Step | Hyp | Ref
| Expression |
1 | | iblabsr.2 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) |
2 | | ifan 4084 |
. . . . . . 7
⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) = if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0), 0) |
3 | | iblabsr.1 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
4 | 1, 3 | mbfmptcl 23210 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
5 | 4 | adantlr 747 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
6 | | elfzelz 12213 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (0...3) → 𝑘 ∈
ℤ) |
7 | 6 | ad2antlr 759 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 𝑘 ∈ ℤ) |
8 | | ax-icn 9874 |
. . . . . . . . . . . . . . 15
⊢ i ∈
ℂ |
9 | | ine0 10344 |
. . . . . . . . . . . . . . 15
⊢ i ≠
0 |
10 | | expclz 12747 |
. . . . . . . . . . . . . . 15
⊢ ((i
∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈
ℂ) |
11 | 8, 9, 10 | mp3an12 1406 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℤ →
(i↑𝑘) ∈
ℂ) |
12 | 7, 11 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (i↑𝑘) ∈ ℂ) |
13 | | expne0i 12754 |
. . . . . . . . . . . . . . 15
⊢ ((i
∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0) |
14 | 8, 9, 13 | mp3an12 1406 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℤ →
(i↑𝑘) ≠
0) |
15 | 7, 14 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (i↑𝑘) ≠ 0) |
16 | 5, 12, 15 | divcld 10680 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (𝐵 / (i↑𝑘)) ∈ ℂ) |
17 | 16 | recld 13782 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 / (i↑𝑘))) ∈ ℝ) |
18 | | 0re 9919 |
. . . . . . . . . . 11
⊢ 0 ∈
ℝ |
19 | | ifcl 4080 |
. . . . . . . . . . 11
⊢
(((ℜ‘(𝐵 /
(i↑𝑘))) ∈ ℝ
∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ∈ ℝ) |
20 | 17, 18, 19 | sylancl 693 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ∈ ℝ) |
21 | 20 | rexrd 9968 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ∈
ℝ*) |
22 | | max1 11890 |
. . . . . . . . . 10
⊢ ((0
∈ ℝ ∧ (ℜ‘(𝐵 / (i↑𝑘))) ∈ ℝ) → 0 ≤ if(0 ≤
(ℜ‘(𝐵 /
(i↑𝑘))),
(ℜ‘(𝐵 /
(i↑𝑘))),
0)) |
23 | 18, 17, 22 | sylancr 694 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤
(ℜ‘(𝐵 /
(i↑𝑘))),
(ℜ‘(𝐵 /
(i↑𝑘))),
0)) |
24 | | elxrge0 12152 |
. . . . . . . . 9
⊢ (if(0
≤ (ℜ‘(𝐵 /
(i↑𝑘))),
(ℜ‘(𝐵 /
(i↑𝑘))), 0) ∈
(0[,]+∞) ↔ (if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ∈ ℝ* ∧ 0
≤ if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) |
25 | 21, 23, 24 | sylanbrc 695 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ∈
(0[,]+∞)) |
26 | | 0e0iccpnf 12154 |
. . . . . . . . 9
⊢ 0 ∈
(0[,]+∞) |
27 | 26 | a1i 11 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈
(0[,]+∞)) |
28 | 25, 27 | ifclda 4070 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0), 0) ∈
(0[,]+∞)) |
29 | 2, 28 | syl5eqel 2692 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ∈
(0[,]+∞)) |
30 | 29 | adantr 480 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ∈
(0[,]+∞)) |
31 | | eqid 2610 |
. . . . 5
⊢ (𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤
(ℜ‘(𝐵 /
(i↑𝑘)))),
(ℜ‘(𝐵 /
(i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤
(ℜ‘(𝐵 /
(i↑𝑘)))),
(ℜ‘(𝐵 /
(i↑𝑘))),
0)) |
32 | 30, 31 | fmptd 6292 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))),
0)):ℝ⟶(0[,]+∞)) |
33 | | iblabsr.3 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈
𝐿1) |
34 | 4 | abscld 14023 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℝ) |
35 | 4 | absge0d 14031 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘𝐵)) |
36 | 34, 35 | iblpos 23365 |
. . . . . . 7
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ 𝐿1 ↔
((𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ MblFn ∧
(∫2‘(𝑥
∈ ℝ ↦ if(𝑥
∈ 𝐴, (abs‘𝐵), 0))) ∈
ℝ))) |
37 | 33, 36 | mpbid 221 |
. . . . . 6
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ MblFn ∧
(∫2‘(𝑥
∈ ℝ ↦ if(𝑥
∈ 𝐴, (abs‘𝐵), 0))) ∈
ℝ)) |
38 | 37 | simprd 478 |
. . . . 5
⊢ (𝜑 →
(∫2‘(𝑥
∈ ℝ ↦ if(𝑥
∈ 𝐴, (abs‘𝐵), 0))) ∈
