Project-declaredLean 4.32.0 · mathlib@249c48c2

Ultra Product continuous of bdd Above card

UltraProduct.continuous_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 ring homs f i : R →+* R i, the lift R →+* 𝒰(Rᵢ) is also continuous.

Exact Lean statement

lemma UltraProduct.continuous_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)) :
    Continuous ((π R F).comp (RingHom.pi f))

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma UltraProduct.continuous_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)) :    Continuous ((π R F).comp (RingHom.pi f)) := by  suffices IsOpen (X := R₀) (RingHom.ker ((π R F).comp (RingHom.pi f))) by    apply continuous_of_continuousAt_zero    rw [ContinuousAt, map_zero, nhds_discrete (UltraProduct R F), pure_zero, tendsto_zero]    exact this.mem_nhds (x := 0) (map_zero _)  obtain ⟨i, ⟨e, he⟩, hf, hR, H⟩ :=    ((UltraProduct.exists_ringEquiv_of_bddAbove_card F N H f hf).and (hf.and H)).exists  have : e.symm.toRingHom.comp (f i) = (π R F).comp (RingHom.pi f) := by    rw [← he, ← RingHom.comp_assoc]; simp  rw [← this, ← RingHom.comap_ker,    (RingHom.injective_iff_ker_eq_bot e.symm.toRingHom).mp e.symm.injective]  exact hf.isOpen_preimage {0} (isOpen_discrete {0})

Project: Fermat's Last Theorem
License: Apache-2.0
Commit: 8dd808888295
Source: FLT/Patching/Ultraproduct.lean:398-418

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