One Jet Bundle chart target
oneJetBundle_chart_target
Plain-language statement
In J¹(M, M'), the target of a chart has a nice formula
Exact Lean statement
theorem oneJetBundle_chart_target (x₀ : J¹MM') :
(chartAt HJ x₀).target = Prod.fst ⁻¹' (chartAt (ModelProd H H') x₀.proj).targetFormal artifact
Lean source
theorem oneJetBundle_chart_target (x₀ : J¹MM') : (chartAt HJ x₀).target = Prod.fst ⁻¹' (chartAt (ModelProd H H') x₀.proj).target := by rw [FiberBundle.chartedSpace_chartAt] simp only [prodChartedSpace_chartAt, OpenPartialHomeomorph.trans_toPartialEquiv, OpenPartialHomeomorph.prod_toPartialHomeomorph, OpenPartialHomeomorph.refl_partialEquiv, PartialEquiv.trans_target, PartialEquiv.prod_target, PartialEquiv.refl_target] erw [hom_trivializationAt_target] simp only [trivializationAt_pullBack_baseSet, TangentBundle.trivializationAt_baseSet] rcases x₀ with ⟨⟨m, m'⟩, φ⟩ simp only [ContMDiffMap.coe_fst, ContMDiffMap.fst_apply, ContMDiffMap.coe_snd, ContMDiffMap.snd_apply] erw [prod_univ, inter_eq_left, prod_univ, PartialEquiv.prod_symm, PartialEquiv.prod_symm] rw [preimage_preimage, ← Set.prod_eq, PartialEquiv.refl_symm, PartialEquiv.prod_coe, PartialEquiv.refl_coe] have : (fun x : ModelProd (ModelProd H H') (E →SL[σ] E') ↦ ((chartAt H m).toPartialEquiv.symm.prod (chartAt H' m').toPartialEquiv.symm) x.1) = (Prod.map (chartAt H m).symm (chartAt H' m').symm) ∘ Prod.fst := by ext x <;> rfl rw [this, preimage_comp, preimage_prod_map_prod] gcongr · exact (chartAt H m).target_subset_preimage_source · exact (chartAt H' m').target_subset_preimage_source- Project
- Sphere eversion
- License
- Apache-2.0
- Commit
- ded8fda5e76b
- Source
- SphereEversion/Global/OneJetBundle.lean:307-329
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