Theorems · Theorem · linear algebra
LinearMap.IsIdempotentElem.range_eq_ker
∀ {S : Type u_5} [inst : Semiring S] {E : Type u_7} [inst_1 : AddCommGroup E] [inst_2 : Module S E] {p : E →ₗ[S] E},
IsIdempotentElem p → p.range = (LinearMap.id - p).ker- Defined in
- Mathlib.LinearAlgebra.Projection
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 46 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringAddCommGroupModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · 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
- AddCommGroupstatement and proof · cited by 12,871
- LinearMapstatement and proof · cited by 10,215
- Submodulestatement · cited by 7,192
- le_antisymmproof · cited by 2,068
- LinearMap.rangestatement · cited by 893
- LinearMap.kerstatement and proof · cited by 848
- LinearMap.idstatement and proof · cited by 625
- IsIdempotentElemstatement and proof · cited by 217
Cited by2
Results whose statement or proof uses this declaration.
- LinearMap.IsIdempotentElem.range_eq_ker_one_subproof · cited by 1
- ContinuousLinearMap.IsIdempotentElem.isClosed_rangeproof · cited by 0