Theorems · Theorem · global analysis

eqOn_piecewise_of_isMIntegralCurveOn_Ioo

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {H : Type u_2} [inst_2 : TopologicalSpace H]
  {I : ModelWithCorners ℝ E H} {M : Type u_3} [inst_3 : TopologicalSpace M] [inst_4 : ChartedSpace H M]
  [inst_5 : IsManifold I 1 M] [T2Space M] {γ γ' : ℝ → M} {v : (x : M) → TangentSpace I x} {t₀ : ℝ}
  [BoundarylessManifold I M],
  (ContMDiff I I.tangent 1 fun x => ⟨x, v x⟩) →
    ∀ {a b a' b' : ℝ},
      IsMIntegralCurveOn γ v (Set.Ioo a b) →
        IsMIntegralCurveOn γ' v (Set.Ioo a' b') →
          t₀ ∈ Set.Ioo a b ∩ Set.Ioo a' b' → γ t₀ = γ' t₀ → Set.EqOn ((Set.Ioo a b).piecewise γ γ') γ' (Set.Ioo a' b')

Let γ and γ' be integral curves defined on Ioo a b and Ioo a' b', respectively. Then, piecewise (Ioo a b) γ γ' is equal to γ and γ' in their respective domains. Set.piecewise_eqOn shows the equality for γ by definition, while this lemma shows the equality for γ' by the uniqueness of integral curves.

Defined in

Mathlib.Geometry.Manifold.IntegralCurve.UniformTime

Cited by

1 results in Mathlib

Foundations

Depth 218 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceIsManifoldT2SpaceBoundarylessManifold