Theorems · Definition · order theory

Function.Injective.lattice

{α : Type u} →
  {β : Type v} →
    [inst : Max α] →
      [inst_1 : Min α] →
        [inst_2 : LE α] →
          [inst_3 : LT α] →
            [inst_4 : Lattice 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) → Lattice α

A type endowed with ⊔ and ⊓ is a Lattice, if it admits an injective map that preserves ⊔ and ⊓ to a Lattice. See note [reducible non-instances].

Defined in

Mathlib.Order.Lattice

Cited by

0 results in Mathlib

Foundations

Depth 15 from the axioms · uses propext

Assumes

MaxMinLELTLattice