Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Orthogonal To Linear Isometry Equiv right inv

EuclideanGroup.orthogonalToLinearIsometryEquiv_right_inv

physics quantum field theory relativity affine group

Plain-language statement

linearIsometryEquivToOrthogonal is a right inverse of orthogonalToLinearIsometryEquiv; this proves right_inv of toAffineIsometryMulEquiv.

Exact Lean statement

@[simp] lemma orthogonalToLinearIsometryEquiv_right_inv
    (L : EuclideanSpace ℝ (Fin n) ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin n)) :
    orthogonalToLinearIsometryEquiv (linearIsometryEquivToOrthogonal L) = L

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

@[simp] lemma orthogonalToLinearIsometryEquiv_right_inv    (L : EuclideanSpace ℝ (Fin n) ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin n)) :    orthogonalToLinearIsometryEquiv (linearIsometryEquivToOrthogonal L) = L := by    apply LinearIsometryEquiv.ext; intro x    rw [orthogonalToLinearIsometryEquiv_apply]    show Matrix.toEuclideanLin (linearIsometryEquivToOrthogonal L).val x = L x    rw [Matrix.toEuclideanLin_eq_toLin_orthonormal]    show Matrix.toLin _ _        (LinearMap.toMatrix _ _          (L.toLinearEquiv : EuclideanSpace ℝ (Fin n) →ₗ[ℝ] EuclideanSpace ℝ (Fin n))) x = L x    rw [Matrix.toLin_toMatrix]    rfl

Project: Physlib
License: Apache-2.0
Commit: dd43e9e65791
Source: Physlib/SpaceAndTime/Space/EuclideanGroup/AffineGroup.lean:169-180

Reuse this declaration

Bring the exact result into your workflow

The import identifies the source module. Your project still needs the pinned package dependency shown on this page.

What this badge means

This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.

Continue in this project

Related declarations

Project-declaredLean 4.32.0

Adiabatic relation log

adiabatic_relation_log

Plain-language statement

Adiabatic relation in logarithmic form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then c * log (Ua/Ub) + log (Va/Vb) = 0.

physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

View proof record

Project-declaredLean 4.32.0

Adiabatic relation Ua Ub Va Vb

adiabatic_relation_UaUbVaVb

Plain-language statement

Adiabatic relation in product form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then (Ua/Ub)^c * (Va/Vb) = 1.

physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

View proof record

Project-declaredLean 4.32.0

Deriv Within mean Energy Beta eq neg variance

CanonicalEnsemble.derivWithin_meanEnergy_Beta_eq_neg_variance

Plain-language statement

(∂U/∂β) = -Var(E) for finite systems.

physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

View proof record