ℝ) |
39 | 38 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) →
(∫2‘(𝑥
∈ ℝ ↦ if(𝑥
∈ 𝐴, (abs‘𝐵), 0))) ∈
ℝ) |
40 | 34 | rexrd 9968 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈
ℝ*) |
41 | | elxrge0 12152 |
. . . . . . . . . 10
⊢
((abs‘𝐵)
∈ (0[,]+∞) ↔ ((abs‘𝐵) ∈ ℝ* ∧ 0 ≤
(abs‘𝐵))) |
42 | 40, 35, 41 | sylanbrc 695 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ (0[,]+∞)) |
43 | 26 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈
(0[,]+∞)) |
44 | 42, 43 | ifclda 4070 |
. . . . . . . 8
⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ∈
(0[,]+∞)) |
45 | 44 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ∈
(0[,]+∞)) |
46 | | eqid 2610 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) |
47 | 45, 46 | fmptd 6292 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵),
0)):ℝ⟶(0[,]+∞)) |
48 | 47 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵),
0)):ℝ⟶(0[,]+∞)) |
49 | 16 | releabsd 14038 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 / (i↑𝑘))) ≤ (abs‘(𝐵 / (i↑𝑘)))) |
50 | 5, 12, 15 | absdivd 14042 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐵 / (i↑𝑘))) = ((abs‘𝐵) / (abs‘(i↑𝑘)))) |
51 | | elfznn0 12302 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ (0...3) → 𝑘 ∈
ℕ0) |
52 | 51 | ad2antlr 759 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 𝑘 ∈ ℕ0) |
53 | | absexp 13892 |
. . . . . . . . . . . . . . . . 17
⊢ ((i
∈ ℂ ∧ 𝑘
∈ ℕ0) → (abs‘(i↑𝑘)) = ((abs‘i)↑𝑘)) |
54 | 8, 52, 53 | sylancr 694 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘(i↑𝑘)) = ((abs‘i)↑𝑘)) |
55 | | absi 13874 |
. . . . . . . . . . . . . . . . . 18
⊢
(abs‘i) = 1 |
56 | 55 | oveq1i 6559 |
. . . . . . . . . . . . . . . . 17
⊢
((abs‘i)↑𝑘) = (1↑𝑘) |
57 | | 1exp 12751 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℤ →
(1↑𝑘) =
1) |
58 | 7, 57 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (1↑𝑘) = 1) |
59 | 56, 58 | syl5eq 2656 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘i)↑𝑘) = 1) |
60 | 54, 59 | eqtrd 2644 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘(i↑𝑘)) = 1) |
61 | 60 | oveq2d 6565 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘𝐵) / (abs‘(i↑𝑘))) = ((abs‘𝐵) / 1)) |
62 | 34 | recnd 9947 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℂ) |
63 | 62 | adantlr 747 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℂ) |
64 | 63 | div1d 10672 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘𝐵) / 1) = (abs‘𝐵)) |
65 | 50, 61, 64 | 3eqtrd 2648 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐵 / (i↑𝑘))) = (abs‘𝐵)) |
66 | 49, 65 | breqtrd 4609 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 / (i↑𝑘))) ≤ (abs‘𝐵)) |
67 | 5 | absge0d 14031 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘𝐵)) |
68 | | breq1 4586 |
. . . . . . . . . . . . 13
⊢
((ℜ‘(𝐵 /
(i↑𝑘))) = if(0 ≤
(ℜ‘(𝐵 /
(i↑𝑘))),
(ℜ‘(𝐵 /
(i↑𝑘))), 0) →
((ℜ‘(𝐵 /
(i↑𝑘))) ≤
(abs‘𝐵) ↔ if(0
≤ (ℜ‘(𝐵 /
(i↑𝑘))),
(ℜ‘(𝐵 /
(i↑𝑘))), 0) ≤
(abs‘𝐵))) |
69 | | breq1 4586 |
. . . . . . . . . . . . 13
⊢ (0 = if(0
≤ (ℜ‘(𝐵 /
(i↑𝑘))),
(ℜ‘(𝐵 /
(i↑𝑘))), 0) → (0
≤ (abs‘𝐵) ↔
if(0 ≤ (ℜ‘(𝐵
/ (i↑𝑘))),
(ℜ‘(𝐵 /
(i↑𝑘))), 0) ≤
(abs‘𝐵))) |
70 | 68, 69 | ifboth 4074 |
. . . . . . . . . . . 12
⊢
(((ℜ‘(𝐵 /
(i↑𝑘))) ≤
(abs‘𝐵) ∧ 0 ≤
(abs‘𝐵)) → if(0
≤ (ℜ‘(𝐵 /
(i↑𝑘))),
(ℜ‘(𝐵 /
(i↑𝑘))), 0) ≤
(abs‘𝐵)) |
71 | 66, 67, 70 | syl2anc 691 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ≤ (abs‘𝐵)) |
72 | | iftrue 4042 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0), 0) = if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0)) |
73 | 72 | adantl 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0), 0) = if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0)) |
74 | | iftrue 4042 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) = (abs‘𝐵)) |
75 | 74 | adantl 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) = (abs‘𝐵)) |
76 | 71, 73, 75 | 3brtr4d 4615 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0), 0) ≤ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) |
