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Metamath Proof Explorer

Theorem matgsumcl

Description: Closure of a group sum over the diagonal coefficients of a square matrix over a commutative ring. (Contributed by AV, 29-Dec-2018) (Proof shortened by AV, 23-Jul-2019)

Ref Expression
Hypotheses madetsumid.a 
|- A = ( N Mat R )
madetsumid.b 
|- B = ( Base ` A )
madetsumid.u 
|- U = ( mulGrp ` R )
Assertion matgsumcl 
|- ( ( R e. CRing /\ M e. B ) -> ( U gsum ( r e. N |-> ( r M r ) ) ) e. ( Base ` R ) )

Proof

Step Hyp Ref Expression
1 madetsumid.a 
 |-  A = ( N Mat R )
2 madetsumid.b 
 |-  B = ( Base ` A )
3 madetsumid.u 
 |-  U = ( mulGrp ` R )
4 eqid 
 |-  ( Base ` R ) = ( Base ` R )
5 3 4 mgpbas 
 |-  ( Base ` R ) = ( Base ` U )
6 3 crngmgp 
 |-  ( R e. CRing -> U e. CMnd )
7 6 adantr 
 |-  ( ( R e. CRing /\ M e. B ) -> U e. CMnd )
8 1 2 matrcl 
 |-  ( M e. B -> ( N e. Fin /\ R e. _V ) )
9 8 adantl 
 |-  ( ( R e. CRing /\ M e. B ) -> ( N e. Fin /\ R e. _V ) )
10 9 simpld 
 |-  ( ( R e. CRing /\ M e. B ) -> N e. Fin )
11 simpr 
 |-  ( ( R e. CRing /\ M e. B ) -> M e. B )
12 1 4 2 matbas2i 
 |-  ( M e. B -> M e. ( ( Base ` R ) ^m ( N X. N ) ) )
13 elmapi 
 |-  ( M e. ( ( Base ` R ) ^m ( N X. N ) ) -> M : ( N X. N ) --> ( Base ` R ) )
14 11 12 13 3syl 
 |-  ( ( R e. CRing /\ M e. B ) -> M : ( N X. N ) --> ( Base ` R ) )
15 14 adantr 
 |-  ( ( ( R e. CRing /\ M e. B ) /\ r e. N ) -> M : ( N X. N ) --> ( Base ` R ) )
16 simpr 
 |-  ( ( ( R e. CRing /\ M e. B ) /\ r e. N ) -> r e. N )
17 15 16 16 fovcdmd 
 |-  ( ( ( R e. CRing /\ M e. B ) /\ r e. N ) -> ( r M r ) e. ( Base ` R ) )
18 17 ralrimiva 
 |-  ( ( R e. CRing /\ M e. B ) -> A. r e. N ( r M r ) e. ( Base ` R ) )
19 5 7 10 18 gsummptcl 
 |-  ( ( R e. CRing /\ M e. B ) -> ( U gsum ( r e. N |-> ( r M r ) ) ) e. ( Base ` R ) )