Step |
Hyp |
Ref |
Expression |
1 |
|
mgpsumunsn.m |
|- M = ( mulGrp ` R ) |
2 |
|
mgpsumunsn.t |
|- .x. = ( .r ` R ) |
3 |
|
mgpsumunsn.r |
|- ( ph -> R e. CRing ) |
4 |
|
mgpsumunsn.n |
|- ( ph -> N e. Fin ) |
5 |
|
mgpsumunsn.i |
|- ( ph -> I e. N ) |
6 |
|
mgpsumunsn.a |
|- ( ( ph /\ k e. N ) -> A e. ( Base ` R ) ) |
7 |
|
mgpsumz.z |
|- .0. = ( 0g ` R ) |
8 |
|
mgpsumz.0 |
|- ( k = I -> A = .0. ) |
9 |
|
crngring |
|- ( R e. CRing -> R e. Ring ) |
10 |
|
ringmnd |
|- ( R e. Ring -> R e. Mnd ) |
11 |
9 10
|
syl |
|- ( R e. CRing -> R e. Mnd ) |
12 |
3 11
|
syl |
|- ( ph -> R e. Mnd ) |
13 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
14 |
13 7
|
mndidcl |
|- ( R e. Mnd -> .0. e. ( Base ` R ) ) |
15 |
12 14
|
syl |
|- ( ph -> .0. e. ( Base ` R ) ) |
16 |
1 2 3 4 5 6 15 8
|
mgpsumunsn |
|- ( ph -> ( M gsum ( k e. N |-> A ) ) = ( ( M gsum ( k e. ( N \ { I } ) |-> A ) ) .x. .0. ) ) |
17 |
3 9
|
syl |
|- ( ph -> R e. Ring ) |
18 |
1 13
|
mgpbas |
|- ( Base ` R ) = ( Base ` M ) |
19 |
1
|
crngmgp |
|- ( R e. CRing -> M e. CMnd ) |
20 |
3 19
|
syl |
|- ( ph -> M e. CMnd ) |
21 |
|
diffi |
|- ( N e. Fin -> ( N \ { I } ) e. Fin ) |
22 |
4 21
|
syl |
|- ( ph -> ( N \ { I } ) e. Fin ) |
23 |
|
eldifi |
|- ( k e. ( N \ { I } ) -> k e. N ) |
24 |
23 6
|
sylan2 |
|- ( ( ph /\ k e. ( N \ { I } ) ) -> A e. ( Base ` R ) ) |
25 |
24
|
ralrimiva |
|- ( ph -> A. k e. ( N \ { I } ) A e. ( Base ` R ) ) |
26 |
18 20 22 25
|
gsummptcl |
|- ( ph -> ( M gsum ( k e. ( N \ { I } ) |-> A ) ) e. ( Base ` R ) ) |
27 |
13 2 7
|
ringrz |
|- ( ( R e. Ring /\ ( M gsum ( k e. ( N \ { I } ) |-> A ) ) e. ( Base ` R ) ) -> ( ( M gsum ( k e. ( N \ { I } ) |-> A ) ) .x. .0. ) = .0. ) |
28 |
17 26 27
|
syl2anc |
|- ( ph -> ( ( M gsum ( k e. ( N \ { I } ) |-> A ) ) .x. .0. ) = .0. ) |
29 |
16 28
|
eqtrd |
|- ( ph -> ( M gsum ( k e. N |-> A ) ) = .0. ) |