import Mathlib
import OAI.Analysis.BiholderTransport.Convexity.UniformEnvelope

namespace OAI


noncomputable section
open Set Filter Manifold Bundle
open scoped Topology ContDiff

namespace WeakMTWTransport
variable {n : ℕ} {M : Type*} [MetricSpace M]
  [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M]

lemma chartFiberInverse_at_center (a : M) (p : Model n) :
    chartFiberInverse a ((extChartAt 𝓘(ℝ,Model n) a) a) p=p := by
  dsimp only [chartFiberInverse]
  rw [(extChartAt 𝓘(ℝ,Model n) a).left_inv (mem_extChartAt_source a)]
  rw [TangentBundle.symmL_trivializationAt_eq_core (mem_chart_source (Model n) a)]
  exact tangentCoordChange_self (I := 𝓘(ℝ,Model n)) (mem_extChartAt_source a)
end WeakMTWTransport

end

end OAI