Extension finite
IsDedekindDomain.HeightOneSpectrum.Extension.finite
Plain-language statement
There are only finitely many nonzero primes of B above a nonzero prime of A.
Exact Lean statement
theorem Extension.finite (v : HeightOneSpectrum A) : Finite (v.Extension B)
Formal artifact
Lean source
theorem Extension.finite (v : HeightOneSpectrum A) : Finite (v.Extension B) := by have := isTorsionFree A K L B rw [Extension, ← Set.coe_setOf] rw [@Set.finite_coe_iff] have := primesOver_finite v.asIdeal B refine Set.Finite.of_finite_image (f := HeightOneSpectrum.asIdeal) ?_ ?_ · refine Set.Finite.subset this ?_ simp only [Set.subset_def, Set.mem_image, Set.mem_setOf_eq, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂] rintro w rfl simp only [Ideal.primesOver, Set.mem_setOf_eq, isPrime, true_and] constructor simp [Ideal.under_def, under] · intro x hx y hy hxy rwa [← @HeightOneSpectrum.ext_iff] at hxy- Project
- Fermat's Last Theorem
- License
- Apache-2.0
- Commit
- 8dd808888295
- Source
- FLT/DedekindDomain/IntegralClosure.lean:132-146
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Plain-language statement
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