Theorems · Theorem · linear algebra
LinearMap.IsIdempotentElem.ker_eq_range
∀ {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.ker = (LinearMap.id - p).range- Defined in
- Mathlib.LinearAlgebra.Projection
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 48 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.
Cites13
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
- 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
- LinearMap.rangestatement and proof · cited by 893
- LinearMap.kerstatement and proof · cited by 848
- LinearMap.idstatement · cited by 625
- IsIdempotentElemstatement and proof · cited by 217
- sub_sub_cancelproof · cited by 105
- IsIdempotentElem.one_subproof · cited by 20
Cited by3
Results whose statement or proof uses this declaration.
- LinearMap.IsIdempotentElem.comp_eq_left_iffproof · cited by 1
- LinearMap.IsIdempotentElem.ker_eq_range_one_subproof · cited by 0
- Submodule.ker_starProjectionproof · cited by 0