Basis eq mfderiv model Diffeo single
Space.basis_eq_mfderiv_modelDiffeo_single
Plain-language statement
The derivative of modelDiffeo provides an equivalence between Space d and EuclideanSpace ℝ (Fin d). This equivalences takes the basis of EuclideanSpace ℝ (Fin d) to the basis of Space d, and vice versa.
Exact Lean statement
lemma basis_eq_mfderiv_modelDiffeo_single (d : ℕ) (μ : Fin d) (x : Space d) :
basis μ = mfderiv (𝓡 d) 𝓘(ℝ, Space d) (modelDiffeo (d := d)) x
(EuclideanSpace.single μ 1)Formal artifact
Lean source
lemma basis_eq_mfderiv_modelDiffeo_single (d : ℕ) (μ : Fin d) (x : Space d) : basis μ = mfderiv (𝓡 d) 𝓘(ℝ, Space d) (modelDiffeo (d := d)) x (EuclideanSpace.single μ 1) := by simp only [modelDiffeo_apply, mfderiv, writtenInExtChartAt, extChartAt, OpenPartialHomeomorph.extend, OpenPartialHomeomorph.refl_partialEquiv, PartialEquiv.refl_source, OpenPartialHomeomorph.singletonChartedSpace_chartAt_eq, modelWithCornersSelf_partialEquiv, PartialEquiv.trans_refl, PartialEquiv.refl_coe, Homeomorph.symm_toOpenPartialHomeomorph, OpenPartialHomeomorph.symm_toPartialEquiv, PartialEquiv.symm_symm, OpenPartialHomeomorph.toFun_eq_coe, Homeomorph.toOpenPartialHomeomorph_apply, CompTriple.comp_eq, modelWithCornersSelf_coe, Set.range_id, OpenPartialHomeomorph.coe_toPartialEquiv_symm, Homeomorph.toOpenPartialHomeomorph_symm_apply, fderivWithin_univ] rw [if_pos (modelDiffeo.mdifferentiable (WithTop.top_ne_zero)).mdifferentiableAt] ext i have h := fderiv_space_components i ((⇑modelDiffeo ∘ ⇑(homEuclideanSpaceSpace d))) (by simpa [Function.comp_def, homEuclideanSpaceSpace] using by fun_prop) (((homEuclideanSpaceSpace d).symm x)) ((EuclideanSpace.single μ 1)) convert! h.symm simp only [basis_apply, homEuclideanSpaceSpace, PiLp.continuousLinearEquiv_symm_apply, Homeomorph.homeomorph_mk_coe, Equiv.coe_fn_mk, Function.comp_apply, modelDiffeo_apply, PiLp.continuousLinearEquiv_apply, Homeomorph.homeomorph_mk_coe_symm, Equiv.symm_mk] change _ = fderiv ℝ (EuclideanSpace.proj i) _ (EuclideanSpace.single μ 1) simp only [ContinuousLinearMap.fderiv, PiLp.proj_apply, PiLp.single_apply] congr 1 exact Eq.propIntro (fun a => Eq.symm a) fun a => (Eq.symm a)- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/SpaceAndTime/Space/Module.lean:683-707
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