| Step | Hyp | Ref
| Expression |
| 1 | | fconstmpt 5747 |
. 2
⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵) |
| 2 | | mbfconst 25668 |
. . . . 5
⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℂ) → (𝐴 × {𝐵}) ∈ MblFn) |
| 3 | 2 | 3adant2 1132 |
. . . 4
⊢ ((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) → (𝐴 ×
{𝐵}) ∈
MblFn) |
| 4 | 1, 3 | eqeltrrid 2846 |
. . 3
⊢ ((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) → (𝑥 ∈
𝐴 ↦ 𝐵) ∈ MblFn) |
| 5 | | ifan 4579 |
. . . . . . . 8
⊢ 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) |
| 6 | 5 | mpteq2i 5247 |
. . . . . . 7
⊢ (𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤
(ℜ‘(𝐵 /
(i↑𝑘)))),
(ℜ‘(𝐵 /
(i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0), 0)) |
| 7 | 6 | fveq2i 6909 |
. . . . . 6
⊢
(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0), 0))) |
| 8 | | simpl1 1192 |
. . . . . . 7
⊢ (((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
(0...3)) → 𝐴 ∈
dom vol) |
| 9 | | simpl2 1193 |
. . . . . . 7
⊢ (((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
(0...3)) → (vol‘𝐴) ∈ ℝ) |
| 10 | | simpl3 1194 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
(0...3)) → 𝐵 ∈
ℂ) |
| 11 | | ax-icn 11214 |
. . . . . . . . . . . 12
⊢ i ∈
ℂ |
| 12 | | ine0 11698 |
. . . . . . . . . . . 12
⊢ i ≠
0 |
| 13 | | elfzelz 13564 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ (0...3) → 𝑘 ∈
ℤ) |
| 14 | 13 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
(0...3)) → 𝑘 ∈
ℤ) |
| 15 | | expclz 14125 |
. . . . . . . . . . . 12
⊢ ((i
∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈
ℂ) |
| 16 | 11, 12, 14, 15 | mp3an12i 1467 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
(0...3)) → (i↑𝑘)
∈ ℂ) |
| 17 | | expne0i 14135 |
. . . . . . . . . . . 12
⊢ ((i
∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0) |
| 18 | 11, 12, 14, 17 | mp3an12i 1467 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
(0...3)) → (i↑𝑘)
≠ 0) |
| 19 | 10, 16, 18 | divcld 12043 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
(0...3)) → (𝐵 /
(i↑𝑘)) ∈
ℂ) |
| 20 | 19 | recld 15233 |
. . . . . . . . 9
⊢ (((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
(0...3)) → (ℜ‘(𝐵 / (i↑𝑘))) ∈ ℝ) |
| 21 | | 0re 11263 |
. . . . . . . . 9
⊢ 0 ∈
ℝ |
| 22 | | ifcl 4571 |
. . . . . . . . 9
⊢
(((ℜ‘(𝐵 /
(i↑𝑘))) ∈ ℝ
∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ∈ ℝ) |
| 23 | 20, 21, 22 | sylancl 586 |
. . . . . . . 8
⊢ (((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
(0...3)) → if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ∈ ℝ) |
| 24 | | max1 13227 |
. . . . . . . . 9
⊢ ((0
∈ ℝ ∧ (ℜ‘(𝐵 / (i↑𝑘))) ∈ ℝ) → 0 ≤ if(0 ≤
(ℜ‘(𝐵 /
(i↑𝑘))),
(ℜ‘(𝐵 /
(i↑𝑘))),
0)) |
| 25 | 21, 20, 24 | sylancr 587 |
. . . . . . . 8
⊢ (((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
(0...3)) → 0 ≤ if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0)) |
| 26 | | elrege0 13494 |
. . . . . . . 8
⊢ (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))) |
| 27 | 23, 25, 26 | sylanbrc 583 |
. . . . . . 7
⊢ (((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
(0...3)) → if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ∈
(0[,)+∞)) |
| 28 | | itg2const 25775 |
. . . . . . 7
⊢ ((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ∈ (0[,)+∞)) →
(∫2‘(𝑥
∈ ℝ ↦ 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) ·
(vol‘𝐴))) |
| 29 | 8, 9, 27, 28 | syl3anc 1373 |
. . . . . 6
⊢ (((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
(0...3)) → (∫2‘(𝑥 ∈ ℝ ↦ 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) ·
(vol‘𝐴))) |
| 30 | 7, 29 | eqtrid 2789 |
. . . . 5
⊢ (((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
(0...3)) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) = (if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) · (vol‘𝐴))) |
| 31 | 23, 9 | remulcld 11291 |
. . . . 5
⊢ (((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
(0...3)) → (if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) · (vol‘𝐴)) ∈ ℝ) |
| 32 | 30, 31 | eqeltrd 2841 |
. . . 4
⊢ (((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) ∧ 𝑘 ∈
(0...3)) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ) |
| 33 | 32 | ralrimiva 3146 |
. . 3
⊢ ((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) → ∀𝑘
∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ) |
| 34 | | eqidd 2738 |
. . . 4
⊢ ((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) → (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))) |
| 35 | | eqidd 2738 |
. . . 4
⊢ (((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) ∧ 𝑥 ∈
𝐴) →
(ℜ‘(𝐵 /
(i↑𝑘))) =
(ℜ‘(𝐵 /
(i↑𝑘)))) |
| 36 | | simpl3 1194 |
. . . 4
⊢ (((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) ∧ 𝑥 ∈
𝐴) → 𝐵 ∈ ℂ) |
| 37 | 34, 35, 36 | isibl2 25801 |
. . 3
⊢ ((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) → ((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))) |
| 38 | 4, 33, 37 | mpbir2and 713 |
. 2
⊢ ((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) → (𝑥 ∈
𝐴 ↦ 𝐵) ∈
𝐿1) |
| 39 | 1, 38 | eqeltrid 2845 |
1
⊢ ((𝐴 ∈ dom vol ∧
(vol‘𝐴) ∈
ℝ ∧ 𝐵 ∈
ℂ) → (𝐴 ×
{𝐵}) ∈
𝐿1) |