Proof of Theorem itgabsnc
| Step | Hyp | Ref
| Expression |
| 1 | | itgabsnc.1 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
| 2 | | itgabsnc.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 | | itgabsnc.m2 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) ∈ MblFn) |
| 21 | 4, 16, 19, 20 | iblmulc2nc 37692 |
. . . . . . . . 9
⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) ∈
𝐿1) |
| 22 | 4 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (∗‘∫𝐴𝐵 d𝑥) ∈ ℂ) |
| 23 | 22, 16 | mulcld 11281 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵) ∈ ℂ) |
| 24 | 23 | 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))) |
| 25 | 21, 24 | mpbid 232 |
. . . . . . . 8
⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦
(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵))) ∈ 𝐿1 ∧
(𝑦 ∈ 𝐴 ↦
(ℑ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵))) ∈
𝐿1)) |
| 26 | 25 | simpld 494 |
. . . . . . 7
⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦
(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵))) ∈
𝐿1) |
| 27 | 22, 16 | absmuld 15493 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) →
(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) = ((abs‘(∗‘∫𝐴𝐵 d𝑥)) · (abs‘⦋𝑦 / 𝑥⦌𝐵))) |
| 28 | 27 | mpteq2dva 5242 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦
(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵))) = (𝑦 ∈ 𝐴 ↦
((abs‘(∗‘∫𝐴𝐵 d𝑥)) · (abs‘⦋𝑦 / 𝑥⦌𝐵)))) |
| 29 | 6, 1 | mbfdm2 25672 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ∈ dom vol) |
| 30 | 22 | abscld 15475 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) →
(abs‘(∗‘∫𝐴𝐵 d𝑥)) ∈ ℝ) |
| 31 | 16 | abscld 15475 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (abs‘⦋𝑦 / 𝑥⦌𝐵) ∈ ℝ) |
| 32 | | fconstmpt 5747 |
. . . . . . . . . . . 12
⊢ (𝐴 ×
{(abs‘(∗‘∫𝐴𝐵 d𝑥))}) = (𝑦 ∈ 𝐴 ↦
(abs‘(∗‘∫𝐴𝐵 d𝑥))) |
| 33 | 32 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐴 ×
{(abs‘(∗‘∫𝐴𝐵 d𝑥))}) = (𝑦 ∈ 𝐴 ↦
(abs‘(∗‘∫𝐴𝐵 d𝑥)))) |
| 34 | | nfcv 2905 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑦(abs‘𝐵) |
| 35 | | nfcv 2905 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑥abs |
| 36 | 35, 10 | nffv 6916 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥(abs‘⦋𝑦 / 𝑥⦌𝐵) |
| 37 | 12 | fveq2d 6910 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑦 → (abs‘𝐵) = (abs‘⦋𝑦 / 𝑥⦌𝐵)) |
| 38 | 34, 36, 37 | cbvmpt 5253 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) = (𝑦 ∈ 𝐴 ↦ (abs‘⦋𝑦 / 𝑥⦌𝐵)) |
| 39 | 38 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) = (𝑦 ∈ 𝐴 ↦ (abs‘⦋𝑦 / 𝑥⦌𝐵))) |
| 40 | 29, 30, 31, 33, 39 | offval2 7717 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐴 ×
{(abs‘(∗‘∫𝐴𝐵 d𝑥))}) ∘f · (𝑥 ∈ 𝐴 ↦ (abs‘𝐵))) = (𝑦 ∈ 𝐴 ↦
((abs‘(∗‘∫𝐴𝐵 d𝑥)) · (abs‘⦋𝑦 / 𝑥⦌𝐵)))) |
