Theorem: mul_le_mul_iff_of_ge
File: Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1
a₁ a₂ b₁ b₂ : α
ha : a₁ ≤ a₂
hb : b₁ ≤ b₂
this✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
this : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
⊢ a₂ * b₂ ≤ a₁ * b₁ ↔ a₁ = a₂ ∧ b₁ = b₂
state_after: α : Type u_1
β : Type u_2
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1
a₁ a₂ b₁ b₂ : α
ha : a₁ ≤ a₂
hb : b₁ ≤ b₂
this✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
this : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
h : a₂ * b₂ ≤ a₁ * b₁
⊢ a₁ = a₂ ∧ b₁ = b₂
tactic: refine ⟨fun h ↦ ?_, by rintro ⟨rfl, rfl⟩; rfl⟩
state_before: α : Type u_1
β : Type u_2
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1
a₁ a₂ b₁ b₂ : α
ha : a₁ ≤ a₂
hb : b₁ ≤ b₂
this✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
this : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
h : a₂ * b₂ ≤ a₁ * b₁
⊢ a₁ = a₂ ∧ b₁ = b₂
state_after: α : Type u_1
β : Type u_2
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1
a₁ a₂ b₁ b₂ : α
ha : a₁ ≤ a₂
hb : b₁ ≤ b₂
this✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
this : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
h : a₂ * b₂ ≤ a₁ * b₁
⊢ ¬a₁ < a₂ ∧ ¬b₁ < b₂
tactic: simp only [eq_iff_le_not_lt, ha, hb, true_and]
state_before: α : Type u_1
β : Type u_2
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1
a₁ a₂ b₁ b₂ : α
ha : a₁ ≤ a₂
hb : b₁ ≤ b₂
this✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
this : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
h : a₂ * b₂ ≤ a₁ * b₁
⊢ ¬a₁ < a₂ ∧ ¬b₁ < b₂
state_after: case refine_1
α : Type u_1
β : Type u_2
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1
a₁ a₂ b₁ b₂ : α
ha✝ : a₁ ≤ a₂
hb : b₁ ≤ b₂
this✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
this : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
h : a₂ * b₂ ≤ a₁ * b₁
ha : a₁ < a₂
⊢ a₁ * b₁ < a₂ * b₂

case refine_2
α : Type u_1
β : Type u_2
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1
a₁ a₂ b₁ b₂ : α
ha : a₁ ≤ a₂
hb✝ : b₁ ≤ b₂
this✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
this : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
h : a₂ * b₂ ≤ a₁ * b₁
hb : b₁ < b₂
⊢ a₁ * b₁ < a₂ * b₂
tactic: refine ⟨fun ha ↦ h.not_lt ?_, fun hb ↦ h.not_lt ?_⟩
state_before: case refine_1
α : Type u_1
β : Type u_2
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1
a₁ a₂ b₁ b₂ : α
ha✝ : a₁ ≤ a₂
hb : b₁ ≤ b₂
this✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
this : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
h : a₂ * b₂ ≤ a₁ * b₁
ha : a₁ < a₂
⊢ a₁ * b₁ < a₂ * b₂

case refine_2
α : Type u_1
β : Type u_2
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1
a₁ a₂ b₁ b₂ : α
ha : a₁ ≤ a₂
hb✝ : b₁ ≤ b₂
this✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
this : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
h : a₂ * b₂ ≤ a₁ * b₁
hb : b₁ < b₂
⊢ a₁ * b₁ < a₂ * b₂
state_after: no goals
tactic: exacts [mul_lt_mul_of_lt_of_le ha hb, mul_lt_mul_of_le_of_lt ha hb]