mlieBracketWithin_apply — Mathlib

English

The application formula for mlieBracketWithin refines to the standard formula with pullbacks and board-level extChartAt.

Русский

Применение mlieBracketWithin сводится к стандартной формуле через вытягивания и extChartAt.

LaTeX

$$mlieBracketWithin_apply = ... -- formula in terms of mfderiv and lieBracketWithin with pullbacks$$

Lean4

theorem mlieBracketWithin_apply :
    mlieBracketWithin I V W s x₀ =
      (mfderiv I 𝓘(𝕜, E) (extChartAt I x₀) x₀).inverse
        ((lieBracketWithin 𝕜 (mpullbackWithin 𝓘(𝕜, E) I (extChartAt I x₀).symm V (range I))
            (mpullbackWithin 𝓘(𝕜, E) I (extChartAt I x₀).symm W (range I)) ((extChartAt I x₀).symm ⁻¹' s ∩ range I))
          ((extChartAt I x₀ x₀))) :=
  rfl

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