| 41 | 28, 40 | eqtr4d 2780 |
. . . . . . . . 9
⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦
(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵))) = ((𝐴 ×
{(abs‘(∗‘∫𝐴𝐵 d𝑥))}) ∘f · (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)))) |
| 42 | | itgabsnc.m1 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ MblFn) |
| 43 | 4 | abscld 15475 |
. . . . . . . . . 10
⊢ (𝜑 →
(abs‘(∗‘∫𝐴𝐵 d𝑥)) ∈ ℝ) |
| 44 | 7 | abscld 15475 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℝ) |
| 45 | 44 | recnd 11289 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℂ) |
| 46 | 45 | fmpttd 7135 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)):𝐴⟶ℂ) |
| 47 | 42, 43, 46 | mbfmulc2re 25683 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐴 ×
{(abs‘(∗‘∫𝐴𝐵 d𝑥))}) ∘f · (𝑥 ∈ 𝐴 ↦ (abs‘𝐵))) ∈ MblFn) |
| 48 | 41, 47 | eqeltrd 2841 |
. . . . . . . 8
⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦
(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵))) ∈ MblFn) |
| 49 | 23, 21, 48 | iblabsnc 37691 |
. . . . . . 7
⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦
(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵))) ∈
𝐿1) |
| 50 | 23 | recld 15233 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) →
(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) ∈ ℝ) |
| 51 | 23 | abscld 15475 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) →
(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) ∈ ℝ) |
| 52 | 23 | releabsd 15490 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) →
(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) ≤
(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵))) |
| 53 | 26, 49, 50, 51, 52 | itgle 25845 |
. . . . . 6
⊢ (𝜑 → ∫𝐴(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦 ≤ ∫𝐴(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦) |
| 54 | 3 | abscld 15475 |
. . . . . . . . 9
⊢ (𝜑 → (abs‘∫𝐴𝐵 d𝑥) ∈ ℝ) |
| 55 | 54 | recnd 11289 |
. . . . . . . 8
⊢ (𝜑 → (abs‘∫𝐴𝐵 d𝑥) ∈ ℂ) |
| 56 | 55 | sqvald 14183 |
. . . . . . 7
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥)↑2) = ((abs‘∫𝐴𝐵 d𝑥) · (abs‘∫𝐴𝐵 d𝑥))) |
| 57 | 3 | absvalsqd 15481 |
. . . . . . . . . 10
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥)↑2) = (∫𝐴𝐵 d𝑥 · (∗‘∫𝐴𝐵 d𝑥))) |
| 58 | 3, 4 | mulcomd 11282 |
. . . . . . . . . 10
⊢ (𝜑 → (∫𝐴𝐵 d𝑥 · (∗‘∫𝐴𝐵 d𝑥)) = ((∗‘∫𝐴𝐵 d𝑥) · ∫𝐴𝐵 d𝑥)) |
| 59 | 12, 17, 10 | cbvitg 25811 |
. . . . . . . . . . . 12
⊢
∫𝐴𝐵 d𝑥 = ∫𝐴⦋𝑦 / 𝑥⦌𝐵 d𝑦 |
| 60 | 59 | oveq2i 7442 |
. . . . . . . . . . 11
⊢
((∗‘∫𝐴𝐵 d𝑥) · ∫𝐴𝐵 d𝑥) = ((∗‘∫𝐴𝐵 d𝑥) · ∫𝐴⦋𝑦 / 𝑥⦌𝐵 d𝑦) |
| 61 | 4, 16, 19, 20 | itgmulc2nc 37695 |
. . . . . . . . . . 11
⊢ (𝜑 →
((∗‘∫𝐴𝐵 d𝑥) · ∫𝐴⦋𝑦 / 𝑥⦌𝐵 d𝑦) = ∫𝐴((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵) d𝑦) |
| 62 | 60, 61 | eqtrid 2789 |
. . . . . . . . . 10
