topologicalSemiring_of_smooth

theorem topologicalSemiring_of_smooth : ∀ {𝕜 : Type u_1} {R : Type u_2} {E : Type u_3} {H : Type u_4} [inst : TopologicalSpace R] [inst_1 : TopologicalSpace H] [inst_2 : NontriviallyNormedField 𝕜] [inst_3 : NormedAddCommGroup E] [inst_4 : NormedSpace 𝕜 E] [inst_5 : ChartedSpace H R] (I : ModelWithCorners 𝕜 E H) [inst_6 : Semiring R] [inst_7 : SmoothRing I R], TopologicalSemiring R

This theorem states that a smooth semiring is also a topological semiring. Given a range of types and spaces, including a Nontrivially Normed Field, a Normed Additive Commutative Group, a Normed Space, a Charted Space, a Model With Corners, and a Semiring, if the Semiring is smooth (in the sense of differential geometry), then it can be considered as a topological semiring. The proof of this statement is not provided. This theorem does not instantiate the general case due to specific design and technical considerations.

More concisely: A smooth semiring is a topological semiring.