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 = ( Base ` G )
	gsumsn.s	|- ( k = M -> A = C )
Assertion	gsumsn	|- ( ( G e. Mnd /\ M e. V /\ C e. B ) -> ( G gsum ( k e. { M } |-> A ) ) = C )

Proof

Step	Hyp	Ref	Expression
1		gsumsn.b	|- B = ( Base ` G )
2		gsumsn.s	|- ( k = M -> A = C )
3		nfcv	|- F/_ k C
4	3 1 2	gsumsnf	|- ( ( G e. Mnd /\ M e. V /\ C e. B ) -> ( G gsum ( k e. { M } |-> A ) ) = C )