Metamath Proof Explorer

Theorem m2cpmghm

Description: The transformation of matrices into constant polynomial matrices is an additive group homomorphism. (Contributed by AV, 18-Nov-2019)

Ref Expression
Hypotheses m2cpm.s 
|- S = ( N ConstPolyMat R )
m2cpm.t 
|- T = ( N matToPolyMat R )
m2cpm.a 
|- A = ( N Mat R )
m2cpm.b 
|- B = ( Base ` A )
m2cpmghm.p 
|- P = ( Poly1 ` R )
m2cpmghm.c 
|- C = ( N Mat P )
m2cpmghm.u 
|- U = ( C |`s S )
Assertion m2cpmghm 
|- ( ( N e. Fin /\ R e. Ring ) -> T e. ( A GrpHom U ) )

Proof

Step Hyp Ref Expression
1 m2cpm.s 
 |-  S = ( N ConstPolyMat R )
2 m2cpm.t 
 |-  T = ( N matToPolyMat R )
3 m2cpm.a 
 |-  A = ( N Mat R )
4 m2cpm.b 
 |-  B = ( Base ` A )
5 m2cpmghm.p 
 |-  P = ( Poly1 ` R )
6 m2cpmghm.c 
 |-  C = ( N Mat P )
7 m2cpmghm.u 
 |-  U = ( C |`s S )
8 eqid 
 |-  ( Base ` C ) = ( Base ` C )
9 2 3 4 5 6 8 mat2pmatghm 
 |-  ( ( N e. Fin /\ R e. Ring ) -> T e. ( A GrpHom C ) )
10 1 5 6 cpmatsubgpmat 
 |-  ( ( N e. Fin /\ R e. Ring ) -> S e. ( SubGrp ` C ) )
11 1 2 3 4 m2cpmf 
 |-  ( ( N e. Fin /\ R e. Ring ) -> T : B --> S )
12 11 frnd 
 |-  ( ( N e. Fin /\ R e. Ring ) -> ran T C_ S )
13 7 resghm2b 
 |-  ( ( S e. ( SubGrp ` C ) /\ ran T C_ S ) -> ( T e. ( A GrpHom C ) <-> T e. ( A GrpHom U ) ) )
14 13 bicomd 
 |-  ( ( S e. ( SubGrp ` C ) /\ ran T C_ S ) -> ( T e. ( A GrpHom U ) <-> T e. ( A GrpHom C ) ) )
15 10 12 14 syl2anc 
 |-  ( ( N e. Fin /\ R e. Ring ) -> ( T e. ( A GrpHom U ) <-> T e. ( A GrpHom C ) ) )
16 9 15 mpbird 
 |-  ( ( N e. Fin /\ R e. Ring ) -> T e. ( A GrpHom U ) )