Theorems · Theorem · ring theory
IsLinearMap.map_smul
∀ {R : Type u} {M : Type v} {M₂ : Type w} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid M₂]
[inst_3 : Module R M] [inst_4 : Module R M₂] {f : M → M₂}, IsLinearMap R f → ∀ (c : R) (x : M), f (c • x) = c • f xA linear map preserves scalar multiplication.
- Defined in
- Mathlib.Algebra.Module.LinearMap.Defs
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 8 from the axioms · uses no axioms
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
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
- IsLinearMapstatement and proof · cited by 41
Cited by4
Results whose statement or proof uses this declaration.
- IsLinearMap.mk'proof · cited by 17
- ContMDiffWithinAt.smul_sectionproof · cited by 3
- MDifferentiableWithinAt.smul_sectionproof · cited by 3
- Matrix.cramer_is_linearproof · cited by 0