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 23550 |
. . . . . . . . . . 11
⊢ (𝜑 → ∫𝐴𝐵 d𝑥 ∈ ℂ) |
4 | 3 | cjcld 13936 |
. . . . . . . . . 10
⊢ (𝜑 → (∗‘∫𝐴𝐵 d𝑥) ∈ ℂ) |
5 | | iblmbf 23534 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) |
6 | 2, 5 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) |
7 | 6, 1 | mbfmptcl 23404 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
8 | 7 | ralrimiva 2966 |
. . . . . . . . . . . 12
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ ℂ) |
9 | | nfv 1843 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑦 𝐵 ∈ ℂ |
10 | | nfcsb1v 3549 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 |
11 | 10 | nfel1 2779 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ |
12 | | csbeq1a 3542 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) |
13 | 12 | eleq1d 2686 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑦 → (𝐵 ∈ ℂ ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ)) |
14 | 9, 11, 13 | cbvral 3167 |
. . . . . . . . . . . 12
⊢
(∀𝑥 ∈
𝐴 𝐵 ∈ ℂ ↔ ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ) |
15 | 8, 14 | sylib 208 |
. . . . . . . . . . 11
⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ) |
16 | 15 | r19.21bi 2932 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ) |
17 | | nfcv 2764 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑦𝐵 |
18 | 17, 10, 12 | cbvmpt 4749 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
19 | 18, 2 | syl5eqelr 2706 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) ∈
𝐿1) |
20 | 4, 16, 19 | iblmulc2 23597 |
. . . . . . . . 9
⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) ∈
𝐿1) |
21 | 4 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (∗‘∫𝐴𝐵 d𝑥) ∈ ℂ) |
22 | 21, 16 | mulcld 10060 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵) ∈ ℂ) |
23 | 22 | iblcn 23565 |
. . . . . . . . 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 222 |
. . . . . . . 8
⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦
(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵))) ∈ 𝐿1 ∧
(𝑦 ∈ 𝐴 ↦
(ℑ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵))) ∈
𝐿1)) |
25 | 24 | simpld 475 |
. . . . . . 7
⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦
(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵))) ∈
𝐿1) |
26 | | ovexd 6680 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵) ∈ V) |
27 | 26, 20 | iblabs 23595 |
. . . . . . 7
⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦
(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵))) ∈
𝐿1) |
28 | 22 | recld 13934 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) →
(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) ∈ ℝ) |
29 | 22 | abscld 14175 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) →
(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) ∈ ℝ) |
30 | 22 | releabsd 14190 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) →
(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) ≤
(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵))) |
31 | 25, 27, 28, 29, 30 | itgle 23576 |
. . . . . 6
