rootSet_mapsTo — Mathlib

English

The polynomial X lies in the lifts f subsemiring.

Русский

Полином X лежит в подпол semiring lifts f.

LaTeX

$$$$ (X : S[X]) \in \mathrm{lifts}(f) $$$$

Lean4

theorem rootSet_mapsTo {p : T[X]} {S S'} [CommRing S] [IsDomain S] [Algebra T S] [CommRing S'] [IsDomain S']
    [Algebra T S'] [NoZeroSMulDivisors T S'] (f : S →ₐ[T] S') : (p.rootSet S).MapsTo f (p.rootSet S') :=
  by
  refine rootSet_maps_to' (fun h₀ => ?_) f
  obtain rfl : p = 0 := map_injective _ (FaithfulSMul.algebraMap_injective T S') (by rwa [Polynomial.map_zero])
  exact Polynomial.map_zero _

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