Theorems · Theorem · order theory
sup_inf_assoc_of_le
∀ {α : Type u_1} [inst : Lattice α] [IsModularLattice α] {x : α} (y : α) {z : α}, x ≤ z → (x ⊔ y) ⊓ z = x ⊔ y ⊓ z- Defined in
- Mathlib.Order.ModularLattice
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 7 from the axioms · uses no axioms
- Assumes
- LatticeIsModularLattice
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- le_antisymmproof · cited by 2,068
- Latticestatement and proof · cited by 916
- inf_le_leftproof · cited by 286
- inf_le_rightproof · cited by 238
- sup_leproof · cited by 159
- le_infproof · cited by 107
- IsModularLatticestatement and proof · cited by 86
- sup_le_sup_leftproof · cited by 20
- sup_inf_le_assoc_of_leproof · cited by 3
Cited by5
Results whose statement or proof uses this declaration.
- IsModularLattice.inf_sup_inf_assocproof · cited by 3
- IsModularLattice.exists_inf_eq_and_sup_eqproof · cited by 2
- LieAlgebra.InvariantForm.atomisticproof · cited by 1
- eq_of_le_of_inf_le_of_le_supproof · cited by 1
- LinearMap.IsSymmetric.isFinitelySemisimpleproof · cited by 1