preimage_image_mk_eq_iUnion_image

English

A refined expression replaces a big image by an iUnion over cosets of a subgroup; equivalently, the preimage equals the union of images under left multiplication by representatives.

Русский

Уточнённая формула заменяет образ на объединение по индексам косет; прeeобраз равен объединению по левой умноженной косете.

LaTeX

$$$\text{preimage}(\text{QuotientGroup.mk}, \text{image}(\text{QuotientGroup.mk} \, s)) = \bigcup_{x \in N} (\cdot x)^{-1}(s)$$

Lean4

@[to_additive]
theorem preimage_image_mk_eq_iUnion_image (N : Subgroup α) (s : Set α) :
    mk ⁻¹' ((mk : α → α ⧸ N) '' s) = ⋃ x : N, (· * (x : α)) '' s :=
  by
  rw [preimage_image_mk, iUnion_congr_of_surjective (·⁻¹) inv_surjective]
  exact fun x ↦ image_mul_right'

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