Step | Hyp | Ref
| Expression |
1 | | iblss.1 |
. . . 4
⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
2 | 1 | resmptd 5702 |
. . 3
⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
3 | | iblss.4 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈
𝐿1) |
4 | | iblmbf 23971 |
. . . . 5
⊢ ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿1 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn) |
5 | 3, 4 | syl 17 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn) |
6 | | iblss.2 |
. . . 4
⊢ (𝜑 → 𝐴 ∈ dom vol) |
7 | | mbfres 23848 |
. . . 4
⊢ (((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn ∧ 𝐴 ∈ dom vol) → ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) ∈ MblFn) |
8 | 5, 6, 7 | syl2anc 579 |
. . 3
⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) ∈ MblFn) |
9 | 2, 8 | eqeltrrd 2860 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn) |
10 | | ifan 4358 |
. . . . . 6
⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) = if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) |
11 | | simpll 757 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → 𝜑) |
12 | 1 | sselda 3821 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) |
13 | 11, 12 | sylan 575 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) |
14 | | iblss.3 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉) |
15 | 5, 14 | mbfmptcl 23840 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℂ) |
16 | 11, 15 | sylan 575 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℂ) |
17 | | elfzelz 12659 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (0...3) → 𝑘 ∈
ℤ) |
18 | 17 | ad3antlr 721 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → 𝑘 ∈ ℤ) |
19 | | ax-icn 10331 |
. . . . . . . . . . . . . . 15
⊢ i ∈
ℂ |
20 | | ine0 10810 |
. . . . . . . . . . . . . . 15
⊢ i ≠
0 |
21 | | expclz 13203 |
. . . . . . . . . . . . . . 15
⊢ ((i
∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈
ℂ) |
22 | 19, 20, 21 | mp3an12 1524 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℤ →
(i↑𝑘) ∈
ℂ) |
23 | 18, 22 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → (i↑𝑘) ∈ ℂ) |
24 | | expne0i 13210 |
. . . . . . . . . . . . . . 15
⊢ ((i
∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0) |
25 | 19, 20, 24 | mp3an12 1524 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℤ →
(i↑𝑘) ≠
0) |
26 | 18, 25 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → (i↑𝑘) ≠ 0) |
27 | 16, 23, 26 | divcld 11151 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → (𝐶 / (i↑𝑘)) ∈ ℂ) |
28 | 27 | recld 14341 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ) |
29 | | 0re 10378 |
. . . . . . . . . . 11
⊢ 0 ∈
ℝ |
30 | | ifcl 4351 |
. . . . . . . . . . 11
⊢
(((ℜ‘(𝐶 /
(i↑𝑘))) ∈ ℝ
∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ) |
31 | 28, 29, 30 | sylancl 580 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ) |
32 | 31 | rexrd 10426 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈
ℝ*) |
33 | | max1 12328 |
. . . . . . . . . 10
⊢ ((0
∈ ℝ ∧ (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ) → 0 ≤ if(0 ≤
(ℜ‘(𝐶 /
(i↑𝑘))),
(ℜ‘(𝐶 /
(i↑𝑘))),
0)) |
34 | 29, 28, 33 | sylancr 581 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → 0 ≤ if(0 ≤
(ℜ‘(𝐶 /
(i↑𝑘))),
(ℜ‘(𝐶 /
(i↑𝑘))),
0)) |
35 | | elxrge0 12595 |
. . . . . . . . 9
⊢ (if(0
≤ (ℜ‘(𝐶 /
(i↑𝑘))),
(ℜ‘(𝐶 /
(i↑𝑘))), 0) ∈
(0[,]+∞) ↔ (if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ* ∧ 0
≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) |
36 | 32, 34, 35 | sylanbrc 578 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈
(0[,]+∞)) |
37 | 13, 36 | syldan 585 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈
(0[,]+∞)) |
38 | | 0e0iccpnf 12597 |
. . . . . . . 8
⊢ 0 ∈
(0[,]+∞) |
39 | 38 | a1i 11 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈
(0[,]+∞)) |
40 | 37, 39 | ifclda 4341 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ∈
(0[,]+∞)) |
41 | 10, 40 | syl5eqel 2863 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈
(0[,]+∞)) |
42 | 41 | fmpttd 6649 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))),
0)):ℝ⟶(0[,]+∞)) |
43 | | eqidd 2779 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) |
44 | | eqidd 2779 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘)))) |
45 | 43, 44, 3, 14 | iblitg 23972 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) →
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ) |
46 | 17, 45 | sylan2 586 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) →
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ) |
47 | | ifan 4358 |
. . . . . . 7
⊢ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) = if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) |
48 | 38 | a1i 11 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ 𝐵) → 0 ∈
(0[,]+∞)) |
49 | 36, 48 | ifclda 4341 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ∈
(0[,]+∞)) |
50 | 47, 49 | syl5eqel 2863 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈
(0[,]+∞)) |
51 | 50 | fmpttd 6649 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))),
0)):ℝ⟶(0[,]+∞)) |
52 | 31 | leidd 10941 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) |
53 | | breq1 4889 |
. . . . . . . . . . . 12
⊢ (if(0
≤ (ℜ‘(𝐶 /
(i↑𝑘))),
(ℜ‘(𝐶 /
(i↑𝑘))), 0) = if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) → (if(0 ≤
