α : Type u_1
β : Type u_2
γ : Type ?u.455830
δ : Type ?u.455833
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeSup β
inst✝ : ContinuousSup β
f g : α →ₘ[μ] β
⊢ ↑g ≤ᵐ[μ] ↑(f ⊔ g)
α : Type u_1
β : Type u_2
γ : Type ?u.455830
δ : Type ?u.455833
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeSup β
inst✝ : ContinuousSup β
f g : α →ₘ[μ] β
⊢ g ≤ f ⊔ g
rw [← coeFn_le]
case h
α : Type u_1
β : Type u_2
γ : Type ?u.455830
δ : Type ?u.455833
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeSup β
inst✝ : ContinuousSup β
f g : α →ₘ[μ] β
a✝ : α
ha : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝
⊢ ↑g a✝ ≤ ↑(f ⊔ g) a✝
α : Type u_1
β : Type u_2
γ : Type ?u.455830
δ : Type ?u.455833
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeSup β
inst✝ : ContinuousSup β
f g : α →ₘ[μ] β
⊢ ↑g ≤ᵐ[μ] ↑(f ⊔ g)
filter_upwards [coeFn_sup f g] with _ ha
case h
α : Type u_1
β : Type u_2
γ : Type ?u.455830
δ : Type ?u.455833
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeSup β
inst✝ : ContinuousSup β
f g : α →ₘ[μ] β
a✝ : α
ha : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝
⊢ ↑g a✝ ≤ ↑f a✝ ⊔ ↑g a✝
case h
α : Type u_1
β : Type u_2
γ : Type ?u.455830
δ : Type ?u.455833
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeSup β
inst✝ : ContinuousSup β
f g : α →ₘ[μ] β
a✝ : α
ha : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝
⊢ ↑g a✝ ≤ ↑(f ⊔ g) a✝
rw [ha]
no goals
case h
α : Type u_1
β : Type u_2
γ : Type ?u.455830
δ : Type ?u.455833
inst✝⁵ : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
inst✝¹ : SemilatticeSup β
inst✝ : ContinuousSup β
f g : α →ₘ[μ] β
a✝ : α
ha : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝
⊢ ↑g a✝ ≤ ↑f a✝ ⊔ ↑g a✝
exact le_sup_right