3
⊢ (0 =
(abs‘∫𝐴𝐵 d𝑥) → (0 ≤ ∫𝐴(abs‘𝐵) d𝑥 ↔ (abs‘∫𝐴𝐵 d𝑥) ≤ ∫𝐴(abs‘𝐵) d𝑥)) | 
| 80 | 78, 79 | syl5ibcom 245 | . 2
⊢ (𝜑 → (0 = (abs‘∫𝐴𝐵 d𝑥) → (abs‘∫𝐴𝐵 d𝑥) ≤ ∫𝐴(abs‘𝐵) d𝑥)) | 
| 81 | 3 | absge0d 15483 | . . 3
⊢ (𝜑 → 0 ≤
(abs‘∫𝐴𝐵 d𝑥)) | 
| 82 |  | 0re 11263 | . . . 4
⊢ 0 ∈
ℝ | 
| 83 |  | leloe 11347 | . . . 4
⊢ ((0
∈ ℝ ∧ (abs‘∫𝐴𝐵 d𝑥) ∈ ℝ) → (0 ≤
(abs‘∫𝐴𝐵 d𝑥) ↔ (0 < (abs‘∫𝐴𝐵 d𝑥) ∨ 0 = (abs‘∫𝐴𝐵 d𝑥)))) | 
| 84 | 82, 32, 83 | sylancr 587 | . . 3
⊢ (𝜑 → (0 ≤
(abs‘∫𝐴𝐵 d𝑥) ↔ (0 < (abs‘∫𝐴𝐵 d𝑥) ∨ 0 = (abs‘∫𝐴𝐵 d𝑥)))) | 
| 85 | 81, 84 | mpbid 232 | . 2
⊢ (𝜑 → (0 <
(abs‘∫𝐴𝐵 d𝑥) ∨ 0 = (abs‘∫𝐴𝐵 d𝑥))) | 
| 86 | 76, 80, 85 | mpjaod 861 | 1
⊢ (𝜑 → (abs‘∫𝐴𝐵 d𝑥) ≤ ∫𝐴(abs‘𝐵) d𝑥) |