Theorems · Theorem · commutative algebra
Submodule.neg_le
∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
{S T : Submodule R M}, -S ≤ T ↔ S ≤ -T- Cited by
- 3 results in Mathlib
- Foundations
- Depth 28 from the axioms · uses propext, Quot.sound
- Assumes
- SemiringAddCommGroupModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommGroupstatement and proof · cited by 12,871
- Submodulestatement and proof · cited by 7,192
- SetLike.coe_subset_coeproof · cited by 53
- Submodule.pointwiseNegstatement · cited by 20
- Set.neg_subsetproof · cited by 6
Cited by3
Results whose statement or proof uses this declaration.
- Submodule.sup_inf_assoc_of_le_of_neg_leproof · cited by 2
- Submodule.span_neg_eq_negproof · cited by 1
- Submodule.neg_eq_self_iff_neg_leproof · cited by 0