Metamath Proof Explorer

Theorem cbvitgdavw

Description: Change bound variable in an integral. Deduction form. (Contributed by GG, 14-Aug-2025)

Ref Expression
Hypothesis cbvitgdavw.1 ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐵 = 𝐶 )
Assertion cbvitgdavw ( 𝜑 → ∫ 𝐴 𝐵 d 𝑥 = ∫ 𝐴 𝐶 d 𝑦 )

Proof

Step Hyp Ref Expression
1 cbvitgdavw.1 ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐵 = 𝐶 )
2 1 fvoveq1d ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑡 ) ) ) = ( ℜ ‘ ( 𝐶 / ( i ↑ 𝑡 ) ) ) )
3 eleq1w ⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
4 3 adantl ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
5 4 anbi1d ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣 ) ↔ ( 𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣 ) ) )
6 5 ifbid ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣 ) , 𝑣 , 0 ) = if ( ( 𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣 ) , 𝑣 , 0 ) )
7 2 6 csbeq12dv ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ⦋ ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑡 ) ) ) / 𝑣 ⦌ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣 ) , 𝑣 , 0 ) = ⦋ ( ℜ ‘ ( 𝐶 / ( i ↑ 𝑡 ) ) ) / 𝑣 ⦌ if ( ( 𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣 ) , 𝑣 , 0 ) )
8 7 cbvmptdavw ⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑡 ) ) ) / 𝑣 ⦌ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣 ) , 𝑣 , 0 ) ) = ( 𝑦 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐶 / ( i ↑ 𝑡 ) ) ) / 𝑣 ⦌ if ( ( 𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣 ) , 𝑣 , 0 ) ) )
9 8 fveq2d ⊢ ( 𝜑 → ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑡 ) ) ) / 𝑣 ⦌ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣 ) , 𝑣 , 0 ) ) ) = ( ∫2 ‘ ( 𝑦 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐶 / ( i ↑ 𝑡 ) ) ) / 𝑣 ⦌ if ( ( 𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣 ) , 𝑣 , 0 ) ) ) )
10 9 oveq2d ⊢ ( 𝜑 → ( ( i ↑ 𝑡 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑡 ) ) ) / 𝑣 ⦌ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣 ) , 𝑣 , 0 ) ) ) ) = ( ( i ↑ 𝑡 ) · ( ∫2 ‘ ( 𝑦 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐶 / ( i ↑ 𝑡 ) ) ) / 𝑣 ⦌ if ( ( 𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣 ) , 𝑣 , 0 ) ) ) ) )
11 10 sumeq2sdv ⊢ ( 𝜑 → Σ 𝑡 ∈ ( 0 ... 3 ) ( ( i ↑ 𝑡 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑡 ) ) ) / 𝑣 ⦌ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣 ) , 𝑣 , 0 ) ) ) ) = Σ 𝑡 ∈ ( 0 ... 3 ) ( ( i ↑ 𝑡 ) · ( ∫2 ‘ ( 𝑦 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐶 / ( i ↑ 𝑡 ) ) ) / 𝑣 ⦌ if ( ( 𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣 ) , 𝑣 , 0 ) ) ) ) )
12 df-itg ⊢ ∫ 𝐴 𝐵 d 𝑥 = Σ 𝑡 ∈ ( 0 ... 3 ) ( ( i ↑ 𝑡 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐵 / ( i ↑ 𝑡 ) ) ) / 𝑣 ⦌ if ( ( 𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣 ) , 𝑣 , 0 ) ) ) )
13 df-itg ⊢ ∫ 𝐴 𝐶 d 𝑦 = Σ 𝑡 ∈ ( 0 ... 3 ) ( ( i ↑ 𝑡 ) · ( ∫2 ‘ ( 𝑦 ∈ ℝ ↦ ⦋ ( ℜ ‘ ( 𝐶 / ( i ↑ 𝑡 ) ) ) / 𝑣 ⦌ if ( ( 𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣 ) , 𝑣 , 0 ) ) ) )
14 11 12 13 3eqtr4g ⊢ ( 𝜑 → ∫ 𝐴 𝐵 d 𝑥 = ∫ 𝐴 𝐶 d 𝑦 )