Step | Hyp | Ref
| Expression |
1 | | eqid 2739 |
. . . . . 6
⊢ ℝ =
ℝ |
2 | | simpl 486 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))) → 𝑥 ∈ 𝐴) |
3 | 2 | con3i 157 |
. . . . . . . . . . 11
⊢ (¬
𝑥 ∈ 𝐴 → ¬ (𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘))))) |
4 | 3 | iffalsed 4435 |
. . . . . . . . . 10
⊢ (¬
𝑥 ∈ 𝐴 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) = 0) |
5 | | simpl 486 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))) → 𝑥 ∈ 𝐴) |
6 | 5 | con3i 157 |
. . . . . . . . . . 11
⊢ (¬
𝑥 ∈ 𝐴 → ¬ (𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘))))) |
7 | 6 | iffalsed 4435 |
. . . . . . . . . 10
⊢ (¬
𝑥 ∈ 𝐴 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) = 0) |
8 | 4, 7 | eqtr4d 2777 |
. . . . . . . . 9
⊢ (¬
𝑥 ∈ 𝐴 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) |
9 | | fvoveq1 7205 |
. . . . . . . . . . . 12
⊢ (𝐵 = 𝐶 → (ℜ‘(𝐵 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘)))) |
10 | 9 | breq2d 5052 |
. . . . . . . . . . 11
⊢ (𝐵 = 𝐶 → (0 ≤ (ℜ‘(𝐵 / (i↑𝑘))) ↔ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘))))) |
11 | 10 | anbi2d 632 |
. . . . . . . . . 10
⊢ (𝐵 = 𝐶 → ((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))) ↔ (𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))))) |
12 | 11, 9 | ifbieq1d 4448 |
. . . . . . . . 9
⊢ (𝐵 = 𝐶 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) |
13 | 8, 12 | ja 189 |
. . . . . . . 8
⊢ ((𝑥 ∈ 𝐴 → 𝐵 = 𝐶) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) |
14 | 13 | a1d 25 |
. . . . . . 7
⊢ ((𝑥 ∈ 𝐴 → 𝐵 = 𝐶) → (𝑥 ∈ ℝ → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) |
15 | 14 | ralimi2 3073 |
. . . . . 6
⊢
(∀𝑥 ∈
𝐴 𝐵 = 𝐶 → ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) |
16 | | mpteq12 5127 |
. . . . . 6
⊢ ((ℝ
= ℝ ∧ ∀𝑥
∈ ℝ if((𝑥 ∈
𝐴 ∧ 0 ≤
(ℜ‘(𝐵 /
(i↑𝑘)))),
(ℜ‘(𝐵 /
(i↑𝑘))), 0) =
if((𝑥 ∈ 𝐴 ∧ 0 ≤
(ℜ‘(𝐶 /
(i↑𝑘)))),
(ℜ‘(𝐶 /
(i↑𝑘))), 0)) →
(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤
(ℜ‘(𝐵 /
(i↑𝑘)))),
(ℜ‘(𝐵 /
(i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤
(ℜ‘(𝐶 /
(i↑𝑘)))),
(ℜ‘(𝐶 /
(i↑𝑘))),
0))) |
17 | 1, 15, 16 | sylancr 590 |
. . . . 5
⊢
(∀𝑥 ∈
𝐴 𝐵 = 𝐶 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) |
18 | 17 | fveq2d 6690 |
. . . 4
⊢
(∀𝑥 ∈
𝐴 𝐵 = 𝐶 → (∫2‘(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤
(ℜ‘(𝐵 /
(i↑𝑘)))),
(ℜ‘(𝐵 /
(i↑𝑘))), 0))) =
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))) |
19 | 18 | oveq2d 7198 |
. . 3
⊢
(∀𝑥 ∈
𝐴 𝐵 = 𝐶 → ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤
(ℜ‘(𝐵 /
(i↑𝑘)))),
(ℜ‘(𝐵 /
(i↑𝑘))), 0)))) =
((i↑𝑘) ·
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))) |
20 | 19 | sumeq2sdv 15166 |
. 2
⊢
(∀𝑥 ∈
𝐴 𝐵 = 𝐶 → Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤
(ℜ‘(𝐵 /
(i↑𝑘)))),
(ℜ‘(𝐵 /
(i↑𝑘))), 0)))) =
Σ𝑘 ∈
(0...3)((i↑𝑘) ·
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))) |
21 | | eqid 2739 |
. . 3
⊢
(ℜ‘(𝐵 /
(i↑𝑘))) =
(ℜ‘(𝐵 /
(i↑𝑘))) |
22 | 21 | dfitg 24534 |
. 2
⊢
∫𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤
(ℜ‘(𝐵 /
(i↑𝑘)))),
(ℜ‘(𝐵 /
(i↑𝑘))),
0)))) |
23 | | eqid 2739 |
. . 3
⊢
(ℜ‘(𝐶 /
(i↑𝑘))) =
(ℜ‘(𝐶 /
(i↑𝑘))) |
24 | 23 | dfitg 24534 |
. 2
⊢
∫𝐴𝐶 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤
(ℜ‘(𝐶 /
(i↑𝑘)))),
(ℜ‘(𝐶 /
(i↑𝑘))),
0)))) |
25 | 20, 22, 24 | 3eqtr4g 2799 |
1
⊢
(∀𝑥 ∈
𝐴 𝐵 = 𝐶 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥) |