Theorems · Theorem · commutative algebra
Submodule.sub_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
- 2 results in Mathlib
- Foundations
- Depth 19 from the axioms · uses propext
- Assumes
- RingAddCommGroup
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
- AddCommGroupstatement and proof · cited by 12,871
- Ringstatement and proof · cited by 7,463
- Submodulestatement and proof · cited by 7,192
- sub_eq_add_negproof · cited by 1,023
- Submodule.neg_memproof · cited by 31
- Submodule.add_mem_iff_leftproof · cited by 8
Cited by2
Results whose statement or proof uses this declaration.
- Polynomial.Sequence.span_degreeLTproof · cited by 3
- Algebra.exists_etale_isIdempotentElem_forall_liesOver_eq_auxproof · cited by 1