⊢ (𝜑 → ∫𝐴(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦 ≤ ∫𝐴(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦) |
32 | 3 | abscld 14175 |
. . . . . . . . 9
⊢ (𝜑 → (abs‘∫𝐴𝐵 d𝑥) ∈ ℝ) |
33 | 32 | recnd 10068 |
. . . . . . . 8
⊢ (𝜑 → (abs‘∫𝐴𝐵 d𝑥) ∈ ℂ) |
34 | 33 | sqvald 13005 |
. . . . . . 7
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥)↑2) = ((abs‘∫𝐴𝐵 d𝑥) · (abs‘∫𝐴𝐵 d𝑥))) |
35 | 3 | absvalsqd 14181 |
. . . . . . . . . 10
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥)↑2) = (∫𝐴𝐵 d𝑥 · (∗‘∫𝐴𝐵 d𝑥))) |
36 | 3, 4 | mulcomd 10061 |
. . . . . . . . . 10
⊢ (𝜑 → (∫𝐴𝐵 d𝑥 · (∗‘∫𝐴𝐵 d𝑥)) = ((∗‘∫𝐴𝐵 d𝑥) · ∫𝐴𝐵 d𝑥)) |
37 | 12, 17, 10 | cbvitg 23542 |
. . . . . . . . . . . 12
⊢
∫𝐴𝐵 d𝑥 = ∫𝐴⦋𝑦 / 𝑥⦌𝐵 d𝑦 |
38 | 37 | oveq2i 6661 |
. . . . . . . . . . 11
⊢
((∗‘∫𝐴𝐵 d𝑥) · ∫𝐴𝐵 d𝑥) = ((∗‘∫𝐴𝐵 d𝑥) · ∫𝐴⦋𝑦 / 𝑥⦌𝐵 d𝑦) |
39 | 4, 16, 19 | itgmulc2 23600 |
. . . . . . . . . . 11
⊢ (𝜑 →
((∗‘∫𝐴𝐵 d𝑥) · ∫𝐴⦋𝑦 / 𝑥⦌𝐵 d𝑦) = ∫𝐴((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵) d𝑦) |
40 | 38, 39 | syl5eq 2668 |
. . . . . . . . . 10
⊢ (𝜑 →
((∗‘∫𝐴𝐵 d𝑥) · ∫𝐴𝐵 d𝑥) = ∫𝐴((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵) d𝑦) |
41 | 35, 36, 40 | 3eqtrd 2660 |
. . . . . . . . 9
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥)↑2) = ∫𝐴((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵) d𝑦) |
42 | 41 | fveq2d 6195 |
. . . . . . . 8
⊢ (𝜑 →
(ℜ‘((abs‘∫𝐴𝐵 d𝑥)↑2)) = (ℜ‘∫𝐴((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵) d𝑦)) |
43 | 32 | resqcld 13035 |
. . . . . . . . 9
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥)↑2) ∈ ℝ) |
44 | 43 | rered 13964 |
. . . . . . . 8
⊢ (𝜑 →
(ℜ‘((abs‘∫𝐴𝐵 d𝑥)↑2)) = ((abs‘∫𝐴𝐵 d𝑥)↑2)) |
45 | 26, 20 | itgre 23567 |
. . . . . . . 8
⊢ (𝜑 → (ℜ‘∫𝐴((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵) d𝑦) = ∫𝐴(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦) |
46 | 42, 44, 45 | 3eqtr3d 2664 |
. . . . . . 7
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥)↑2) = ∫𝐴(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦) |
47 | 34, 46 | eqtr3d 2658 |
. . . . . 6
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥) · (abs‘∫𝐴𝐵 d𝑥)) = ∫𝐴(ℜ‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦) |
48 | 12 | fveq2d 6195 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (abs‘𝐵) = (abs‘⦋𝑦 / 𝑥⦌𝐵)) |
49 | | nfcv 2764 |
. . . . . . . . 9
⊢
Ⅎ𝑦(abs‘𝐵) |
50 | | nfcv 2764 |
. . . . . . . . . 10
⊢
Ⅎ𝑥abs |
51 | 50, 10 | nffv 6198 |
. . . . . . . . 9
⊢
Ⅎ𝑥(abs‘⦋𝑦 / 𝑥⦌𝐵) |
52 | 48, 49, 51 | cbvitg 23542 |
. . . . . . . 8
⊢
∫𝐴(abs‘𝐵) d𝑥 = ∫𝐴(abs‘⦋𝑦 / 𝑥⦌𝐵) d𝑦 |
53 | 52 | oveq2i 6661 |
. . . . . . 7
⊢
((abs‘∫𝐴𝐵 d𝑥) · ∫𝐴(abs‘𝐵) d𝑥) = ((abs‘∫𝐴𝐵 d𝑥) · ∫𝐴(abs‘⦋𝑦 / 𝑥⦌𝐵) d𝑦) |
54 | 16 | abscld 14175 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (abs‘⦋𝑦 / 𝑥⦌𝐵) ∈ ℝ) |
55 | 16, 19 | iblabs 23595 |
. . . . . . . . 9
⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ (abs‘⦋𝑦 / 𝑥⦌𝐵)) ∈
𝐿1) |
56 | 33, 54, 55 | itgmulc2 23600 |
. . . . . . . 8
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥) · ∫𝐴(abs‘⦋𝑦 / 𝑥⦌𝐵) d𝑦) = ∫𝐴((abs‘∫𝐴𝐵 d𝑥) · (abs‘⦋𝑦 / 𝑥⦌𝐵)) d𝑦) |
