map_closureS — Mathlib

English

The image under a ring homomorphism of the subsemiring generated by a set equals the subsemiring generated by the image of the set.

Русский

Образ подпол semiring через гомоморфизм кольца от подгенератора множества равен подгенератору образа множества.

LaTeX

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

Lean4

/-- The image under a ring homomorphism of the subsemiring generated by a set equals
the subsemiring generated by the image of the set. -/
theorem map_closureS (f : R →+* S) (s : Set R) : (closure s).map f = closure (f '' s) :=
  Set.image_preimage.l_comm_of_u_comm (gc_map_comap f) (Subsemiring.gi S).gc (Subsemiring.gi R).gc fun _ ↦ coe_comap _ _

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