case inl
α : Type u_1
inst✝ : LinearOrderedField α
a✝ a b c : α
hab : a ≤ b
hc : 0 ≤ c
h✝ : 0 = c
⊢ (fun x => x * c) '' Icc a b = Icc (a * c) (b * c)

case inr
α : Type u_1
inst✝ : LinearOrderedField α
a✝ a b c : α
hab : a ≤ b
hc : 0 ≤ c
h✝ : 0 < c
⊢ (fun x => x * c) '' Icc a b = Icc (a * c) (b * c)
α : Type u_1
inst✝ : LinearOrderedField α
a✝ a b c : α
hab : a ≤ b
hc : 0 ≤ c
⊢ (fun x => x * c) '' Icc a b = Icc (a * c) (b * c)
cases eq_or_lt_of_le hc
no goals
case inr
α : Type u_1
inst✝ : LinearOrderedField α
a✝ a b c : α
hab : a ≤ b
hc : 0 ≤ c
h✝ : 0 < c
⊢ (fun x => x * c) '' Icc a b = Icc (a * c) (b * c)
exact image_mul_right_Icc' a b ‹0 < c›
case inl
α : Type u_1
inst✝ : LinearOrderedField α
a✝ a b : α
hab : a ≤ b
hc : 0 ≤ 0
⊢ (fun x => x * 0) '' Icc a b = Icc (a * 0) (b * 0)
case inl
α : Type u_1
inst✝ : LinearOrderedField α
a✝ a b c : α
hab : a ≤ b
hc : 0 ≤ c
h✝ : 0 = c
⊢ (fun x => x * c) '' Icc a b = Icc (a * c) (b * c)
subst c
no goals
case inl
α : Type u_1
inst✝ : LinearOrderedField α
a✝ a b : α
hab : a ≤ b
hc : 0 ≤ 0
⊢ (fun x => x * 0) '' Icc a b = Icc (a * 0) (b * 0)
simp [(nonempty_Icc.2 hab).image_const]