Theorem: Lean4Axiomatic.Rational.lt_subst_neg (Lean4Axiomatic/Rational/Order.lean)

State before:
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p₁ p₂ : ℚ
⊢ p₁ < p₂ → -p₂ < -p₁
Tactic:
intro (_ : p₁ < p₂)
State after:
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p₁ p₂ : ℚ
x✝ : p₁ < p₂
⊢ -p₂ < -p₁

State before:
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p₁ p₂ : ℚ
x✝ : p₁ < p₂
⊢ -p₂ < -p₁
Tactic:
calc
  _ ≃ -p₂     := eqv_refl
  _ ≃ -1 * p₂ := eqv_symm mul_neg_one
  _ < -1 * p₁ := lt_substR_mul_neg neg_one_lt_zero ‹p₁ < p₂›
  _ ≃ -p₁     := mul_neg_one
State after:
no goals