Plain-language statement
Evaluation of integral power series at a point of the open unit disc is multiplicative. Together with evalInt_add this makes evalInt q a ring homomorphism ℤ⟦X⟧ → k for each |q| < 1. Proof: on the open unit disc, evalInt is the coercion to k of mathlib's topological power-series evaluation over the ring of integers (key below): q is an inte...
Exact Lean statement
theorem evalInt_mul (q : k) (hq : valuation k q < 1) (F G : ℤ⟦X⟧) :
evalInt q (F * G) = evalInt q F * evalInt q GFormal artifact
Lean source
theorem evalInt_mul (q : k) (hq : valuation k q < 1) (F G : ℤ⟦X⟧) : evalInt q (F * G) = evalInt q F * evalInt q G := by letI : UniformSpace k := IsTopologicalAddGroup.rightUniformSpace k haveI : IsUniformAddGroup k := isUniformAddGroup_of_addCommGroup haveI : IsUniformAddGroup 𝒪[k] := inferInstanceAs (IsUniformAddGroup 𝒪[k].toAddSubgroup) have hind : Topology.IsInducing ((↑) : 𝒪[k] → k) := ⟨rfl⟩ have hφ : Continuous (Int.castRingHom 𝒪[k]) := continuous_of_discreteTopology have ha : PowerSeries.HasEval (⟨q, hq.le⟩ : 𝒪[k]) := hind.tendsto_nhds_iff.mpr (by simpa [Function.comp_def] using tendsto_pow_nhds_zero hq) have key : ∀ H : ℤ⟦X⟧, evalInt q H = (PowerSeries.eval₂Hom hφ ha H : k) := by intro H change (∑' n : ℕ, ((PowerSeries.coeff n H : ℤ) : k) * q ^ n) = _ rw [PowerSeries.coe_eval₂Hom hφ ha] refine HasSum.tsum_eq ?_ simpa [Function.comp_def] using (PowerSeries.hasSum_eval₂ hφ ha H).map (Subring.subtype 𝒪[k]).toAddMonoidHom continuous_subtype_val simp only [key, map_mul] push_cast ring- Project
- Fermat's Last Theorem
- License
- Apache-2.0
- Commit
- 8dd808888295
- Source
- FLT/KnownIn1980s/EllipticCurves/TateParameter.lean:330-348
Reuse this declaration
Bring the exact result into your workflow
The import identifies the source module. Your project still needs the pinned package dependency shown on this page.
What this badge means
This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.
Continue in this project
Related declarations
Eq finsum quotient out of bij On
AbstractHeckeOperator.eq_finsum_quotient_out_of_bijOn'
Plain-language statement
If a is fixed by V then ∑ᶠ g ∈ s, g • a is independent of the choice s of coset representatives in G for a subset of G ⧸ V
Source project: Fermat's Last Theorem
Person-level attribution pending.
Comm Group no compact automorphisms
CommGroup.no_compact_automorphisms
Plain-language statement
A connected compact Hausdorff abelian topological group does not admit a nontrivial compact group of automorphisms.
Source project: Fermat's Last Theorem
Person-level attribution pending.
Fermat Last Theorem of p ge 5
FermatLastTheorem.of_p_ge_5
Plain-language statement
If Fermat's Last Theorem is true for primes p ≥ 5, then FLT is true.
Source project: Fermat's Last Theorem
Person-level attribution pending.