Theorem: Measurable.isGLB
File: Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean

tactic: change ∀ b, IsGLB (range fun i => f i b) (g b) at hg
state_before: α : Type u_3
β : Type ?u.1210770
γ : Type ?u.1210773
γ₂ : Type ?u.1210776
δ : Type u_2
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_1
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsGLB {a | ∃ i, f i b = a} (g b)
⊢ Measurable g

tactic: rw [‹BorelSpace α›.measurable_eq, borel_eq_generateFrom_Iio α]
state_before: α : Type u_3
β : Type ?u.1210770
γ : Type ?u.1210773
γ₂ : Type ?u.1210776
δ : Type u_2
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_1
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsGLB (range fun i => f i b) (g b)
⊢ Measurable g

tactic: apply measurable_generateFrom
state_before: α : Type u_3
β : Type ?u.1210770
γ : Type ?u.1210773
γ₂ : Type ?u.1210776
δ : Type u_2
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_1
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsGLB (range fun i => f i b) (g b)
⊢ Measurable g

tactic: rintro _ ⟨a, rfl⟩
state_before: case h
α : Type u_3
β : Type ?u.1210770
γ : Type ?u.1210773
γ₂ : Type ?u.1210776
δ : Type u_2
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_1
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsGLB (range fun i => f i b) (g b)
⊢ ∀ (t : Set α), t ∈ range Iio → MeasurableSet (g ⁻¹' t)

tactic: simp_rw [Set.preimage, mem_Iio, isGLB_lt_iff (hg _), exists_range_iff, setOf_exists]
state_before: case h.intro
α : Type u_3
β : Type ?u.1210770
γ : Type ?u.1210773
γ₂ : Type ?u.1210776
δ : Type u_2
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_1
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsGLB (range fun i => f i b) (g b)
a : α
⊢ MeasurableSet (g ⁻¹' Iio a)

tactic: exact MeasurableSet.iUnion fun i => hf i (isOpen_gt' _).measurableSet
state_before: case h.intro
α : Type u_3
β : Type ?u.1210770
γ : Type ?u.1210773
γ₂ : Type ?u.1210776
δ : Type u_2
ι✝ : Sort y
s t u : Set α
inst✝¹³ : TopologicalSpace α
inst✝¹² : MeasurableSpace α
inst✝¹¹ : BorelSpace α
inst✝¹⁰ : TopologicalSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_1
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsGLB (range fun i => f i b) (g b)
a : α
⊢ MeasurableSet (⋃ (i : ι), {x | f i x < a})