Range base Change comp tensor Adic Completion To eq pi
IsDedekindDomain.HeightOneSpectrum.range_baseChange_comp_tensorAdicCompletionTo_eq_pi
Plain-language statement
The image of B โ[A] ๐_v in โ_w L_w is โ_w ๐_w.
Exact Lean statement
theorem range_baseChange_comp_tensorAdicCompletionTo_eq_pi [FiniteDimensional K L] :
Set.range (baseChange K L B v โ tensorAdicCompletionIntegersTo K L B v) =
Set.univ.pi (fun w โฆ (w.1.adicCompletionIntegers L).carrier)Formal artifact
Lean source
theorem range_baseChange_comp_tensorAdicCompletionTo_eq_pi [FiniteDimensional K L] : Set.range (baseChange K L B v โ tensorAdicCompletionIntegersTo K L B v) = Set.univ.pi (fun w โฆ (w.1.adicCompletionIntegers L).carrier) := by have hrange : Set.range (algebraMap B ((w : Extension B v) โ adicCompletion L w.1)) = (baseChange K L B v) '' (Set.range (algebraMap B (L โ[K] adicCompletion K v))) := by ext x simp [Algebra.algebraMap_eq_smul_one] have hrange' := isClosed_baseChange_image_closure_range_algebraMap K L B v rw [Set.range_comp, range_tensorAdicCompletionIntegersTo_eq_closure_range_algebraMap, pi_adicCompletionIntegers_eq_closure_range_algebraMap, hrange, โ IsClosed.closure_eq hrange'] exact closure_image_closure (baseChangeRight_isOpenQuotientMap K L B v).continuous- Project
- Fermat's Last Theorem
- License
- Apache-2.0
- Commit
- 8dd808888295
- Source
- FLT/DedekindDomain/Completion/BaseChange.lean:539-551
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