Theorem: Lean4Axiomatic.Rational.le_substL_eqv (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) (_ : p₁ ≤ 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
p₁ p₂ q : ℚ
x✝¹ : p₁ ≃ p₂
x✝ : 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✝¹ : p₁ ≃ p₂
x✝ : p₁ ≤ q
⊢ p₂ ≤ q
Tactic:
have : sgn (p₁ - q) ≄ 1 := le_sgn.mp ‹p₁ ≤ 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
p₁ p₂ q : ℚ
x✝¹ : p₁ ≃ p₂
x✝ : p₁ ≤ q
this : sgn (p₁ - q) ≄ 1
⊢ 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✝¹ : p₁ ≃ p₂
x✝ : p₁ ≤ q
this : sgn (p₁ - q) ≄ 1
⊢ p₂ ≤ q
Tactic:
have : sgn (p₁ - q) ≃ sgn (p₂ - q) := sgn_subst (sub_substL ‹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✝¹ : p₁ ≃ p₂
x✝ : p₁ ≤ q
this✝ : sgn (p₁ - q) ≄ 1
this : sgn (p₁ - q) ≃ sgn (p₂ - 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✝¹ : p₁ ≃ p₂
x✝ : p₁ ≤ q
this✝ : sgn (p₁ - q) ≄ 1
this : sgn (p₁ - q) ≃ sgn (p₂ - q)
⊢ p₂ ≤ q
Tactic:
have : sgn (p₂ - q) ≄ 1 := AA.neqv_substL this ‹sgn (p₁ - 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
p₁ p₂ q : ℚ
x✝¹ : p₁ ≃ p₂
x✝ : p₁ ≤ q
this✝¹ : sgn (p₁ - q) ≄ 1
this✝ : sgn (p₁ - q) ≃ sgn (p₂ - q)
this : sgn (p₂ - q) ≄ 1
⊢ 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✝¹ : p₁ ≃ p₂
x✝ : p₁ ≤ q
this✝¹ : sgn (p₁ - q) ≄ 1
this✝ : sgn (p₁ - q) ≃ sgn (p₂ - q)
this : sgn (p₂ - q) ≄ 1
⊢ p₂ ≤ q
Tactic:
have : 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✝¹ : p₁ ≃ p₂
x✝ : p₁ ≤ q
this✝² : sgn (p₁ - q) ≄ 1
this✝¹ : sgn (p₁ - q) ≃ sgn (p₂ - q)
this✝ : sgn (p₂ - q) ≄ 1
this : p₂ ≤ 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✝¹ : p₁ ≃ p₂
x✝ : p₁ ≤ q
this✝² : sgn (p₁ - q) ≄ 1
this✝¹ : sgn (p₁ - q) ≃ sgn (p₂ - q)
this✝ : sgn (p₂ - q) ≄ 1
this : p₂ ≤ q
⊢ p₂ ≤ q
Tactic:
exact this
State after:
no goals