Generalized Sum Restricted to Non-zero Summands/Corollary

Theorem
Let $G$ be a commutative topological semigroup with identity $0_G$.

Let $\family{g_i}_{i \in I}$ be an indexed family of elements of $G$.

Let $K \subseteq I : \set{i \in I : g_i \ne 0_G} \subseteq K$

Let $g^* = g \restriction_K$ be the restriction of $g$ to $K$.

Let $h \in G$.

Then:
 * the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converges to $h$


 * the generalized sum $\ds \sum_{k \mathop \in K} \paren{g {\restriction_K} }_k$ converges to $h$
 * the generalized sum $\ds \sum_{k \mathop \in K} \paren{g {\restriction_K} }_k$ converges to $h$

Proof
Let $J = \set{i \in I : g_i \ne 0_G}$.

Let $g' = g \restriction_J$ be the restriction of $g$ to $J$.

From Restriction of Restriction is Restriction:

From User:Leigh.Samphier/Topology/Generalized Sum Restricted to Non-zero Summands:
 * the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converges to $h$


 * the generalized sum $\ds \sum_{j \mathop \in J} g'_j$ converges to $h$
 * the generalized sum $\ds \sum_{j \mathop \in J} g'_j$ converges to $h$


 * the generalized sum $\ds \sum_{k \mathop \in K} g^*_k$ converges to $h$
 * the generalized sum $\ds \sum_{k \mathop \in K} g^*_k$ converges to $h$