Orthogonal To Linear Isometry Equiv left inv
EuclideanGroup.orthogonalToLinearIsometryEquiv_left_inv
Plain-language statement
linearIsometryEquivToOrthogonal is a left inverse of orthogonalToLinearIsometryEquiv. Together with linearIsometryEquiv_constVAdd_mul, this proves left_inv of toAffineIsometryMulEquiv.
Exact Lean statement
@[simp] lemma orthogonalToLinearIsometryEquiv_left_inv
(Q : Matrix.orthogonalGroup (Fin n) ℝ) :
linearIsometryEquivToOrthogonal (orthogonalToLinearIsometryEquiv Q) = QFormal artifact
Lean source
@[simp] lemma orthogonalToLinearIsometryEquiv_left_inv (Q : Matrix.orthogonalGroup (Fin n) ℝ) : linearIsometryEquivToOrthogonal (orthogonalToLinearIsometryEquiv Q) = Q := by apply Subtype.ext have hlin : ((orthogonalToLinearIsometryEquiv Q).toLinearEquiv : EuclideanSpace ℝ (Fin n) →ₗ[ℝ] EuclideanSpace ℝ (Fin n)) = Matrix.toLin (EuclideanSpace.basisFun (Fin n) ℝ).toBasis (EuclideanSpace.basisFun (Fin n) ℝ).toBasis Q.val := by ext x; rfl show LinearMap.toMatrix _ _ ((orthogonalToLinearIsometryEquiv Q).toLinearEquiv : EuclideanSpace ℝ (Fin n) →ₗ[ℝ] EuclideanSpace ℝ (Fin n)) = Q.val rw [hlin, LinearMap.toMatrix_toLin]- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/SpaceAndTime/Space/EuclideanGroup/AffineGroup.lean:137-150
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