[GOAL]
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : CommSemigroup β
f : α → β
n m : α
⊢ f (min n m) * f (max n m) = f n * f m
[PROOFSTEP]
cases' le_total n m with h h
[GOAL]
case inl
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : CommSemigroup β
f : α → β
n m : α
h : n ≤ m
⊢ f (min n m) * f (max n m) = f n * f m
[PROOFSTEP]
simp [h, mul_comm]
[GOAL]
case inr
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : CommSemigroup β
f : α → β
n m : α
h : m ≤ n
⊢ f (min n m) * f (max n m) = f n * f m
[PROOFSTEP]
simp [h, mul_comm]
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : Mul 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
⊢ a₁ * b₁ < a₂ * b₂ → a₁ < a₂ ∨ b₁ < b₂
[PROOFSTEP]
contrapose!
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : Mul 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
⊢ a₂ ≤ a₁ ∧ b₂ ≤ b₁ → a₂ * b₂ ≤ a₁ * b₁
[PROOFSTEP]
exact fun h => mul_le_mul' h.1 h.2
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : Mul 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
⊢ a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ < b₂
[PROOFSTEP]
contrapose!
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : Mul 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
⊢ a₂ < a₁ ∧ b₂ ≤ b₁ → a₂ * b₂ < a₁ * b₁
[PROOFSTEP]
exact fun h => mul_lt_mul_of_lt_of_le h.1 h.2
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : Mul 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
⊢ a₁ * b₁ ≤ a₂ * b₂ → a₁ < a₂ ∨ b₁ ≤ b₂
[PROOFSTEP]
contrapose!
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : Mul 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
⊢ a₂ ≤ a₁ ∧ b₂ < b₁ → a₂ * b₂ < a₁ * b₁
[PROOFSTEP]
exact fun h => mul_lt_mul_of_le_of_lt h.1 h.2
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : Mul 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
⊢ a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ ≤ b₂
[PROOFSTEP]
contrapose!
[GOAL]
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : Mul 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
⊢ a₂ < a₁ ∧ b₂ < b₁ → a₂ * b₂ < a₁ * b₁
[PROOFSTEP]
exact fun h => mul_lt_mul_of_lt_of_lt h.1 h.2
[GOAL]
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : Mul α
inst✝³ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
inst✝² : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
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
⊢ a₁ * b₁ < a₂ * b₂ ↔ a₁ < a₂ ∨ b₁ < b₂
[PROOFSTEP]
refine' ⟨lt_or_lt_of_mul_lt_mul, fun h => _⟩
[GOAL]
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : Mul α
inst✝³ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
inst✝² : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
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
h : a₁ < a₂ ∨ b₁ < b₂
⊢ a₁ * b₁ < a₂ * b₂
[PROOFSTEP]
cases' h with ha' hb'
[GOAL]
case inl
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : Mul α
inst✝³ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
inst✝² : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
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
ha' : a₁ < a₂
⊢ a₁ * b₁ < a₂ * b₂
[PROOFSTEP]
exact mul_lt_mul_of_lt_of_le ha' hb
[GOAL]
case inr
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : Mul α
inst✝³ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
inst✝² : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
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
hb' : b₁ < b₂
⊢ a₁ * b₁ < a₂ * b₂
[PROOFSTEP]
exact mul_lt_mul_of_le_of_lt ha hb'
