Theorems · Definition · order theory

Function.Injective.completeAtomicBooleanAlgebra

{α : Type u} →
  {β : Type v} →
    [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 : Compl α] →
                      [inst_9 : HImp α] →
                        [inst_10 : HNot α] →
                          [inst_11 : SDiff α] →
                            [inst_12 : CompleteAtomicBooleanAlgebra 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 ⊥ = ⊥ →
                                                  (∀ (a : α), f aᶜ = (f a)ᶜ) →
                                                    (∀ (a b : α), f (a ⇨ b) = f a ⇨ f b) →
                                                      (∀ (a : α), f (￢a) = ￢f a) →
                                                        (∀ (a b : α), f (a \ b) = f a \ f b) →
                                                          CompleteAtomicBooleanAlgebra α

Pullback a CompleteAtomicBooleanAlgebra along an injection.

Defined in

Mathlib.Order.CompleteBooleanAlgebra

Cited by

0 results in Mathlib

Foundations

Depth 60 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MaxMinLELTSupSetInfSetTopBotComplHImpHNotSDiffCompleteAtomicBooleanAlgebra