Metamath Proof Explorer


Theorem dfitg

Description: Evaluate the class substitution in df-itg . (Contributed by Mario Carneiro, 28-Jun-2014) (Revised by Mario Carneiro, 23-Aug-2014)

Ref Expression
Hypothesis dfitg.1 𝑇 = ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) )
Assertion dfitg 𝐴 𝐵 d 𝑥 = Σ 𝑘 ∈ ( 0 ... 3 ) ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( ( 𝑥𝐴 ∧ 0 ≤ 𝑇 ) , 𝑇 , 0 ) ) ) )

Proof

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 ( ( 𝑥𝐴 ∧ 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 ) ) ) → ( ( 𝑥𝐴 ∧ 0 ≤ 𝑦 ) ↔ ( 𝑥𝐴 ∧ 0 ≤ 𝑇 ) ) )
8 7 5 ifbieq1d ( 𝑦 = ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) → if ( ( 𝑥𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) = if ( ( 𝑥𝐴 ∧ 0 ≤ 𝑇 ) , 𝑇 , 0 ) )
9 3 8 csbie ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) / 𝑦 if ( ( 𝑥𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) = if ( ( 𝑥𝐴 ∧ 0 ≤ 𝑇 ) , 𝑇 , 0 )
10 9 mpteq2i ( 𝑥 ∈ ℝ ↦ ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) / 𝑦 if ( ( 𝑥𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) ) = ( 𝑥 ∈ ℝ ↦ if ( ( 𝑥𝐴 ∧ 0 ≤ 𝑇 ) , 𝑇 , 0 ) )
11 10 fveq2i ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) / 𝑦 if ( ( 𝑥𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) ) ) = ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( ( 𝑥𝐴 ∧ 0 ≤ 𝑇 ) , 𝑇 , 0 ) ) )
12 11 oveq2i ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) / 𝑦 if ( ( 𝑥𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) ) ) ) = ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( ( 𝑥𝐴 ∧ 0 ≤ 𝑇 ) , 𝑇 , 0 ) ) ) )
13 12 a1i ( 𝑘 ∈ ( 0 ... 3 ) → ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) / 𝑦 if ( ( 𝑥𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) ) ) ) = ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( ( 𝑥𝐴 ∧ 0 ≤ 𝑇 ) , 𝑇 , 0 ) ) ) ) )
14 13 sumeq2i Σ 𝑘 ∈ ( 0 ... 3 ) ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑘 ) ) ) / 𝑦 if ( ( 𝑥𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) ) ) ) = Σ 𝑘 ∈ ( 0 ... 3 ) ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( ( 𝑥𝐴 ∧ 0 ≤ 𝑇 ) , 𝑇 , 0 ) ) ) )
15 2 14 eqtri 𝐴 𝐵 d 𝑥 = Σ 𝑘 ∈ ( 0 ... 3 ) ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( ( 𝑥𝐴 ∧ 0 ≤ 𝑇 ) , 𝑇 , 0 ) ) ) )