Theorem gsumsn

Description: Group sum of a singleton. (Contributed by Mario Carneiro, 19-Dec-2014) (Revised by Mario Carneiro, 24-Apr-2016) (Proof shortened by AV, 11-Dec-2019)

	Ref	Expression
Hypotheses	gsumsn.b	$$⊢B={\mathrm{Base}}_G$$
	gsumsn.s	$$⊢k=M\to A=C$$
Assertion	gsumsn	$$⊢\left(G\in \mathrm{Mnd}\wedge M\in V\wedge C\in B\right)\to \underset{k\in \left\{M\right\}}{{\sum }_G}A=C$$

Proof

Step	Hyp	Ref	Expression
1		gsumsn.b	$$⊢B={\mathrm{Base}}_G$$
2		gsumsn.s	$$⊢k=M\to A=C$$
3		nfcv	$$⊢\underset{\_}{Ⅎ}kC$$
4	3 1 2	gsumsnf	$$⊢\left(G\in \mathrm{Mnd}\wedge M\in V\wedge C\in B\right)\to \underset{k\in \left\{M\right\}}{{\sum }_G}A=C$$