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

Proof steps:
state_before: u v : ℂ
⊢ v.betaIntegral u = u.betaIntegral v
state_after: u v : ℂ
⊢ ∫ (x : ℝ) in 0 ..1, ↑x ^ (v - 1) * (1 - ↑x) ^ (u - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
tactic: rw [betaIntegral, betaIntegral]
state_before: u v : ℂ
⊢ ∫ (x : ℝ) in 0 ..1, ↑x ^ (v - 1) * (1 - ↑x) ^ (u - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
state_after: u v : ℂ
this :
  ∫ (x : ℝ) in 0 ..1, ↑(-1 * x + 1) ^ (u - 1) * (1 - ↑(-1 * x + 1)) ^ (v - 1) =
    (-1)⁻¹ • ∫ (x : ℝ) in -1 * 0 + 1 ..-1 * 1 + 1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
⊢ ∫ (x : ℝ) in 0 ..1, ↑x ^ (v - 1) * (1 - ↑x) ^ (u - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
tactic: have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
    (fun x : ℝ => (x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1)) neg_one_lt_zero.ne 1
state_before: u v : ℂ
this :
  ∫ (x : ℝ) in 0 ..1, ↑(-1 * x + 1) ^ (u - 1) * (1 - ↑(-1 * x + 1)) ^ (v - 1) =
    (-1)⁻¹ • ∫ (x : ℝ) in -1 * 0 + 1 ..-1 * 1 + 1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
⊢ ∫ (x : ℝ) in 0 ..1, ↑x ^ (v - 1) * (1 - ↑x) ^ (u - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
state_after: u v : ℂ
this :
  ∫ (x : ℝ) in 0 ..1, ↑(-1 * x + 1) ^ (u - 1) * (1 - ↑(-1 * x + 1)) ^ (v - 1) =
    ∫ (x : ℝ) in -1 * 1 + 1 ..-1 * 0 + 1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
⊢ ∫ (x : ℝ) in 0 ..1, ↑x ^ (v - 1) * (1 - ↑x) ^ (u - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
tactic: rw [inv_neg, inv_one, neg_one_smul, ← intervalIntegral.integral_symm] at this
state_before: u v : ℂ
this :
  ∫ (x : ℝ) in 0 ..1, ↑(-1 * x + 1) ^ (u - 1) * (1 - ↑(-1 * x + 1)) ^ (v - 1) =
    ∫ (x : ℝ) in -1 * 1 + 1 ..-1 * 0 + 1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
⊢ ∫ (x : ℝ) in 0 ..1, ↑x ^ (v - 1) * (1 - ↑x) ^ (u - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
state_after: u v : ℂ
this : ∫ (x : ℝ) in 0 ..1, (-↑x + 1) ^ (u - 1) * ↑x ^ (v - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
⊢ ∫ (x : ℝ) in 0 ..1, ↑x ^ (v - 1) * (1 - ↑x) ^ (u - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
tactic: simp? at this says
    simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel_right, neg_neg,
      mul_one, add_left_neg, mul_zero, zero_add] at this
state_before: u v : ℂ
this : ∫ (x : ℝ) in 0 ..1, (-↑x + 1) ^ (u - 1) * ↑x ^ (v - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
⊢ ∫ (x : ℝ) in 0 ..1, ↑x ^ (v - 1) * (1 - ↑x) ^ (u - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
state_after: u v : ℂ
this : ∫ (x : ℝ) in 0 ..1, ↑x ^ (v - 1) * (1 - ↑x) ^ (u - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
⊢ ∫ (x : ℝ) in 0 ..1, ↑x ^ (v - 1) * (1 - ↑x) ^ (u - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
tactic: conv_lhs at this => arg 1; intro x; rw [add_comm, ← sub_eq_add_neg, mul_comm]
state_before: u v : ℂ
this : ∫ (x : ℝ) in 0 ..1, ↑x ^ (v - 1) * (1 - ↑x) ^ (u - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
⊢ ∫ (x : ℝ) in 0 ..1, ↑x ^ (v - 1) * (1 - ↑x) ^ (u - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
state_after: no goals
tactic: exact this