Theorems · Theorem · commutative algebra
Submodule.add_mem_iff_left
∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] {module_M : Module R M} (p : Submodule R M)
{x y : M}, y ∈ p → (x + y ∈ p ↔ x ∈ p)- Defined in
- Mathlib.Algebra.Module.Submodule.Defs
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 18 from the axioms · uses propext
- Assumes
- RingAddCommGroup
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
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
- AddCommGroupstatement and proof · cited by 12,871
- Ringstatement and proof · cited by 7,463
- Submodulestatement and proof · cited by 7,192
- add_mem_cancel_rightproof · cited by 10
Cited by8
Results whose statement or proof uses this declaration.
- Submodule.sub_mem_iff_leftproof · cited by 2
- Ideal.add_mem_iff_leftproof · cited by 2
- Submodule.wcovBy_span_singleton_supproof · cited by 1
- ball_subset_range_iff_surjectiveproof · cited by 1
- IsLocalization.AtPrime.under_map_eq_mapproof · cited by 1
- FractionalIdeal.isPrincipal.of_finite_maximals_of_invproof · cited by 1
- Submodule.prodEquivOfIsCompl_symm_apply_fst_eq_zeroproof · cited by 0