map_modByMonic — Mathlib

English

Unicity of liftHom: if g is an R-algebra hom S → T with g(h.root) = x, then g = h.liftHom x hx'.

Русский

Уникальность liftHom: если g — R-алгебра-гомоморфизм S → T и g(h.root) = x, то g = h.liftHom x hx'.

LaTeX

$$$ g = h.liftHom x hx' $$$

Lean4

theorem map_modByMonic (g : R[X]) : h.map (g %ₘ f) = h.map g :=
  by
  rw [← RingHom.sub_mem_ker_iff, mem_ker_map, modByMonic_eq_sub_mul_div _ h.monic, sub_right_comm, sub_self, zero_sub,
    dvd_neg]
  exact ⟨_, rfl⟩

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