Theorem gsumsnd

Description: Group sum of a singleton, deduction form. (Contributed by Thierry Arnoux, 30-Jan-2017) (Proof shortened by AV, 11-Dec-2019)

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

Proof

Step	Hyp	Ref	Expression
1		gsumsnd.b	$$⊢B={\mathrm{Base}}_G$$
2		gsumsnd.g	$$⊢\phi \to G\in \mathrm{Mnd}$$
3		gsumsnd.m	$$⊢\phi \to M\in V$$
4		gsumsnd.c	$$⊢\phi \to C\in B$$
5		gsumsnd.s	$$⊢\left(\phi \wedge k=M\right)\to A=C$$
6		nfv	$$⊢Ⅎk\phi$$
7		nfcv	$$⊢\underset{\_}{Ⅎ}kC$$
8	1 2 3 4 5 6 7	gsumsnfd	$$⊢\phi \to \underset{k\in \left\{M\right\}}{{\sum }_G}A=C$$