Theorems · Definition · order theory

Function.Injective.completeLattice

{α : Type u_1} →
  {β : Type u_2} →
    [inst : Max α] →
      [inst_1 : Min α] →
        [inst_2 : LE α] →
          [inst_3 : LT α] →
            [inst_4 : SupSet α] →
              [inst_5 : InfSet α] →
                [inst_6 : Top α] →
                  [inst_7 : Bot α] →
                    [inst_8 : CompleteLattice β] →
                      (f : α → β) →
                        Function.Injective f →
                          (∀ {x y : α}, f x ≤ f y ↔ x ≤ y) →
                            (∀ {x y : α}, f x < f y ↔ x < y) →
                              (∀ (a b : α), f (a ⊔ b) = f a ⊔ f b) →
                                (∀ (a b : α), f (a ⊓ b) = f a ⊓ f b) →
                                  (∀ (s : Set α), f (sSup s) = ⨆ a ∈ s, f a) →
                                    (∀ (s : Set α), f (sInf s) = ⨅ a ∈ s, f a) → f ⊤ = ⊤ → f ⊥ = ⊥ → CompleteLattice α

Pullback a CompleteLattice along an injection.

Defined in

Mathlib.Order.CompleteLattice.Basic

Cited by

0 results in Mathlib

Foundations

Depth 21 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MaxMinLELTSupSetInfSetTopBotCompleteLattice