disjoint_rootSpan_ker_rootForm — Mathlib

English

The flip of a crystallographic root pairing preserves anisotropy: disjoint of coroot and kernel persists under flip.

Русский

Разворот кристаллографической пары сохраняет анизотропию.

LaTeX

$$disjoint_corootSpan_ker_corootForm$$

Lean4

theorem disjoint_rootSpan_ker_rootForm : Disjoint (P.rootSpan R) (LinearMap.ker P.RootForm) :=
  by
  have : IsReflexive R M := .of_isPerfPair P.toLinearMap
  rw [← P.ker_polarization_eq_ker_rootForm]
  refine Submodule.disjoint_ker_of_finrank_le (L := P.rootSpan R) P.Polarization ?_
  rw [P.finrank_rootSpan_map_polarization_eq_finrank_corootSpan, P.finrank_corootSpan_eq']

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