Theorems · Theorem · order theory
IsCompl.isAtom_iff_isCoatom
∀ {α : Type u_2} [inst : Lattice α] [inst_1 : BoundedOrder α] [IsModularLattice α] {a b : α},
IsCompl a b → (IsAtom a ↔ IsCoatom b)- Defined in
- Mathlib.Order.Atoms
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 31 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Latticestatement and proof · cited by 916
- IsComplstatement and proof · cited by 351
- BoundedOrderstatement and proof · cited by 270
- IsAtomstatement · cited by 130
- IsCoatomstatement · cited by 114
- IsModularLatticestatement and proof · cited by 86
- OrderIso.isSimpleOrder_iffproof · cited by 12
- Set.isSimpleOrder_Ici_iff_isCoatomproof · cited by 4
- Set.isSimpleOrder_Iic_iff_isAtomproof · cited by 3
- IsCompl.IicOrderIsoIciproof · cited by 2
Cited by3
Results whose statement or proof uses this declaration.
- isAtom_complproof · cited by 2
- IsCompl.isCoatom_iff_isAtomproof · cited by 2
- isCoatomic_of_isAtomic_of_complementedLattice_of_isModularproof · cited by 2