Step |
Hyp |
Ref |
Expression |
1 |
|
dfitg.1 |
⊢ 𝑇 = ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) |
2 |
|
df-itg |
⊢ ∫ 𝐴 𝐵 d 𝑥 = Σ 𝑘 ∈ ( 0 ... 3 ) ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) / 𝑦 ⦌ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) ) ) ) |
3 |
|
fvex |
⊢ ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) ∈ V |
4 |
|
id |
⊢ ( 𝑦 = ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) → 𝑦 = ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) ) |
5 |
4 1
|
eqtr4di |
⊢ ( 𝑦 = ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) → 𝑦 = 𝑇 ) |
6 |
5
|
breq2d |
⊢ ( 𝑦 = ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) → ( 0 ≤ 𝑦 ↔ 0 ≤ 𝑇 ) ) |
7 |
6
|
anbi2d |
⊢ ( 𝑦 = ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) → ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇 ) ) ) |
8 |
7 5
|
ifbieq1d |
⊢ ( 𝑦 = ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) → if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) = if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇 ) , 𝑇 , 0 ) ) |
9 |
3 8
|
csbie |
⊢ ⦋ ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) / 𝑦 ⦌ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) = if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇 ) , 𝑇 , 0 ) |
10 |
9
|
mpteq2i |
⊢ ( 𝑥 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) / 𝑦 ⦌ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) ) = ( 𝑥 ∈ ℝ ↦ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇 ) , 𝑇 , 0 ) ) |
11 |
10
|
fveq2i |
⊢ ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) / 𝑦 ⦌ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) ) ) = ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇 ) , 𝑇 , 0 ) ) ) |
12 |
11
|
oveq2i |
⊢ ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) / 𝑦 ⦌ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) ) ) ) = ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇 ) , 𝑇 , 0 ) ) ) ) |
13 |
12
|
a1i |
⊢ ( 𝑘 ∈ ( 0 ... 3 ) → ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) / 𝑦 ⦌ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) ) ) ) = ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇 ) , 𝑇 , 0 ) ) ) ) ) |
14 |
13
|
sumeq2i |
⊢ Σ 𝑘 ∈ ( 0 ... 3 ) ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) / 𝑦 ⦌ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) ) ) ) = Σ 𝑘 ∈ ( 0 ... 3 ) ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇 ) , 𝑇 , 0 ) ) ) ) |
15 |
2 14
|
eqtri |
⊢ ∫ 𝐴 𝐵 d 𝑥 = Σ 𝑘 ∈ ( 0 ... 3 ) ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇 ) , 𝑇 , 0 ) ) ) ) |