57 | 21, 16 | absmuld 14193 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) →
(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) = ((abs‘(∗‘∫𝐴𝐵 d𝑥)) · (abs‘⦋𝑦 / 𝑥⦌𝐵))) |
58 | 3 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∫𝐴𝐵 d𝑥 ∈ ℂ) |
59 | 58 | abscjd 14189 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) →
(abs‘(∗‘∫𝐴𝐵 d𝑥)) = (abs‘∫𝐴𝐵 d𝑥)) |
60 | 59 | oveq1d 6665 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) →
((abs‘(∗‘∫𝐴𝐵 d𝑥)) · (abs‘⦋𝑦 / 𝑥⦌𝐵)) = ((abs‘∫𝐴𝐵 d𝑥) · (abs‘⦋𝑦 / 𝑥⦌𝐵))) |
61 | 57, 60 | eqtrd 2656 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) →
(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) = ((abs‘∫𝐴𝐵 d𝑥) · (abs‘⦋𝑦 / 𝑥⦌𝐵))) |
62 | 61 | itgeq2dv 23548 |
. . . . . . . 8
⊢ (𝜑 → ∫𝐴(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦 = ∫𝐴((abs‘∫𝐴𝐵 d𝑥) · (abs‘⦋𝑦 / 𝑥⦌𝐵)) d𝑦) |
63 | 56, 62 | eqtr4d 2659 |
. . . . . . 7
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥) · ∫𝐴(abs‘⦋𝑦 / 𝑥⦌𝐵) d𝑦) = ∫𝐴(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦) |
64 | 53, 63 | syl5eq 2668 |
. . . . . 6
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥) · ∫𝐴(abs‘𝐵) d𝑥) = ∫𝐴(abs‘((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) d𝑦) |
65 | 31, 47, 64 | 3brtr4d 4685 |
. . . . 5
⊢ (𝜑 → ((abs‘∫𝐴𝐵 d𝑥) · (abs‘∫𝐴𝐵 d𝑥)) ≤ ((abs‘∫𝐴𝐵 d𝑥) · ∫𝐴(abs‘𝐵) d𝑥)) |
66 | 65 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ 0 <
(abs‘∫𝐴𝐵 d𝑥)) → ((abs‘∫𝐴𝐵 d𝑥) · (abs‘∫𝐴𝐵 d𝑥)) ≤ ((abs‘∫𝐴𝐵 d𝑥) · ∫𝐴(abs‘𝐵) d𝑥)) |
67 | 32 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 0 <
(abs‘∫𝐴𝐵 d𝑥)) → (abs‘∫𝐴𝐵 d𝑥) ∈ ℝ) |
68 | 7 | abscld 14175 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℝ) |
69 | 1, 2 | iblabs 23595 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈
𝐿1) |
70 | 68, 69 | itgrecl 23564 |
. . . . . 6
⊢ (𝜑 → ∫𝐴(abs‘𝐵) d𝑥 ∈ ℝ) |
71 | 70 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 0 <
(abs‘∫𝐴𝐵 d𝑥)) → ∫𝐴(abs‘𝐵) d𝑥 ∈ ℝ) |
72 | | simpr 477 |
. . . . 5
⊢ ((𝜑 ∧ 0 <
(abs‘∫𝐴𝐵 d𝑥)) → 0 < (abs‘∫𝐴𝐵 d𝑥)) |
73 | | lemul2 10876 |
. . . . 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 1330 |
. . . 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 247 |
. . 3
⊢ ((𝜑 ∧ 0 <
(abs‘∫𝐴𝐵 d𝑥)) → (abs‘∫𝐴𝐵 d𝑥) ≤ ∫𝐴(abs‘𝐵) d𝑥) |
76 | 75 | ex 450 |
. 2
⊢ (𝜑 → (0 <
(abs‘∫𝐴𝐵 d𝑥) → (abs‘∫𝐴𝐵 d𝑥) ≤ ∫𝐴(abs‘𝐵) d𝑥)) |
77 | 7 | absge0d 14183 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘𝐵)) |
78 | 69, 68, 77 | itgge0 23577 |
. . 3
⊢ (𝜑 → 0 ≤ ∫𝐴(abs‘𝐵) d𝑥) |
79 | | breq1 4656 |
. . 3
⊢ (0 =
(abs‘∫𝐴𝐵 d𝑥) → (0 ≤ ∫𝐴(abs‘𝐵) d𝑥 ↔ (abs‘∫𝐴𝐵 d𝑥) ≤ ∫𝐴(abs‘𝐵) d𝑥)) |
80 | 78, 79 | syl5ibcom 235 |
. 2
⊢ (𝜑 → (0 = (abs‘∫𝐴𝐵 d𝑥) → (abs‘∫𝐴𝐵 d𝑥) ≤ ∫𝐴(abs‘𝐵) d𝑥)) |
81 | 3 | absge0d 14183 |
. . 3
⊢ (𝜑 → 0 ≤
(abs‘∫𝐴𝐵 d𝑥)) |
82 | | 0re 10040 |
. . . 4
⊢ 0 ∈
ℝ |
83 | | leloe 10124 |
. . . 4
⊢ ((0
∈ ℝ ∧ (abs‘∫𝐴𝐵 d𝑥) ∈ ℝ) → (0 ≤
(abs‘∫𝐴𝐵 d𝑥) ↔ (0 < (abs‘∫𝐴𝐵 d𝑥) ∨ 0 = (abs‘∫𝐴𝐵 d𝑥)))) |
84 | 82, 32, 83 | sylancr 695 |
. . 3
⊢ (𝜑 → (0 ≤
(abs‘∫𝐴𝐵 d𝑥) ↔ (0 < (abs‘∫𝐴𝐵 d𝑥) ∨ 0 = (abs‘∫𝐴𝐵 d𝑥)))) |
85 | 81, 84 | mpbid 222 |
. 2
⊢ (𝜑 → (0 <
(abs‘∫𝐴𝐵 d𝑥) ∨ 0 = (abs‘∫𝐴𝐵 d𝑥))) |
86 | 76, 80, 85 | mpjaod 396 |
1
⊢ (𝜑 → (abs‘∫𝐴𝐵 d𝑥) ≤ ∫𝐴(abs‘𝐵) d𝑥) |