Theorem: Complex.betaIntegral_scaled
File: Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean

Proof steps:
state_before: s t : ℂ
a : ℝ
ha : 0 < a
⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s + t - 1) * s.betaIntegral t
state_after: s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s + t - 1) * s.betaIntegral t
tactic: have ha' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne'
state_before: s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s + t - 1) * s.betaIntegral t
state_after: s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) =
    ↑a ^ (s + t - 1) * ∫ (x : ℝ) in 0 ..1, ↑x ^ (s - 1) * (1 - ↑x) ^ (t - 1)
tactic: rw [betaIntegral]
state_before: s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) =
    ↑a ^ (s + t - 1) * ∫ (x : ℝ) in 0 ..1, ↑x ^ (s - 1) * (1 - ↑x) ^ (t - 1)
state_after: s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) =
    ↑a ^ (s + t - 1) * ∫ (x : ℝ) in 0 ..1, ↑x ^ (s - 1) * (1 - ↑x) ^ (t - 1)
tactic: have A : (a : ℂ) ^ (s + t - 1) = a * ((a : ℂ) ^ (s - 1) * (a : ℂ) ^ (t - 1)) := by
    rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
      mul_assoc]
state_before: s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) =
    ↑a ^ (s + t - 1) * ∫ (x : ℝ) in 0 ..1, ↑x ^ (s - 1) * (1 - ↑x) ^ (t - 1)
state_after: s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) =
    ∫ (x : ℝ) in 0 ..a, ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑(x / a) ^ (s - 1) * (1 - ↑(x / a)) ^ (t - 1))
tactic: rw [A, mul_assoc, ← intervalIntegral.integral_const_mul, ← real_smul, ← zero_div a, ←
    div_self ha.ne', ← intervalIntegral.integral_comp_div _ ha.ne', zero_div]
state_before: s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) =
    ∫ (x : ℝ) in 0 ..a, ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑(x / a) ^ (s - 1) * (1 - ↑(x / a)) ^ (t - 1))
state_after: s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
⊢ ∫ (x : ℝ) in Ioc 0 a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) ∂volume =
    ∫ (x : ℝ) in Ioc 0 a, ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑(x / a) ^ (s - 1) * (1 - ↑(x / a)) ^ (t - 1)) ∂volume
tactic: simp_rw [intervalIntegral.integral_of_le ha.le]
state_before: s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
⊢ ∫ (x : ℝ) in Ioc 0 a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) ∂volume =
    ∫ (x : ℝ) in Ioc 0 a, ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑(x / a) ^ (s - 1) * (1 - ↑(x / a)) ^ (t - 1)) ∂volume
state_after: s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
x : ℝ
hx : x ∈ Ioc 0 a
⊢ ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑(x / a) ^ (s - 1) * (1 - ↑(x / a)) ^ (t - 1))
tactic: refine setIntegral_congr measurableSet_Ioc fun x hx => ?_
state_before: s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
x : ℝ
hx : x ∈ Ioc 0 a
⊢ ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑(x / a) ^ (s - 1) * (1 - ↑(x / a)) ^ (t - 1))
state_after: s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
x : ℝ
hx : x ∈ Ioc 0 a
⊢ ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s - 1) * ↑(x / a) ^ (s - 1) * (↑a ^ (t - 1) * (1 - ↑(x / a)) ^ (t - 1))
tactic: rw [mul_mul_mul_comm]
state_before: s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
x : ℝ
hx : x ∈ Ioc 0 a
⊢ ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s - 1) * ↑(x / a) ^ (s - 1) * (↑a ^ (t - 1) * (1 - ↑(x / a)) ^ (t - 1))
state_after: case e_a
s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
x : ℝ
hx : x ∈ Ioc 0 a
⊢ ↑x ^ (s - 1) = ↑a ^ (s - 1) * ↑(x / a) ^ (s - 1)

case e_a
s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
x : ℝ
hx : x ∈ Ioc 0 a
⊢ (↑a - ↑x) ^ (t - 1) = ↑a ^ (t - 1) * (1 - ↑(x / a)) ^ (t - 1)
tactic: congr 1