Project-declaredLean 4.32.0 · mathlib@249c48c2

Ultra Product surjective of bdd Above card

UltraProduct.surjective_of_bddAbove_card

number theory arithmetic geometry Fermat's Last Theorem ultraproduct

Plain-language statement

Let R₀ be a topological ring, topologically of finite type (over ℤ). Consider a family of (cardinality) finite rings R i with the discrete topology whose cardinalites are unifomly bounded. Given a family of continuous surjective ring homs f i : R →+* R i, the lift R →+* 𝒰(Rᵢ) is also surjective.

Exact Lean statement

lemma UltraProduct.surjective_of_bddAbove_card
    [TopologicalSpace R₀]
    [IsTopologicalRing R₀]
    [Algebra.TopologicallyFG ℤ R₀]
    [∀ i, TopologicalSpace (R i)]
    [∀ i, IsTopologicalRing (R i)]
    [∀ i, T2Space (R i)] (F : Ultrafilter ι)
    (N : ℕ) (H : ∀ᶠ i in F, Finite (R i) ∧ Nat.card (R i) < N)
    (f : ∀ i, R₀ →+* R i) (hf : ∀ᶠ i in F, Continuous (f i))
    (hf' : ∀ᶠ i in F, Function.Surjective (f i)) :
    Function.Surjective ((π R F).comp (RingHom.pi f))

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma UltraProduct.surjective_of_bddAbove_card    [TopologicalSpace R₀]    [IsTopologicalRing R₀]    [Algebra.TopologicallyFG ℤ R₀]    [∀ i, TopologicalSpace (R i)]    [∀ i, IsTopologicalRing (R i)]    [∀ i, T2Space (R i)] (F : Ultrafilter ι)    (N : ℕ) (H : ∀ᶠ i in F, Finite (R i) ∧ Nat.card (R i) < N)    (f : ∀ i, R₀ →+* R i) (hf : ∀ᶠ i in F, Continuous (f i))    (hf' : ∀ᶠ i in F, Function.Surjective (f i)) :    Function.Surjective ((π R F).comp (RingHom.pi f)) := by  obtain ⟨i, ⟨e, he⟩, hf⟩ :=    ((UltraProduct.exists_ringEquiv_of_bddAbove_card F N H f hf).and hf').exists  have : e.symm.toRingHom.comp (f i) = (π R F).comp (RingHom.pi f) := by    rw [← he, ← RingHom.comp_assoc]; simp  rw [← this]  exact e.symm.surjective.comp hf

Project: Fermat's Last Theorem
License: Apache-2.0
Commit: 8dd808888295
Source: FLT/Patching/Ultraproduct.lean:429-445

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