map_dualAnnihilator_linearEquiv_flip_symm

English

The map induced by flips and symmetries exchanges dual annihilators and dual coannihilators compatibly with the base equivalence e: N ≃ Dual M.

Русский

Отображение, индуцируемое flip и симметрии, корректно меняет dual annihilator и dual coannihilator совместимо с базовым эквиваленсом e: N ≃ Dual M.

LaTeX

$$$\\text{map}_{e} : p \\mapsto p.dualAnnihilator \\text{ and } (p.flip) \\mapsto p.dualCoannihilator$ commute appropriately.$$

Lean4

@[simp]
theorem map_dualAnnihilator_linearEquiv_flip_symm (p : Submodule R N) :
    p.dualAnnihilator.map e.flip.symm = (p.map e).dualCoannihilator :=
  by
  have : IsReflexive R N := e.isReflexive_of_equiv_dual_of_isReflexive
  rw [← dualCoannihilator_map_linearEquiv_flip, flip_flip]

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