α : Type u_1
β : Type u_2
γ : Type ?u.463632
δ : Type ?u.463635
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeInf β
inst✝ : ContinuousInf β
f g : α →ₘ[μ] β
⊢ ↑(f ⊓ g) ≤ᵐ[μ] ↑f
α : Type u_1
β : Type u_2
γ : Type ?u.463632
δ : Type ?u.463635
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeInf β
inst✝ : ContinuousInf β
f g : α →ₘ[μ] β
⊢ f ⊓ g ≤ f
rw [← coeFn_le]
case h
α : Type u_1
β : Type u_2
γ : Type ?u.463632
δ : Type ?u.463635
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeInf β
inst✝ : ContinuousInf β
f g : α →ₘ[μ] β
a✝ : α
ha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝
⊢ ↑(f ⊓ g) a✝ ≤ ↑f a✝
α : Type u_1
β : Type u_2
γ : Type ?u.463632
δ : Type ?u.463635
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeInf β
inst✝ : ContinuousInf β
f g : α →ₘ[μ] β
⊢ ↑(f ⊓ g) ≤ᵐ[μ] ↑f
filter_upwards [coeFn_inf f g] with _ ha
case h
α : Type u_1
β : Type u_2
γ : Type ?u.463632
δ : Type ?u.463635
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeInf β
inst✝ : ContinuousInf β
f g : α →ₘ[μ] β
a✝ : α
ha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝
⊢ ↑f a✝ ⊓ ↑g a✝ ≤ ↑f a✝
case h
α : Type u_1
β : Type u_2
γ : Type ?u.463632
δ : Type ?u.463635
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeInf β
inst✝ : ContinuousInf β
f g : α →ₘ[μ] β
a✝ : α
ha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝
⊢ ↑(f ⊓ g) a✝ ≤ ↑f a✝
rw [ha]
no goals
case h
α : Type u_1
β : Type u_2
γ : Type ?u.463632
δ : Type ?u.463635
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeInf β
inst✝ : ContinuousInf β
f g : α →ₘ[μ] β
a✝ : α
ha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝
⊢ ↑f a✝ ⊓ ↑g a✝ ≤ ↑f a✝
exact inf_le_left