surjective_extensionEmbeddingOfIsReal

English

If the infinite place v is real, then the embedding extension from v.Completion to ℝ is surjective.

Русский

Если бесконечнаяPlacement v реален, то вложение v.Completion в ℝ является сюръективным.

LaTeX

$$$\text{IsReal}(v) \Rightarrow \text{Function.Surjective}(\text{extensionEmbeddingOfIsReal}(hv))$$$

Lean4

/-- If `v` is a real infinite place, then the embedding `v.Completion →+* ℝ` is surjective. -/
theorem surjective_extensionEmbeddingOfIsReal {v : InfinitePlace K} (hv : IsReal v) :
    Function.Surjective (extensionEmbeddingOfIsReal hv) :=
  by
  rw [← RingHom.fieldRange_eq_top_iff, ← Real.subfield_eq_of_closed]
  exact isClosed_image_extensionEmbeddingOfIsReal hv

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