Theorems · Theorem · linear algebra
Submodule.span_insert_zero
∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {s : Set M},
Submodule.span R (insert 0 s) = Submodule.span R s- Defined in
- Mathlib.LinearAlgebra.Span.Defs
- Cited by
- 11 results in Mathlib
- Foundations
- Depth 26 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringAddCommMonoidModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Submodulestatement · cited by 7,192
- le_antisymmproof · cited by 2,068
- Submodule.spanstatement and proof · cited by 1,504
- Submodule.subset_spanproof · cited by 234
- Submodule.span_leproof · cited by 164
- Set.subset_insertproof · cited by 96
- Submodule.span_monoproof · cited by 85
- Set.insert_subset_iffproof · cited by 35
Cited by11
Results whose statement or proof uses this declaration.
- vectorSpan_pairproof · cited by 3
- Submodule.span_sdiff_singleton_zeroproof · cited by 1
- Ideal.span_insert_zeroproof · cited by 1
- Matrix.cRank_diagonalproof · cited by 1
- Module.torsion_by_prime_power_decompositionproof · cited by 1
- affineSpan_insert_zeroproof · cited by 1
- exists_affineIndependentproof · cited by 1
- Polynomial.contentIdeal_map_eq_map_contentIdealproof · cited by 0
- LieAlgebra.IsKilling.span_weight_isNonZero_eq_topproof · cited by 0
- InnerProductGeometry.angle_eq_angle_add_angle_iffproof · cited by 0
- MonomialOrder.span_leadingTerm_insert_zeroproof · cited by 0