Theorems · Theorem · commutative algebra
Submodule.add_mem_iff_right
∀ {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}, x ∈ p → (x + y ∈ p ↔ y ∈ 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_leftproof · cited by 8
Cited by8
Results whose statement or proof uses this declaration.
- Submodule.sub_mem_iff_rightproof · cited by 4
- Ideal.add_mem_iff_rightproof · cited by 4
- InfIrred.isPrimaryproof · cited by 1
- Submodule.iUnion_ssubset_of_forall_ne_top_of_card_ltproof · cited by 1
- Submodule.eq_top_of_nonempty_interior'proof · cited by 1
- Submodule.prodEquivOfIsCompl_symm_apply_snd_eq_zeroproof · cited by 0
- AmpleSet.of_one_lt_codimproof · cited by 0
- RootPairing.root_mem_submodule_iff_of_add_mem_invtSubmoduleproof · cited by 0