Theorem: Lean4Axiomatic.Rational.le_substL_add (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 : ℚ
⊢ p₁ ≤ p₂ → p₁ + q ≤ p₂ + q
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 : ℚ
x✝ : p₁ ≤ p₂
⊢ p₁ + q ≤ p₂ + q

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 : ℚ
x✝ : p₁ ≤ p₂
⊢ p₁ + q ≤ p₂ + q
Tactic:
have : sgn (p₁ - p₂) ≄ 1 := le_sgn.mp ‹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 : ℚ
x✝ : p₁ ≤ p₂
this : sgn (p₁ - p₂) ≄ 1
⊢ p₁ + q ≤ p₂ + q

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 : ℚ
x✝ : p₁ ≤ p₂
this : sgn (p₁ - p₂) ≄ 1
⊢ p₁ + q ≤ p₂ + q
Tactic:
have : sgn ((p₁ + q) - (p₂ + q)) ≃ sgn (p₁ - p₂) := sgn_subst sub_cancelR_add
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 : ℚ
x✝ : p₁ ≤ p₂
this✝ : sgn (p₁ - p₂) ≄ 1
this : sgn (p₁ + q - (p₂ + q)) ≃ sgn (p₁ - p₂)
⊢ p₁ + q ≤ p₂ + q

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 : ℚ
x✝ : p₁ ≤ p₂
this✝ : sgn (p₁ - p₂) ≄ 1
this : sgn (p₁ + q - (p₂ + q)) ≃ sgn (p₁ - p₂)
⊢ p₁ + q ≤ p₂ + q
Tactic:
have : sgn ((p₁ + q) - (p₂ + q)) ≄ 1 :=
  AA.neqv_substL (Rel.symm this) ‹sgn (p₁ - p₂) ≄ 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
p₁ p₂ q : ℚ
x✝ : p₁ ≤ p₂
this✝¹ : sgn (p₁ - p₂) ≄ 1
this✝ : sgn (p₁ + q - (p₂ + q)) ≃ sgn (p₁ - p₂)
this : sgn (p₁ + q - (p₂ + q)) ≄ 1
⊢ p₁ + q ≤ p₂ + q

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 : ℚ
x✝ : p₁ ≤ p₂
this✝¹ : sgn (p₁ - p₂) ≄ 1
this✝ : sgn (p₁ + q - (p₂ + q)) ≃ sgn (p₁ - p₂)
this : sgn (p₁ + q - (p₂ + q)) ≄ 1
⊢ p₁ + q ≤ p₂ + q
Tactic:
have : p₁ + q ≤ p₂ + q := le_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
p₁ p₂ q : ℚ
x✝ : p₁ ≤ p₂
this✝² : sgn (p₁ - p₂) ≄ 1
this✝¹ : sgn (p₁ + q - (p₂ + q)) ≃ sgn (p₁ - p₂)
this✝ : sgn (p₁ + q - (p₂ + q)) ≄ 1
this : p₁ + q ≤ p₂ + q
⊢ p₁ + q ≤ p₂ + q

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 : ℚ
x✝ : p₁ ≤ p₂
this✝² : sgn (p₁ - p₂) ≄ 1
this✝¹ : sgn (p₁ + q - (p₂ + q)) ≃ sgn (p₁ - p₂)
this✝ : sgn (p₁ + q - (p₂ + q)) ≄ 1
this : p₁ + q ≤ p₂ + q
⊢ p₁ + q ≤ p₂ + q
Tactic:
exact this
State after:
no goals