English
Pull back a Coframe.MinimalAxioms along an injection preserving the lattice operations.
Русский
Тянем назад Coframe.MinimalAxioms вдоль инъекции, сохраняющей операции решётки.
LaTeX
$$$[Coframe.MinimalAxioms\; \beta] \rightarrow \text{pullback along } f:\alpha \to \beta \Rightarrow Coframe.MinimalAxioms(\alpha).$$$
Lean4
/-- Pullback an `Order.Coframe.MinimalAxioms` along an injection. -/
protected abbrev coframeMinimalAxioms [Max α] [Min α] [SupSet α] [InfSet α] [Top α] [Bot α]
(minAx : Coframe.MinimalAxioms β) (f : α → β) (hf : Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b)
(map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) (map_sSup : ∀ s, f (sSup s) = ⨆ a ∈ s, f a)
(map_sInf : ∀ s, f (sInf s) = ⨅ a ∈ s, f a) (map_top : f ⊤ = ⊤) (map_bot : f ⊥ = ⊥) : Coframe.MinimalAxioms α
where
__ := hf.completeLattice f map_sup map_inf map_sSup map_sInf map_top map_bot
iInf_sup_le_sup_sInf a
s := by
change f _ ≤ f (a ⊔ sInf s)
rw [← sInf_image, map_sup, map_sInf s, minAx.sup_iInf₂_eq]
simp_rw [← map_sup]
exact ((map_sInf _).trans iInf_image).le
