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Metamath Proof Explorer

Theorem itgeq1

Description: Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014)

Ref Expression
Assertion itgeq1 ( 𝐴 = 𝐵 → ∫ 𝐴 𝐶 d 𝑥 = ∫ 𝐵 𝐶 d 𝑥 )

Proof

Step Hyp Ref Expression
1 eleq2 ⊢ ( 𝐴 = 𝐵 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
2 1 anbi1d ⊢ ( 𝐴 = 𝐵 → ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦 ) ) )
3 2 ifbid ⊢ ( 𝐴 = 𝐵 → if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) = if ( ( 𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) )
4 3 csbeq2dv ⊢ ( 𝐴 = 𝐵 → ⦋ ( ℜ ‘ ( 𝐶 / ( i ↑ 𝑘 ) ) ) / 𝑦 ⦌ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) = ⦋ ( ℜ ‘ ( 𝐶 / ( i ↑ 𝑘 ) ) ) / 𝑦 ⦌ if ( ( 𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) )
5 4 mpteq2dv ⊢ ( 𝐴 = 𝐵 → ( 𝑥 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐶 / ( i ↑ 𝑘 ) ) ) / 𝑦 ⦌ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) ) = ( 𝑥 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐶 / ( i ↑ 𝑘 ) ) ) / 𝑦 ⦌ if ( ( 𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) ) )
6 5 fveq2d ⊢ ( 𝐴 = 𝐵 → ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐶 / ( i ↑ 𝑘 ) ) ) / 𝑦 ⦌ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) ) ) = ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐶 / ( i ↑ 𝑘 ) ) ) / 𝑦 ⦌ if ( ( 𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) ) ) )
7 6 oveq2d ⊢ ( 𝐴 = 𝐵 → ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐶 / ( i ↑ 𝑘 ) ) ) / 𝑦 ⦌ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) ) ) ) = ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐶 / ( i ↑ 𝑘 ) ) ) / 𝑦 ⦌ if ( ( 𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) ) ) ) )
8 7 sumeq2sdv ⊢ ( 𝐴 = 𝐵 → Σ 𝑘 ∈ ( 0 ... 3 ) ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐶 / ( i ↑ 𝑘 ) ) ) / 𝑦 ⦌ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) ) ) ) = Σ 𝑘 ∈ ( 0 ... 3 ) ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐶 / ( i ↑ 𝑘 ) ) ) / 𝑦 ⦌ if ( ( 𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) ) ) ) )
9 df-itg ⊢ ∫ 𝐴 𝐶 d 𝑥 = Σ 𝑘 ∈ ( 0 ... 3 ) ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐶 / ( i ↑ 𝑘 ) ) ) / 𝑦 ⦌ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) ) ) )
10 df-itg ⊢ ∫ 𝐵 𝐶 d 𝑥 = Σ 𝑘 ∈ ( 0 ... 3 ) ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐶 / ( i ↑ 𝑘 ) ) ) / 𝑦 ⦌ if ( ( 𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦 ) , 𝑦 , 0 ) ) ) )
11 8 9 10 3eqtr4g ⊢ ( 𝐴 = 𝐵 → ∫ 𝐴 𝐶 d 𝑥 = ∫ 𝐵 𝐶 d 𝑥 )