Step | Hyp | Ref
| Expression |
1 | | fvex 6201 |
. . . . . . . . 9
⊢
(ℜ‘((𝑓‘𝑥) / (i↑𝑘))) ∈ V |
2 | | breq2 4657 |
. . . . . . . . . . 11
⊢ (𝑦 = (ℜ‘((𝑓‘𝑥) / (i↑𝑘))) → (0 ≤ 𝑦 ↔ 0 ≤ (ℜ‘((𝑓‘𝑥) / (i↑𝑘))))) |
3 | 2 | anbi2d 740 |
. . . . . . . . . 10
⊢ (𝑦 = (ℜ‘((𝑓‘𝑥) / (i↑𝑘))) → ((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦) ↔ (𝑥 ∈ dom 𝑓 ∧ 0 ≤ (ℜ‘((𝑓‘𝑥) / (i↑𝑘)))))) |
4 | | id 22 |
. . . . . . . . . 10
⊢ (𝑦 = (ℜ‘((𝑓‘𝑥) / (i↑𝑘))) → 𝑦 = (ℜ‘((𝑓‘𝑥) / (i↑𝑘)))) |
5 | 3, 4 | ifbieq1d 4109 |
. . . . . . . . 9
⊢ (𝑦 = (ℜ‘((𝑓‘𝑥) / (i↑𝑘))) → if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ (ℜ‘((𝑓‘𝑥) / (i↑𝑘)))), (ℜ‘((𝑓‘𝑥) / (i↑𝑘))), 0)) |
6 | 1, 5 | csbie 3559 |
. . . . . . . 8
⊢
⦋(ℜ‘((𝑓‘𝑥) / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ (ℜ‘((𝑓‘𝑥) / (i↑𝑘)))), (ℜ‘((𝑓‘𝑥) / (i↑𝑘))), 0) |
7 | | dmeq 5324 |
. . . . . . . . . . 11
⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹) |
8 | 7 | eleq2d 2687 |
. . . . . . . . . 10
⊢ (𝑓 = 𝐹 → (𝑥 ∈ dom 𝑓 ↔ 𝑥 ∈ dom 𝐹)) |
9 | | fveq1 6190 |
. . . . . . . . . . . . 13
⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥)) |
10 | 9 | oveq1d 6665 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) / (i↑𝑘)) = ((𝐹‘𝑥) / (i↑𝑘))) |
11 | 10 | fveq2d 6195 |
. . . . . . . . . . 11
⊢ (𝑓 = 𝐹 → (ℜ‘((𝑓‘𝑥) / (i↑𝑘))) = (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))) |
12 | 11 | breq2d 4665 |
. . . . . . . . . 10
⊢ (𝑓 = 𝐹 → (0 ≤ (ℜ‘((𝑓‘𝑥) / (i↑𝑘))) ↔ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘))))) |
13 | 8, 12 | anbi12d 747 |
. . . . . . . . 9
⊢ (𝑓 = 𝐹 → ((𝑥 ∈ dom 𝑓 ∧ 0 ≤ (ℜ‘((𝑓‘𝑥) / (i↑𝑘)))) ↔ (𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))))) |
14 | 13, 11 | ifbieq1d 4109 |
. . . . . . . 8
⊢ (𝑓 = 𝐹 → if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ (ℜ‘((𝑓‘𝑥) / (i↑𝑘)))), (ℜ‘((𝑓‘𝑥) / (i↑𝑘))), 0) = if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0)) |
15 | 6, 14 | syl5eq 2668 |
. . . . . . 7
⊢ (𝑓 = 𝐹 → ⦋(ℜ‘((𝑓‘𝑥) / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0)) |
16 | 15 | mpteq2dv 4745 |
. . . . . 6
⊢ (𝑓 = 𝐹 → (𝑥 ∈ ℝ ↦
⦋(ℜ‘((𝑓‘𝑥) / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0))) |
17 | 16 | fveq2d 6195 |
. . . . 5
⊢ (𝑓 = 𝐹 → (∫2‘(𝑥 ∈ ℝ ↦
⦋(ℜ‘((𝑓‘𝑥) / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0))) = (∫2‘(𝑥 ∈ ℝ ↦
if((𝑥 ∈ dom 𝐹 ∧ 0 ≤
(ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0)))) |
18 | 17 | eleq1d 2686 |
. . . 4
⊢ (𝑓 = 𝐹 → ((∫2‘(𝑥 ∈ ℝ ↦
⦋(ℜ‘((𝑓‘𝑥) / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0))) ∈ ℝ ↔
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0))) ∈ ℝ)) |
