map_field_closure — Mathlib

English

The image under f of the closure of s equals the closure of the image of s: map f (closure s) = closure (f '' s).

Русский

Образ подгуммого отображения f образа closure(s) равен closure(образ s).

LaTeX

$$$\operatorname{map} f (\operatorname{closure} s) = \operatorname{closure}(f'' s)$$$

Lean4

/-- The image under a ring homomorphism of the subfield generated by a set equals
the subfield generated by the image of the set. -/
theorem map_field_closure (f : K →+* L) (s : Set K) : (closure s).map f = closure (f '' s) :=
  Set.image_preimage.l_comm_of_u_comm (gc_map_comap f) (Subfield.gi L).gc (Subfield.gi K).gc fun _ ↦ rfl

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