Chart Euclidean smul
EuclideanGroup.chartEuclidean_smul
Plain-language statement
Under the standard chart, the Euclidean action on Space d is the transport of toAffineIsometryMulEquiv acting on EuclideanSpace: chart (g • p) = (toAffineIsometryMulEquiv g) (chart p).
Exact Lean statement
lemma chartEuclidean_smul (g : EuclideanGroup d) (p : Space d) :
Space.chartEuclidean d (g • p) = toAffineIsometryMulEquiv g (Space.chartEuclidean d p)Formal artifact
Lean source
lemma chartEuclidean_smul (g : EuclideanGroup d) (p : Space d) : Space.chartEuclidean d (g • p) = toAffineIsometryMulEquiv g (Space.chartEuclidean d p) := by rw [Space.chartEuclidean_apply] rw [toAffineIsometryMulEquiv_apply, toAffineIsometryHom_apply] have h_left : g • p -ᵥ (0 : Space d) = g.linear • (p -ᵥ (0 : Space d)) + g.translation := by exact (eq_vadd_iff_vsub_eq (g • p) (g.linear • (p -ᵥ (0 : Space d)) + g.translation) ((0 : Space d))).mp rfl simp [h_left] exact add_comm' (g.linear • (p -ᵥ (0 : Space d))) g.translation- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/SpaceAndTime/Space/EuclideanGroup/Action.lean:136-146
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