Theorems · Theorem · linear algebra
Submodule.toLinearMap_prodEquivOfIsCompl_symm
∀ {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), ↑(p.prodEquivOfIsCompl q hpq).symm = (p.projectionOnto q hpq).prod (q.projectionOnto p ⋯)- Defined in
- Mathlib.LinearAlgebra.Projection
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 73 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.
Cites17
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 · 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
- LinearEquiv.symmstatement · cited by 1,461
- LinearEquiv.toLinearMapstatement · cited by 1,171
- LinearMap.extproof · cited by 844
- IsComplstatement and proof · cited by 351
- Submodule.projectionOntostatement and proof · cited by 81
Cited by5
Results whose statement or proof uses this declaration.
- LinearMap.ofIsCompl_eqproof · cited by 6
- LinearMap.ofIsCompl_apply_leftproof · cited by 5
- LinearMap.ofIsCompl_apply_rightproof · cited by 5
- LinearMap.exists_map_addHaar_eq_smul_addHaar'proof · cited by 1
- Submodule.det_reflectionproof · cited by 1