Theorems · Theorem · functional analysis
Submodule.IsOrtho.orthogonalProjection_comp_subtypeL
Deprecated since 2026-05-05Use Submodule.IsOrtho.orthogonalProjectionOnto_comp_subtypeL instead.
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
{U V : Submodule 𝕜 E} [inst_3 : U.HasOrthogonalProjection], U ⟂ V → U.orthogonalProjectionOnto ∘SL V.subtypeL = 0Alias of Submodule.IsOrtho.orthogonalProjectionOnto_comp_subtypeL.
The projection into U from an orthogonal submodule V is the zero map.
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- Foundations
- Depth 183 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- RingHom.idstatement · cited by 18,349
- NormedAddCommGroupstatement · cited by 15,752
- Submodulestatement · cited by 7,192
- ContinuousLinearMapstatement · cited by 5,352
- InnerProductSpacestatement · cited by 3,523
- RCLikestatement · cited by 2,829
- ContinuousLinearMap.compstatement · cited by 709
- Submodule.HasOrthogonalProjectionstatement · cited by 245
- Submodule.orthogonalProjectionOntostatement · cited by 103
- Submodule.subtypeLstatement · cited by 53
- Submodule.IsOrthostatement · cited by 50
- Submodule.IsOrtho.orthogonalProjectionOnto_comp_subtypeLproof · cited by 3
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