77 | 76 | ex 449 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0), 0) ≤ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) |
78 | | 0le0 10987 |
. . . . . . . . . . 11
⊢ 0 ≤
0 |
79 | 78 | a1i 11 |
. . . . . . . . . 10
⊢ (¬
𝑥 ∈ 𝐴 → 0 ≤ 0) |
80 | | iffalse 4045 |
. . . . . . . . . 10
⊢ (¬
𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0), 0) = 0) |
81 | | iffalse 4045 |
. . . . . . . . . 10
⊢ (¬
𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) = 0) |
82 | 79, 80, 81 | 3brtr4d 4615 |
. . . . . . . . 9
⊢ (¬
𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0), 0) ≤ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) |
83 | 77, 82 | pm2.61d1 170 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0), 0) ≤ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) |
84 | 2, 83 | syl5eqbr 4618 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ≤ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) |
85 | 84 | ralrimivw 2950 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ≤ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) |
86 | | reex 9906 |
. . . . . . . 8
⊢ ℝ
∈ V |
87 | 86 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ℝ ∈
V) |
88 | 40 | adantlr 747 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈
ℝ*) |
89 | 88, 67, 41 | sylanbrc 695 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ (0[,]+∞)) |
90 | 89, 27 | ifclda 4070 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ∈
(0[,]+∞)) |
91 | 90 | adantr 480 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ∈
(0[,]+∞)) |
92 | | eqidd 2611 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) |
93 | | eqidd 2611 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) |
94 | 87, 30, 91, 92, 93 | ofrfval2 6813 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)) ∘𝑟 ≤
(𝑥 ∈ ℝ ↦
if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) ↔ ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ≤ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) |
95 | 85, 94 | mpbird 246 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)) ∘𝑟 ≤
(𝑥 ∈ ℝ ↦
if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) |
96 | | itg2le 23312 |
. . . . 5
⊢ (((𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤
(ℜ‘(𝐵 /
(i↑𝑘)))),
(ℜ‘(𝐵 /
(i↑𝑘))),
0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)):ℝ⟶(0[,]+∞) ∧
(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤
(ℜ‘(𝐵 /
(i↑𝑘)))),
(ℜ‘(𝐵 /
(i↑𝑘))), 0))
∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) →
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ≤
(∫2‘(𝑥
∈ ℝ ↦ if(𝑥
∈ 𝐴, (abs‘𝐵), 0)))) |
97 | 32, 48, 95, 96 | syl3anc 1318 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) →
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ≤
(∫2‘(𝑥
∈ ℝ ↦ if(𝑥
∈ 𝐴, (abs‘𝐵), 0)))) |
98 | | itg2lecl 23311 |
. . . 4
⊢ (((𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤
(ℜ‘(𝐵 /
(i↑𝑘)))),
(ℜ‘(𝐵 /
(i↑𝑘))),
0)):ℝ⟶(0[,]+∞) ∧ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) ∈ ℝ ∧
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ≤
(∫2‘(𝑥
∈ ℝ ↦ if(𝑥
∈ 𝐴, (abs‘𝐵), 0)))) →
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ) |
99 | 32, 39, 97, 98 | syl3anc 1318 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) →
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ) |
100 | 99 | ralrimiva 2949 |
. 2
⊢ (𝜑 → ∀𝑘 ∈
(0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ) |
101 | | eqidd 2611 |
. . 3
⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) |
102 | | eqidd 2611 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘)))) |
103 | 101, 102,
3 | isibl2 23339 |
. 2
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔
((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ ∀𝑘 ∈
(0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ))) |
104 | 1, 100, 103 | mpbir2and 959 |
1
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈
𝐿1) |