existsUnique_preimage_of_mem_integerSet

English

Every element a of integerSet(K) maps to a nonzero mixedSpace element.

Русский

Каждый элемент a из integerSet(K) превращается в не нуль в mixedSpace.

LaTeX

$$For all a ∈ integerSet(K), a ≠ 0 in mixedSpace(K).$$

Lean4

/-- If `a` is in `integerSet`, then there is a *unique* algebraic integer in `𝓞 K` such
that `mixedEmbedding K x = a`. -/
theorem existsUnique_preimage_of_mem_integerSet {a : mixedSpace K} (ha : a ∈ integerSet K) :
    ∃! x : 𝓞 K, mixedEmbedding K x = a :=
  by
  obtain ⟨_, ⟨x, rfl⟩⟩ := mem_integerSet.mp ha
  refine Function.Injective.existsUnique_of_mem_range ?_ (Set.mem_range_self x)
  exact (mixedEmbedding_injective K).comp RingOfIntegers.coe_injective

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