Theorems · Theorem · linear algebra
Submodule.projection_eq_self_sub_projection
∀ {R : Type u_1} [inst : Ring R] {E : Type u_2} [inst_1 : AddCommGroup E] [inst_2 : Module R E] {p q : Submodule R E}
(hpq : IsCompl p q) (x : E), (q.projection p ⋯) x = x - (p.projection q hpq) x- Defined in
- Mathlib.LinearAlgebra.Projection
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 74 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- RingAddCommGroupModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement · cited by 18,349
- AddCommGroupstatement and proof · cited by 12,871
- LinearMapstatement · cited by 10,215
- Ringstatement and proof · cited by 7,463
- Submodulestatement and proof · cited by 7,192
- IsComplstatement and proof · cited by 351
- IsCompl.symmstatement and proof · cited by 80
- eq_sub_iff_add_eqproof · cited by 65
- Submodule.projectionstatement · cited by 63
- Submodule.projection_add_projection_eq_selfproof · cited by 7
Cited by4
Results whose statement or proof uses this declaration.
- Submodule.projection_eq_id_sub_projectionproof · cited by 3
- Submodule.projection_eq_self_iffproof · cited by 2
- Submodule.IsCompl.projection_eq_self_sub_projectionproof · cited by 0
- Submodule.sub_projection_memproof · cited by 0