Theorems · Theorem · linear algebra
LinearMap.IsProj.mem_iff_map_id
∀ {S : Type u_5} [inst : Semiring S] {M : Type u_6} [inst_1 : AddCommMonoid M] [inst_2 : Module S M] {m : Submodule S M}
{f : M →ₗ[S] M}, LinearMap.IsProj m f → ∀ {x : M}, x ∈ m ↔ f x = x- Defined in
- Mathlib.LinearAlgebra.Projection
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 21 from the axioms · uses no axioms
- Assumes
- SemiringAddCommMonoidModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement and proof · cited by 18,349
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- LinearMapstatement and proof · cited by 10,215
- Submodulestatement and proof · cited by 7,192
- LinearMap.IsProjstatement and proof · cited by 23
- LinearMap.IsProj.map_idproof · cited by 6
- LinearMap.IsProj.map_memproof · cited by 3
Cited by2
Results whose statement or proof uses this declaration.
- LinearMap.IsIdempotentElem.mem_range_iffproof · cited by 3
- LinearMap.IsProj.submodule_uniqueproof · cited by 1