Theorem: Lean4Axiomatic.Rational.lt_substL_mul_pos (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
p q r : ℚ
⊢ r > 0 → p < q → p * r < q * r
Tactic:
intro (_ : r > 0) (_ : p < q)
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
p q r : ℚ
x✝¹ : r > 0
x✝ : p < q
⊢ p * r < q * r

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
p q r : ℚ
x✝¹ : r > 0
x✝ : p < q
⊢ p * r < q * r
Tactic:
have : sgn r ≃ 1 := gt_zero_sgn.mp ‹r > 0›
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
p q r : ℚ
x✝¹ : r > 0
x✝ : p < q
this : sgn r ≃ 1
⊢ p * r < q * r

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
p q r : ℚ
x✝¹ : r > 0
x✝ : p < q
this : sgn r ≃ 1
⊢ p * r < q * r
Tactic:
have : sgn (p - q) ≃ -1 := lt_sgn.mp ‹p < q›
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
p q r : ℚ
x✝¹ : r > 0
x✝ : p < q
this✝ : sgn r ≃ 1
this : sgn (p - q) ≃ -1
⊢ p * r < q * r

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
p q r : ℚ
x✝¹ : r > 0
x✝ : p < q
this✝ : sgn r ≃ 1
this : sgn (p - q) ≃ -1
⊢ p * r < q * r
Tactic:
have : sgn (p * r - q * r) ≃ sgn (p - q) :=
  sgn_sub_cancelR_mul_pos ‹sgn r ≃ 1›
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
p q r : ℚ
x✝¹ : r > 0
x✝ : p < q
this✝¹ : sgn r ≃ 1
this✝ : sgn (p - q) ≃ -1
this : sgn (p * r - q * r) ≃ sgn (p - q)
⊢ p * r < q * r

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
p q r : ℚ
x✝¹ : r > 0
x✝ : p < q
this✝¹ : sgn r ≃ 1
this✝ : sgn (p - q) ≃ -1
this : sgn (p * r - q * r) ≃ sgn (p - q)
⊢ p * r < q * r
Tactic:
have : sgn (p * r - q * r) ≃ -1 :=
  AA.eqv_substL (Rel.symm this) ‹sgn (p - q) ≃ -1›
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
p q r : ℚ
x✝¹ : r > 0
x✝ : p < q
this✝² : sgn r ≃ 1
this✝¹ : sgn (p - q) ≃ -1
this✝ : sgn (p * r - q * r) ≃ sgn (p - q)
this : sgn (p * r - q * r) ≃ -1
⊢ p * r < q * r

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
p q r : ℚ
x✝¹ : r > 0
x✝ : p < q
this✝² : sgn r ≃ 1
this✝¹ : sgn (p - q) ≃ -1
this✝ : sgn (p * r - q * r) ≃ sgn (p - q)
this : sgn (p * r - q * r) ≃ -1
⊢ p * r < q * r
Tactic:
have : p * r < q * r := lt_sgn.mpr this
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
p q r : ℚ
x✝¹ : r > 0
x✝ : p < q
this✝³ : sgn r ≃ 1
this✝² : sgn (p - q) ≃ -1
this✝¹ : sgn (p * r - q * r) ≃ sgn (p - q)
this✝ : sgn (p * r - q * r) ≃ -1
this : p * r < q * r
⊢ p * r < q * r

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
p q r : ℚ
x✝¹ : r > 0
x✝ : p < q
this✝³ : sgn r ≃ 1
this✝² : sgn (p - q) ≃ -1
this✝¹ : sgn (p * r - q * r) ≃ sgn (p - q)
this✝ : sgn (p * r - q * r) ≃ -1
this : p * r < q * r
⊢ p * r < q * r
Tactic:
exact this
State after:
no goals