Theorems · Theorem · commutative algebra
Submodule.add_mem
∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M)
{x y : M}, x ∈ p → y ∈ p → x + y ∈ p- Defined in
- Mathlib.Algebra.Module.Submodule.Defs
- Cited by
- 75 results in Mathlib
- Foundations
- Depth 14 from the axioms · uses no axioms
- Assumes
- SemiringAddCommMonoid
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
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Submodulestatement and proof · cited by 7,192
- AddMemClass.add_memproof · cited by 229
Cited by77
Results whose statement or proof uses this declaration.
- Submodule.restrictScalarsproof · cited by 180
- Submodule.span_inductionstatement and proof · cited by 77
- Ideal.IsMaximal.isPrimeproof · cited by 53
- direction_affineSpanproof · cited by 46
- Ideal.add_memproof · cited by 36
- Algebra.adjoin_eq_spanproof · cited by 13
- LieSubmodule.lieIdeal_oper_eq_linear_spanproof · cited by 9
- TensorProduct.span_tmul_eq_topproof · cited by 8
- Submodule.toConvexConeproof · cited by 7
- KaehlerDifferential.span_range_derivationproof · cited by 7
- FractionalIdeal.le_one_iff_exists_coeIdealproof · cited by 5
- InnerProductSpace.mem_span_gramSchmidtproof · cited by 4