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 )