19 | 18 | ralbidv 2986 |
. . 3
⊢ (𝑓 = 𝐹 → (∀𝑘 ∈
(0...3)(∫2‘(𝑥 ∈ ℝ ↦
⦋(ℜ‘((𝑓‘𝑥) / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0))) ∈ ℝ ↔ ∀𝑘 ∈
(0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0))) ∈ ℝ)) |
20 | | df-ibl 23391 |
. . 3
⊢
𝐿1 = {𝑓 ∈ MblFn ∣ ∀𝑘 ∈
(0...3)(∫2‘(𝑥 ∈ ℝ ↦
⦋(ℜ‘((𝑓‘𝑥) / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0))) ∈ ℝ} |
21 | 19, 20 | elrab2 3366 |
. 2
⊢ (𝐹 ∈ 𝐿1
↔ (𝐹 ∈ MblFn
∧ ∀𝑘 ∈
(0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0))) ∈ ℝ)) |
22 | | isibl.3 |
. . . . . . . . . . . 12
⊢ (𝜑 → dom 𝐹 = 𝐴) |
23 | 22 | eleq2d 2687 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴)) |
24 | 23 | anbi1d 741 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))) ↔ (𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))))) |
25 | 24 | ifbid 4108 |
. . . . . . . . 9
⊢ (𝜑 → if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0)) |
26 | | isibl.4 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) |
27 | 26 | oveq1d 6665 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) / (i↑𝑘)) = (𝐵 / (i↑𝑘))) |
28 | 27 | fveq2d 6195 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐹‘𝑥) / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘)))) |
29 | | isibl.2 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘)))) |
30 | 28, 29 | eqtr4d 2659 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐹‘𝑥) / (i↑𝑘))) = 𝑇) |
31 | 30 | ibllem 23531 |
. . . . . . . . 9
⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)) |
32 | 25, 31 | eqtrd 2656 |
. . . . . . . 8
⊢ (𝜑 → if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)) |
33 | 32 | mpteq2dv 4745 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))) |
34 | | isibl.1 |
. . . . . . 7
⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))) |
35 | 33, 34 | eqtr4d 2659 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0)) = 𝐺) |
36 | 35 | fveq2d 6195 |
. . . . 5
⊢ (𝜑 →
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0))) = (∫2‘𝐺)) |
37 | 36 | eleq1d 2686 |
. . . 4
⊢ (𝜑 →
((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0))) ∈ ℝ ↔
(∫2‘𝐺)
∈ ℝ)) |
38 | 37 | ralbidv 2986 |
. . 3
⊢ (𝜑 → (∀𝑘 ∈
(0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0))) ∈ ℝ ↔ ∀𝑘 ∈
(0...3)(∫2‘𝐺) ∈ ℝ)) |
39 | 38 | anbi2d 740 |
. 2
⊢ (𝜑 → ((𝐹 ∈ MblFn ∧ ∀𝑘 ∈
(0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0))) ∈ ℝ) ↔ (𝐹 ∈ MblFn ∧
∀𝑘 ∈
(0...3)(∫2‘𝐺) ∈ ℝ))) |
40 | 21, 39 | syl5bb 272 |
1
⊢ (𝜑 → (𝐹 ∈ 𝐿1 ↔ (𝐹 ∈ MblFn ∧
∀𝑘 ∈
(0...3)(∫2‘𝐺) ∈ ℝ))) |