(ℜ‘(𝐶 /
(i↑𝑘))),
(ℜ‘(𝐶 /
(i↑𝑘))), 0) ≤ if(0
≤ (ℜ‘(𝐶 /
(i↑𝑘))),
(ℜ‘(𝐶 /
(i↑𝑘))), 0) ↔
if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ≤ if(0 ≤
(ℜ‘(𝐶 /
(i↑𝑘))),
(ℜ‘(𝐶 /
(i↑𝑘))),
0))) |
54 | | breq1 4889 |
. . . . . . . . . . . 12
⊢ (0 =
if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) → (0 ≤ if(0 ≤
(ℜ‘(𝐶 /
(i↑𝑘))),
(ℜ‘(𝐶 /
(i↑𝑘))), 0) ↔
if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ≤ if(0 ≤
(ℜ‘(𝐶 /
(i↑𝑘))),
(ℜ‘(𝐶 /
(i↑𝑘))),
0))) |
55 | 53, 54 | ifboth 4345 |
. . . . . . . . . . 11
⊢ ((if(0
≤ (ℜ‘(𝐶 /
(i↑𝑘))),
(ℜ‘(𝐶 /
(i↑𝑘))), 0) ≤ if(0
≤ (ℜ‘(𝐶 /
(i↑𝑘))),
(ℜ‘(𝐶 /
(i↑𝑘))), 0) ∧ 0
≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ≤ if(0 ≤
(ℜ‘(𝐶 /
(i↑𝑘))),
(ℜ‘(𝐶 /
(i↑𝑘))),
0)) |
56 | 52, 34, 55 | syl2anc 579 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ≤ if(0 ≤
(ℜ‘(𝐶 /
(i↑𝑘))),
(ℜ‘(𝐶 /
(i↑𝑘))),
0)) |
57 | | iftrue 4313 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝐵 → if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) |
58 | 57 | adantl 475 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) |
59 | 56, 58 | breqtrrd 4914 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ≤ if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)) |
60 | | 0le0 11483 |
. . . . . . . . . . 11
⊢ 0 ≤
0 |
61 | 60 | a1i 11 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ 𝐵) → 0 ≤ 0) |
62 | 13 | stoic1a 1816 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ 𝐴) |
63 | | iffalse 4316 |
. . . . . . . . . . 11
⊢ (¬
𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = 0) |
64 | 62, 63 | syl 17 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ 𝐵) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = 0) |
65 | | iffalse 4316 |
. . . . . . . . . . 11
⊢ (¬
𝑥 ∈ 𝐵 → if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = 0) |
66 | 65 | adantl 475 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ 𝐵) → if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = 0) |
67 | 61, 64, 66 | 3brtr4d 4918 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ 𝐵) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ≤ if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)) |
68 | 59, 67 | pm2.61dan 803 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ≤ if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)) |
69 | 68, 10, 47 | 3brtr4g 4920 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ≤ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) |
70 | 69 | ralrimiva 3148 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ≤ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) |
71 | | reex 10363 |
. . . . . . . 8
⊢ ℝ
∈ V |
72 | 71 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ℝ ∈
V) |
73 | | eqidd 2779 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) |
74 | | eqidd 2779 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) |
75 | 72, 41, 50, 73, 74 | ofrfval2 7192 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) ∘𝑟 ≤
(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐵 ∧ 0 ≤
(ℜ‘(𝐶 /
(i↑𝑘)))),
(ℜ‘(𝐶 /
(i↑𝑘))), 0)) ↔
∀𝑥 ∈ ℝ
if((𝑥 ∈ 𝐴 ∧ 0 ≤
(ℜ‘(𝐶 /
(i↑𝑘)))),
(ℜ‘(𝐶 /
(i↑𝑘))), 0) ≤
if((𝑥 ∈ 𝐵 ∧ 0 ≤
(ℜ‘(𝐶 /
(i↑𝑘)))),
(ℜ‘(𝐶 /
(i↑𝑘))),
0))) |
76 | 70, 75 | mpbird 249 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) ∘𝑟 ≤
(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐵 ∧ 0 ≤
(ℜ‘(𝐶 /
(i↑𝑘)))),
(ℜ‘(𝐶 /
(i↑𝑘))),
0))) |
77 | | itg2le 23943 |
. . . . 5
⊢ (((𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤
(ℜ‘(𝐶 /
(i↑𝑘)))),
(ℜ‘(𝐶 /
(i↑𝑘))),
0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞) ∧
(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤
(ℜ‘(𝐶 /
(i↑𝑘)))),
(ℜ‘(𝐶 /
(i↑𝑘))), 0))
∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) →
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ≤
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))) |
78 | 42, 51, 76, 77 | syl3anc 1439 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) →
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ≤
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))) |
79 | | itg2lecl 23942 |
. . . 4
⊢ (((𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤
(ℜ‘(𝐶 /
(i↑𝑘)))),
(ℜ‘(𝐶 /
(i↑𝑘))),
0)):ℝ⟶(0[,]+∞) ∧ (∫2‘(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐵 ∧ 0 ≤
(ℜ‘(𝐶 /
(i↑𝑘)))),
(ℜ‘(𝐶 /
(i↑𝑘))), 0))) ∈
ℝ ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ≤
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))) →
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ) |
80 | 42, 46, 78, 79 | syl3anc 1439 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) →
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ) |
81 | 80 | ralrimiva 3148 |
. 2
⊢ (𝜑 → ∀𝑘 ∈
(0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ) |
82 | | eqidd 2779 |
. . 3
⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) |
83 | | eqidd 2779 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘)))) |
84 | 12, 15 | syldan 585 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ) |
85 | 82, 83, 84 | isibl2 23970 |
. 2
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1 ↔
((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn ∧ ∀𝑘 ∈
(0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ))) |
86 | 9, 81, 85 | mpbir2and 703 |
1
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈
𝐿1) |