Theorem: Real.sSup_smul_of_nonneg
File: Mathlib/Data/Real/Pointwise.lean

tactic: obtain rfl | hs := s.eq_empty_or_nonempty
state_before: ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
s : Set ℝ
⊢ sSup (a • s) = a • sSup s
state_after: case inl
ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
⊢ sSup (a • ∅) = a • sSup ∅

case inr
ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
s : Set ℝ
hs : s.Nonempty
⊢ sSup (a • s) = a • sSup s

tactic: obtain rfl | ha' := ha.eq_or_lt
state_before: case inr
ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
s : Set ℝ
hs : s.Nonempty
⊢ sSup (a • s) = a • sSup s
state_after: case inr.inl
ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
s : Set ℝ
hs : s.Nonempty
ha : 0 ≤ 0
⊢ sSup (0 • s) = 0 • sSup s

case inr.inr
ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
s : Set ℝ
hs : s.Nonempty
ha' : 0 < a
⊢ sSup (a • s) = a • sSup s

tactic: by_cases h : BddAbove s
state_before: case inr.inr
ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
s : Set ℝ
hs : s.Nonempty
ha' : 0 < a
⊢ sSup (a • s) = a • sSup s
state_after: case pos
ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
s : Set ℝ
hs : s.Nonempty
ha' : 0 < a
h : BddAbove s
⊢ sSup (a • s) = a • sSup s

case neg
ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
s : Set ℝ
hs : s.Nonempty
ha' : 0 < a
h : ¬BddAbove s
⊢ sSup (a • s) = a • sSup s

tactic: rw [smul_set_empty, Real.sSup_empty, smul_zero]
state_before: case inl
ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
⊢ sSup (a • ∅) = a • sSup ∅
state_after: no goals

tactic: rw [zero_smul_set hs, zero_smul]
state_before: case inr.inl
ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
s : Set ℝ
hs : s.Nonempty
ha : 0 ≤ 0
⊢ sSup (0 • s) = 0 • sSup s
state_after: case inr.inl
ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
s : Set ℝ
hs : s.Nonempty
ha : 0 ≤ 0
⊢ sSup 0 = 0

tactic: exact csSup_singleton 0
state_before: case inr.inl
ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
s : Set ℝ
hs : s.Nonempty
ha : 0 ≤ 0
⊢ sSup 0 = 0
state_after: no goals

tactic: exact ((OrderIso.smulRight ha').map_csSup' hs h).symm
state_before: case pos
ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
s : Set ℝ
hs : s.Nonempty
ha' : 0 < a
h : BddAbove s
⊢ sSup (a • s) = a • sSup s
state_after: no goals

tactic: rw [Real.sSup_of_not_bddAbove (mt (bddAbove_smul_iff_of_pos ha').1 h),
    Real.sSup_of_not_bddAbove h, smul_zero]
state_before: case neg
ι : Sort u_1
α : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : MulActionWithZero α ℝ
inst✝ : OrderedSMul α ℝ
a : α
ha : 0 ≤ a
s : Set ℝ
hs : s.Nonempty
ha' : 0 < a
h : ¬BddAbove s
⊢ sSup (a • s) = a • sSup s
state_after: no goals