forall_mem_support_invtSubmodule_iff

English

A Mem-submodule condition is equivalent when holding for all i in support and all i generally.

Русский

Условие подмодуля памяти эквивалентно для всех i и по всей поддержке.

LaTeX

$$theorem forall_mem_support_invtSubmodule_iff$$

Lean4

theorem forall_mem_support_invtSubmodule_iff (q : Submodule R M) :
    (∀ i ∈ b.support, q ∈ invtSubmodule (P.reflection i)) ↔ (∀ i, q ∈ invtSubmodule (P.reflection i)) :=
  by
  refine ⟨fun hq i ↦ ?_, fun hq i _ ↦ hq i⟩
  letI := P.indexNeg
  have (j : ι) : P.reflection (-j) = P.reflection j := by ext x; simp [reflection_apply, two_smul]
  refine b.induction_reflect i (by simp_all) hq ?_
  clear i
  intro i j hi hj
  rw [reflection_reflectionPerm]
  exact Module.End.invtSubmodule.comp _ (Module.End.invtSubmodule.comp _ (hq j hj) hi) (hq j hj)

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