⊢ (𝜑 →
((∗‘∫𝐴𝐵 d𝑥) · ∫𝐴𝐵 d𝑥) = ∫𝐴((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵) d𝑦) |
| 63 | 57, 58, 62 | 3eqtrd 2781 |
. . . . . . . . 9
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥)↑2) = ∫𝐴((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵) d𝑦) |
| 64 | 63 | fveq2d 6910 |
. . . . . . . 8
⊢ (𝜑 →
(ℜ‘((abs‘∫𝐴𝐵 d𝑥)↑2)) = (ℜ‘∫𝐴((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵) d𝑦)) |
| 65 | 54 | resqcld 14165 |
. . . . . . . . 9
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥)↑2) ∈ ℝ) |
| 66 | 65 | rered 15263 |
. . . . . . . 8
⊢ (𝜑 →
(ℜ‘((abs‘∫𝐴𝐵 d𝑥)↑2)) = ((abs‘∫𝐴𝐵 d𝑥)↑2)) |
| 67 | | ovexd 7466 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵) ∈ V) |
| 68 | 67, 21 | itgre 25836 |
. . . . . . . 8
⊢ (𝜑 → (ℜ‘∫𝐴((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵) d𝑦) = ∫𝐴(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦) |
| 69 | 64, 66, 68 | 3eqtr3d 2785 |
. . . . . . 7
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥)↑2) = ∫𝐴(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦) |
| 70 | 56, 69 | eqtr3d 2779 |
. . . . . 6
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥) · (abs‘∫𝐴𝐵 d𝑥)) = ∫𝐴(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦) |
| 71 | 37, 34, 36 | cbvitg 25811 |
. . . . . . . 8
⊢
∫𝐴(abs‘𝐵) d𝑥 = ∫𝐴(abs‘⦋𝑦 / 𝑥⦌𝐵) d𝑦 |
| 72 | 71 | oveq2i 7442 |
. . . . . . 7
⊢
((abs‘∫𝐴𝐵 d𝑥) · ∫𝐴(abs‘𝐵) d𝑥) = ((abs‘∫𝐴𝐵 d𝑥) · ∫𝐴(abs‘⦋𝑦 / 𝑥⦌𝐵) d𝑦) |
| 73 | 1, 2, 42 | iblabsnc 37691 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈
𝐿1) |
| 74 | 38, 73 | eqeltrrid 2846 |
. . . . . . . . 9
⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ (abs‘⦋𝑦 / 𝑥⦌𝐵)) ∈
𝐿1) |
| 75 | 54 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (abs‘∫𝐴𝐵 d𝑥) ∈ ℝ) |
| 76 | | fconstmpt 5747 |
. . . . . . . . . . . 12
⊢ (𝐴 × {(abs‘∫𝐴𝐵 d𝑥)}) = (𝑦 ∈ 𝐴 ↦ (abs‘∫𝐴𝐵 d𝑥)) |
| 77 | 76 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐴 × {(abs‘∫𝐴𝐵 d𝑥)}) = (𝑦 ∈ 𝐴 ↦ (abs‘∫𝐴𝐵 d𝑥))) |
| 78 | 29, 75, 31, 77, 39 | offval2 7717 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐴 × {(abs‘∫𝐴𝐵 d𝑥)}) ∘f · (𝑥 ∈ 𝐴 ↦ (abs‘𝐵))) = (𝑦 ∈ 𝐴 ↦ ((abs‘∫𝐴𝐵 d𝑥) · (abs‘⦋𝑦 / 𝑥⦌𝐵)))) |
| 79 | 42, 54, 46 | mbfmulc2re 25683 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐴 × {(abs‘∫𝐴𝐵 d𝑥)}) ∘f · (𝑥 ∈ 𝐴 ↦ (abs‘𝐵))) ∈ MblFn) |
| 80 | 78, 79 | eqeltrrd 2842 |
. . . . . . . . 9
⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((abs‘∫𝐴𝐵 d𝑥) · (abs‘⦋𝑦 / 𝑥⦌𝐵))) ∈ MblFn) |
| 81 | 55, 31, 74, 80 | itgmulc2nc 37695 |
. . . . . . . 8
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥) · ∫𝐴(abs‘⦋𝑦 / 𝑥⦌𝐵) d𝑦) = ∫𝐴((abs‘∫𝐴𝐵 d𝑥) · (abs‘⦋𝑦 / 𝑥⦌𝐵)) d𝑦) |
| 82 | 3 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∫𝐴𝐵 d𝑥 ∈ ℂ) |
| 83 | 82 | abscjd 15489 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) →
