Theorem: LinearOrderedField.smul_Ioo
File: Mathlib/Algebra/Order/Pointwise.lean

tactic: ext x
state_before: α : Type u_1
K : Type u_2
inst✝ : LinearOrderedField K
a b r : K
hr : 0 < r
⊢ r • Ioo a b = Ioo (r • a) (r • b)
state_after: case h
α : Type u_1
K : Type u_2
inst✝ : LinearOrderedField K
a b r : K
hr : 0 < r
x : K
⊢ x ∈ r • Ioo a b ↔ x ∈ Ioo (r • a) (r • b)

tactic: simp only [mem_smul_set, smul_eq_mul, mem_Ioo]
state_before: case h
α : Type u_1
K : Type u_2
inst✝ : LinearOrderedField K
a b r : K
hr : 0 < r
x : K
⊢ x ∈ r • Ioo a b ↔ x ∈ Ioo (r • a) (r • b)
state_after: case h
α : Type u_1
K : Type u_2
inst✝ : LinearOrderedField K
a b r : K
hr : 0 < r
x : K
⊢ (∃ y, (a < y ∧ y < b) ∧ r * y = x) ↔ r * a < x ∧ x < r * b

tactic: constructor
state_before: case h
α : Type u_1
K : Type u_2
inst✝ : LinearOrderedField K
a b r : K
hr : 0 < r
x : K
⊢ (∃ y, (a < y ∧ y < b) ∧ r * y = x) ↔ r * a < x ∧ x < r * b
state_after: case h.mp
α : Type u_1
K : Type u_2
inst✝ : LinearOrderedField K
a b r : K
hr : 0 < r
x : K
⊢ (∃ y, (a < y ∧ y < b) ∧ r * y = x) → r * a < x ∧ x < r * b

case h.mpr
α : Type u_1
K : Type u_2
inst✝ : LinearOrderedField K
a b r : K
hr : 0 < r
x : K
⊢ r * a < x ∧ x < r * b → ∃ y, (a < y ∧ y < b) ∧ r * y = x

tactic: rintro ⟨a, ⟨a_h_left_left, a_h_left_right⟩, rfl⟩
state_before: case h.mp
α : Type u_1
K : Type u_2
inst✝ : LinearOrderedField K
a b r : K
hr : 0 < r
x : K
⊢ (∃ y, (a < y ∧ y < b) ∧ r * y = x) → r * a < x ∧ x < r * b
state_after: case h.mp.intro.intro.intro
α : Type u_1
K : Type u_2
inst✝ : LinearOrderedField K
a✝ b r : K
hr : 0 < r
a : K
a_h_left_left : a✝ < a
a_h_left_right : a < b
⊢ r * a✝ < r * a ∧ r * a < r * b

tactic: constructor
state_before: case h.mp.intro.intro.intro
α : Type u_1
K : Type u_2
inst✝ : LinearOrderedField K
a✝ b r : K
hr : 0 < r
a : K
a_h_left_left : a✝ < a
a_h_left_right : a < b
⊢ r * a✝ < r * a ∧ r * a < r * b
state_after: case h.mp.intro.intro.intro.left
α : Type u_1
K : Type u_2
inst✝ : LinearOrderedField K
a✝ b r : K
hr : 0 < r
a : K
a_h_left_left : a✝ < a
a_h_left_right : a < b
⊢ r * a✝ < r * a

case h.mp.intro.intro.intro.right
α : Type u_1
K : Type u_2
inst✝ : LinearOrderedField K
a✝ b r : K
hr : 0 < r
a : K
a_h_left_left : a✝ < a
a_h_left_right : a < b
⊢ r * a < r * b

tactic: exact (mul_lt_mul_left hr).mpr a_h_left_left
state_before: case h.mp.intro.intro.intro.left
α : Type u_1
K : Type u_2
inst✝ : LinearOrderedField K
a✝ b r : K
hr : 0 < r
a : K
a_h_left_left : a✝ < a
a_h_left_right : a < b
⊢ r * a✝ < r * a
state_after: no goals

tactic: exact (mul_lt_mul_left hr).mpr a_h_left_right
state_before: case h.mp.intro.intro.intro.right
α : Type u_1
K : Type u_2
inst✝ : LinearOrderedField K
a✝ b r : K
hr : 0 < r
a : K
a_h_left_left : a✝ < a
a_h_left_right : a < b
⊢ r * a < r * b
state_after: no goals

tactic: rintro ⟨a_left, a_right⟩
state_before: case h.mpr
α : Type u_1
K : Type u_2
inst✝ : LinearOrderedField K
a b r : K
hr : 0 < r
x : K
⊢ r * a < x ∧ x < r * b → ∃ y, (a < y ∧ y < b) ∧ r * y = x
state_after: case h.mpr.intro
α : Type u_1
K : Type u_2
inst✝ : LinearOrderedField K
a b r : K
hr : 0 < r
x : K
a_left : r * a < x
a_right : x < r * b
⊢ ∃ y, (a < y ∧ y < b) ∧ r * y = x

tactic: use x / r
state_before: case h.mpr.intro
α : Type u_1
K : Type u_2
inst✝ : LinearOrderedField K
a b r : K
hr : 0 < r
x : K
a_left : r * a < x
a_right : x < r * b
⊢ ∃ y, (a < y ∧ y < b) ∧ r * y = x
state_after: case h
α : Type u_1
K : Type u_2
inst✝ : LinearOrderedField K
a b r : K
hr : 0 < r
x : K
a_left : r * a < x
a_right : x < r * b
⊢ (a < x / r ∧ x / r < b) ∧ r * (x / r) = x

tactic: refine ⟨⟨(lt_div_iff' hr).mpr a_left, (div_lt_iff' hr).mpr a_right⟩, ?_⟩
state_before: case h
α : Type u_1
K : Type u_2
inst✝ : LinearOrderedField K
a b r : K
hr : 0 < r
x : K
a_left : r * a < x
a_right : x < r * b
⊢ (a < x / r ∧ x / r < b) ∧ r * (x / r) = x
state_after: case h
α : Type u_1
K : Type u_2
inst✝ : LinearOrderedField K
a b r : K
hr : 0 < r
x : K
a_left : r * a < x
a_right : x < r * b
⊢ r * (x / r) = x

tactic: rw [mul_div_cancel₀ _ (ne_of_gt hr)]
state_before: case h
α : Type u_1
K : Type u_2
inst✝ : LinearOrderedField K
a b r : K
hr : 0 < r
x : K
a_left : r * a < x
a_right : x < r * b
⊢ r * (x / r) = x
state_after: no goals