Theorems · Theorem · linear algebra
Submodule.mem_span_singleton
∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {x y : M},
x ∈ R ∙ y ↔ ∃ a, a • y = x- Defined in
- Mathlib.LinearAlgebra.Span.Defs
- Cited by
- 61 results in Mathlib
- Foundations
- Depth 26 from the axioms · uses propext, Quot.sound
- Assumes
- SemiringAddCommMonoidModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · 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
- Submodule.spanstatement and proof · cited by 1,504
- one_smulproof · cited by 1,374
- zero_smulproof · cited by 716
- smul_smulproof · cited by 360
- Submodule.subset_spanproof · cited by 234
- add_smulproof · cited by 204
- Submodule.smul_memproof · cited by 204
Cited by61
Results whose statement or proof uses this declaration.
- Ideal.mem_span_singleton'proof · cited by 15
- FractionalIdeal.coeIdeal_span_singletonproof · cited by 10
- Submodule.mem_orthogonal_singleton_iff_inner_rightproof · cited by 10
- FractionalIdeal.mem_spanSingletonproof · cited by 6
- LinearPMap.supSpanSingleton_apply_mkstatement and proof · cited by 5
- collinear_iff_of_memproof · cited by 4
- Submodule.span_singleton_eq_one_iffproof · cited by 3
- AffineSubspace.mem_affineSpan_insert_iffproof · cited by 3
- Affine.Simplex.mongePoint_mem_mongePlaneproof · cited by 3
- LinearPMap.mkSpanSingleton'_applyproof · cited by 3
- Submodule.mem_smul_span_singletonproof · cited by 2
- LinearPMap.supSpanSingleton_apply_selfproof · cited by 2