(abs‘(∗‘∫𝐴𝐵 d𝑥)) = (abs‘∫𝐴𝐵 d𝑥)) |
| 84 | 83 | oveq1d 7446 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) →
((abs‘(∗‘∫𝐴𝐵 d𝑥)) · (abs‘⦋𝑦 / 𝑥⦌𝐵)) = ((abs‘∫𝐴𝐵 d𝑥) · (abs‘⦋𝑦 / 𝑥⦌𝐵))) |
| 85 | 27, 84 | eqtrd 2777 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) →
(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) = ((abs‘∫𝐴𝐵 d𝑥) · (abs‘⦋𝑦 / 𝑥⦌𝐵))) |
| 86 | 85 | itgeq2dv 25817 |
. . . . . . . 8
⊢ (𝜑 → ∫𝐴(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦 = ∫𝐴((abs‘∫𝐴𝐵 d𝑥) · (abs‘⦋𝑦 / 𝑥⦌𝐵)) d𝑦) |
| 87 | 81, 86 | eqtr4d 2780 |
. . . . . . 7
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥) · ∫𝐴(abs‘⦋𝑦 / 𝑥⦌𝐵) d𝑦) = ∫𝐴(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦) |
| 88 | 72, 87 | eqtrid 2789 |
. . . . . 6
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥) · ∫𝐴(abs‘𝐵) d𝑥) = ∫𝐴(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦) |
| 89 | 53, 70, 88 | 3brtr4d 5175 |
. . . . 5
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥) · (abs‘∫𝐴𝐵 d𝑥)) ≤ ((abs‘∫𝐴𝐵 d𝑥) · ∫𝐴(abs‘𝐵) d𝑥)) |
| 90 | 89 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 0 <
(abs‘∫𝐴𝐵 d𝑥)) → ((abs‘∫𝐴𝐵 d𝑥) · (abs‘∫𝐴𝐵 d𝑥)) ≤ ((abs‘∫𝐴𝐵 d𝑥) · ∫𝐴(abs‘𝐵) d𝑥)) |
| 91 | 54 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 0 <
(abs‘∫𝐴𝐵 d𝑥)) → (abs‘∫𝐴𝐵 d𝑥) ∈ ℝ) |
| 92 | 44, 73 | itgrecl 25833 |
. . . . . 6
⊢ (𝜑 → ∫𝐴(abs‘𝐵) d𝑥 ∈ ℝ) |
| 93 | 92 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 0 <
(abs‘∫𝐴𝐵 d𝑥)) → ∫𝐴(abs‘𝐵) d𝑥 ∈ ℝ) |
| 94 | | simpr 484 |
. . . . 5
⊢ ((𝜑 ∧ 0 <
(abs‘∫𝐴𝐵 d𝑥)) → 0 < (abs‘∫𝐴𝐵 d𝑥)) |
| 95 | | 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))) |
| 96 | 91, 93, 91, 94, 95 | 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))) |
| 97 | 90, 96 | mpbird 257 |
. . 3
⊢ ((𝜑 ∧ 0 <
(abs‘∫𝐴𝐵 d𝑥)) → (abs‘∫𝐴𝐵 d𝑥) ≤ ∫𝐴(abs‘𝐵) d𝑥) |
| 98 | 97 | ex 412 |
. 2
⊢ (𝜑 → (0 <
(abs‘∫𝐴𝐵 d𝑥) → (abs‘∫𝐴𝐵 d𝑥) ≤ ∫𝐴(abs‘𝐵) d𝑥)) |
| 99 | 7 | absge0d 15483 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘𝐵)) |
| 100 | 73, 44, 99 | itgge0 25846 |
. . 3
⊢ (𝜑 → 0 ≤ ∫𝐴(abs‘𝐵) d𝑥) |
| 101 | | breq1 5146 |
. . 3
⊢ (0 =
(abs‘∫𝐴𝐵 d𝑥) → (0 ≤ ∫𝐴(abs‘𝐵) d𝑥 ↔ (abs‘∫𝐴𝐵 d𝑥) ≤ ∫𝐴(abs‘𝐵) d𝑥)) |
| 102 | 100, 101 | syl5ibcom 245 |
. 2
⊢ (𝜑 → (0 = (abs‘∫𝐴𝐵 d𝑥) → (abs‘∫𝐴𝐵 d𝑥) ≤ ∫𝐴(abs‘𝐵) d𝑥)) |
| 103 | 3 | absge0d 15483 |
. . 3
⊢ (𝜑 → 0 ≤
(abs‘∫𝐴𝐵 d𝑥)) |
| 104 | | 0re 11263 |
. . . 4
⊢ 0 ∈
ℝ |
| 105 | | leloe 11347 |
. . . 4
⊢ ((0
∈ ℝ ∧ (abs‘∫𝐴𝐵 d𝑥) ∈ ℝ) → (0 ≤
(abs‘∫𝐴𝐵 d𝑥) ↔ (0 < (abs‘∫𝐴𝐵 d𝑥) ∨ 0 = (abs‘∫𝐴𝐵 d𝑥)))) |
| 106 | 104, 54, 105 | sylancr 587 |
. . 3
⊢ (𝜑 → (0 ≤
(abs‘∫𝐴𝐵 d𝑥) ↔ (0 < (abs‘∫𝐴𝐵 d𝑥) ∨ 0 = (abs‘∫𝐴𝐵 d𝑥)))) |
| 107 | 103, 106 | mpbid 232 |
. 2
⊢ (𝜑 → (0 <
(abs‘∫𝐴𝐵 d𝑥) ∨ 0 = (abs‘∫𝐴𝐵 d𝑥))) |
| 108 | 98, 102, 107 | mpjaod 861 |
1
⊢ (𝜑 → (abs‘∫𝐴𝐵 d𝑥) ≤ ∫𝐴(abs‘𝐵) d𝑥) |