Metamath Proof Explorer


Theorem gsummulc2

Description: A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014) (Revised by AV, 10-Jul-2019) Remove unused hypothesis. (Revised by SN, 7-Mar-2025)

Ref Expression
Hypotheses gsummulc1.b
|- B = ( Base ` R )
gsummulc1.z
|- .0. = ( 0g ` 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 gsummulc2
|- ( ph -> ( R gsum ( k e. A |-> ( Y .x. X ) ) ) = ( Y .x. ( R gsum ( k e. A |-> X ) ) ) )

Proof

Step Hyp Ref Expression
1 gsummulc1.b
 |-  B = ( Base ` R )
2 gsummulc1.z
 |-  .0. = ( 0g ` R )
3 gsummulc1.t
 |-  .x. = ( .r ` R )
4 gsummulc1.r
 |-  ( ph -> R e. Ring )
5 gsummulc1.a
 |-  ( ph -> A e. V )
6 gsummulc1.y
 |-  ( ph -> Y e. B )
7 gsummulc1.x
 |-  ( ( ph /\ k e. A ) -> X e. B )
8 gsummulc1.n
 |-  ( ph -> ( k e. A |-> X ) finSupp .0. )
9 4 ringcmnd
 |-  ( ph -> R e. CMnd )
10 ringmnd
 |-  ( R e. Ring -> R e. Mnd )
11 4 10 syl
 |-  ( ph -> R e. Mnd )
12 1 3 ringlghm
 |-  ( ( R e. Ring /\ Y e. B ) -> ( x e. B |-> ( Y .x. x ) ) e. ( R GrpHom R ) )
13 4 6 12 syl2anc
 |-  ( ph -> ( x e. B |-> ( Y .x. x ) ) e. ( R GrpHom R ) )
14 ghmmhm
 |-  ( ( x e. B |-> ( Y .x. x ) ) e. ( R GrpHom R ) -> ( x e. B |-> ( Y .x. x ) ) e. ( R MndHom R ) )
15 13 14 syl
 |-  ( ph -> ( x e. B |-> ( Y .x. x ) ) e. ( R MndHom R ) )
16 oveq2
 |-  ( x = X -> ( Y .x. x ) = ( Y .x. X ) )
17 oveq2
 |-  ( x = ( R gsum ( k e. A |-> X ) ) -> ( Y .x. x ) = ( Y .x. ( R gsum ( k e. A |-> X ) ) ) )
18 1 2 9 11 5 15 7 8 16 17 gsummhm2
 |-  ( ph -> ( R gsum ( k e. A |-> ( Y .x. X ) ) ) = ( Y .x. ( R gsum ( k e. A |-> X ) ) ) )