Theorem gsummulc1

Description: A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014) (Revised by AV, 10-Jul-2019)

	Ref	Expression
Hypotheses	gsummulc1.b	|- B = ( Base ` R )
	gsummulc1.z	|- .0. = ( 0g ` R )
	gsummulc1.p	|- .+ = ( +g ` R )
	gsummulc1.t	|- .x. = ( .r ` R )
	gsummulc1.r	|- ( ph -> R e. Ring )
	gsummulc1.a	|- ( ph -> A e. V )
	gsummulc1.y	|- ( ph -> Y e. B )
	gsummulc1.x	|- ( ( ph /\ k e. A ) -> X e. B )
	gsummulc1.n	|- ( ph -> ( k e. A |-> X ) finSupp .0. )
Assertion	gsummulc1	|- ( ph -> ( R gsum ( k e. A |-> ( X .x. Y ) ) ) = ( ( R gsum ( k e. A |-> X ) ) .x. Y ) )

Proof

Step	Hyp	Ref	Expression
1		gsummulc1.b	|- B = ( Base ` R )
2		gsummulc1.z	|- .0. = ( 0g ` R )
3		gsummulc1.p	|- .+ = ( +g ` R )
4		gsummulc1.t	|- .x. = ( .r ` R )
5		gsummulc1.r	|- ( ph -> R e. Ring )
6		gsummulc1.a	|- ( ph -> A e. V )
7		gsummulc1.y	|- ( ph -> Y e. B )
8		gsummulc1.x	|- ( ( ph /\ k e. A ) -> X e. B )
9		gsummulc1.n	|- ( ph -> ( k e. A |-> X ) finSupp .0. )
10		ringcmn	|- ( R e. Ring -> R e. CMnd )
11	5 10	syl	|- ( ph -> R e. CMnd )
12		ringmnd	|- ( R e. Ring -> R e. Mnd )
13	5 12	syl	|- ( ph -> R e. Mnd )
14	1 4	ringrghm	|- ( ( R e. Ring /\ Y e. B ) -> ( x e. B |-> ( x .x. Y ) ) e. ( R GrpHom R ) )
15	5 7 14	syl2anc	|- ( ph -> ( x e. B |-> ( x .x. Y ) ) e. ( R GrpHom R ) )
16		ghmmhm	|- ( ( x e. B |-> ( x .x. Y ) ) e. ( R GrpHom R ) -> ( x e. B |-> ( x .x. Y ) ) e. ( R MndHom R ) )
17	15 16	syl	|- ( ph -> ( x e. B |-> ( x .x. Y ) ) e. ( R MndHom R ) )
18		oveq1	|- ( x = X -> ( x .x. Y ) = ( X .x. Y ) )
19		oveq1	|- ( x = ( R gsum ( k e. A |-> X ) ) -> ( x .x. Y ) = ( ( R gsum ( k e. A |-> X ) ) .x. Y ) )
20	1 2 11 13 6 17 8 9 18 19	gsummhm2	|- ( ph -> ( R gsum ( k e. A |-> ( X .x. Y ) ) ) = ( ( R gsum ( k e. A |-> X ) ) .x. Y ) )