Theorem: Lean4Axiomatic.Rational.close_substM_eqv (Lean4Axiomatic/Rational/Metric.lean)

State before:
ℕ ℤ ℚ : Type
inst✝¹² : Natural ℕ
inst✝¹¹ : Integer ℤ
inst✝¹⁰ : Core ℚ
inst✝⁹ : Addition ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Order ℚ
inst✝¹ : MinMax ℚ
inst✝ : Metric ℚ
ε₁ ε₂ p q : ℚ
⊢ ε₁ ≃ ε₂ → p ⊢ε₁⊣ q → p ⊢ε₂⊣ q
Tactic:
intro (_ : ε₁ ≃ ε₂) (_ : p ⊢ε₁⊣ q)
State after:
ℕ ℤ ℚ : Type
inst✝¹² : Natural ℕ
inst✝¹¹ : Integer ℤ
inst✝¹⁰ : Core ℚ
inst✝⁹ : Addition ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Order ℚ
inst✝¹ : MinMax ℚ
inst✝ : Metric ℚ
ε₁ ε₂ p q : ℚ
x✝¹ : ε₁ ≃ ε₂
x✝ : p ⊢ε₁⊣ q
⊢ p ⊢ε₂⊣ q

State before:
ℕ ℤ ℚ : Type
inst✝¹² : Natural ℕ
inst✝¹¹ : Integer ℤ
inst✝¹⁰ : Core ℚ
inst✝⁹ : Addition ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Order ℚ
inst✝¹ : MinMax ℚ
inst✝ : Metric ℚ
ε₁ ε₂ p q : ℚ
x✝¹ : ε₁ ≃ ε₂
x✝ : p ⊢ε₁⊣ q
⊢ p ⊢ε₂⊣ q
Tactic:
have : dist p q ≤ ε₁ := close_dist.mp ‹p ⊢ε₁⊣ q›
State after:
ℕ ℤ ℚ : Type
inst✝¹² : Natural ℕ
inst✝¹¹ : Integer ℤ
inst✝¹⁰ : Core ℚ
inst✝⁹ : Addition ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Order ℚ
inst✝¹ : MinMax ℚ
inst✝ : Metric ℚ
ε₁ ε₂ p q : ℚ
x✝¹ : ε₁ ≃ ε₂
x✝ : p ⊢ε₁⊣ q
this : dist p q ≤ ε₁
⊢ p ⊢ε₂⊣ q

State before:
ℕ ℤ ℚ : Type
inst✝¹² : Natural ℕ
inst✝¹¹ : Integer ℤ
inst✝¹⁰ : Core ℚ
inst✝⁹ : Addition ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Order ℚ
inst✝¹ : MinMax ℚ
inst✝ : Metric ℚ
ε₁ ε₂ p q : ℚ
x✝¹ : ε₁ ≃ ε₂
x✝ : p ⊢ε₁⊣ q
this : dist p q ≤ ε₁
⊢ p ⊢ε₂⊣ q
Tactic:
have : dist p q ≤ ε₂ := le_substR_eqv ‹ε₁ ≃ ε₂› this
State after:
ℕ ℤ ℚ : Type
inst✝¹² : Natural ℕ
inst✝¹¹ : Integer ℤ
inst✝¹⁰ : Core ℚ
inst✝⁹ : Addition ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Order ℚ
inst✝¹ : MinMax ℚ
inst✝ : Metric ℚ
ε₁ ε₂ p q : ℚ
x✝¹ : ε₁ ≃ ε₂
x✝ : p ⊢ε₁⊣ q
this✝ : dist p q ≤ ε₁
this : dist p q ≤ ε₂
⊢ p ⊢ε₂⊣ q

State before:
ℕ ℤ ℚ : Type
inst✝¹² : Natural ℕ
inst✝¹¹ : Integer ℤ
inst✝¹⁰ : Core ℚ
inst✝⁹ : Addition ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Order ℚ
inst✝¹ : MinMax ℚ
inst✝ : Metric ℚ
ε₁ ε₂ p q : ℚ
x✝¹ : ε₁ ≃ ε₂
x✝ : p ⊢ε₁⊣ q
this✝ : dist p q ≤ ε₁
this : dist p q ≤ ε₂
⊢ p ⊢ε₂⊣ q
Tactic:
have : p ⊢ε₂⊣ q := close_dist.mpr this
State after:
ℕ ℤ ℚ : Type
inst✝¹² : Natural ℕ
inst✝¹¹ : Integer ℤ
inst✝¹⁰ : Core ℚ
inst✝⁹ : Addition ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Order ℚ
inst✝¹ : MinMax ℚ
inst✝ : Metric ℚ
ε₁ ε₂ p q : ℚ
x✝¹ : ε₁ ≃ ε₂
x✝ : p ⊢ε₁⊣ q
this✝¹ : dist p q ≤ ε₁
this✝ : dist p q ≤ ε₂
this : p ⊢ε₂⊣ q
⊢ p ⊢ε₂⊣ q

State before:
ℕ ℤ ℚ : Type
inst✝¹² : Natural ℕ
inst✝¹¹ : Integer ℤ
inst✝¹⁰ : Core ℚ
inst✝⁹ : Addition ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Order ℚ
inst✝¹ : MinMax ℚ
inst✝ : Metric ℚ
ε₁ ε₂ p q : ℚ
x✝¹ : ε₁ ≃ ε₂
x✝ : p ⊢ε₁⊣ q
this✝¹ : dist p q ≤ ε₁
this✝ : dist p q ≤ ε₂
this : p ⊢ε₂⊣ q
⊢ p ⊢ε₂⊣ q
Tactic:
exact this
State after:
no goals