Metamath Proof Explorer

Theorem mat2pmatf

Description: The matrix transformation is a function from the matrices to the polynomial matrices. (Contributed by AV, 27-Oct-2019)

Ref Expression
Hypotheses mat2pmatbas.t 
|- T = ( N matToPolyMat R )
mat2pmatbas.a 
|- A = ( N Mat R )
mat2pmatbas.b 
|- B = ( Base ` A )
mat2pmatbas.p 
|- P = ( Poly1 ` R )
mat2pmatbas.c 
|- C = ( N Mat P )
mat2pmatbas0.h 
|- H = ( Base ` C )
Assertion mat2pmatf 
|- ( ( N e. Fin /\ R e. Ring ) -> T : B --> H )

Proof

Step Hyp Ref Expression
1 mat2pmatbas.t 
 |-  T = ( N matToPolyMat R )
2 mat2pmatbas.a 
 |-  A = ( N Mat R )
3 mat2pmatbas.b 
 |-  B = ( Base ` A )
4 mat2pmatbas.p 
 |-  P = ( Poly1 ` R )
5 mat2pmatbas.c 
 |-  C = ( N Mat P )
6 mat2pmatbas0.h 
 |-  H = ( Base ` C )
7 simpl 
 |-  ( ( N e. Fin /\ R e. Ring ) -> N e. Fin )
8 7 7 jca 
 |-  ( ( N e. Fin /\ R e. Ring ) -> ( N e. Fin /\ N e. Fin ) )
9 8 adantr 
 |-  ( ( ( N e. Fin /\ R e. Ring ) /\ m e. B ) -> ( N e. Fin /\ N e. Fin ) )
10 mpoexga 
 |-  ( ( N e. Fin /\ N e. Fin ) -> ( x e. N , y e. N |-> ( ( algSc ` P ) ` ( x m y ) ) ) e. _V )
11 9 10 syl 
 |-  ( ( ( N e. Fin /\ R e. Ring ) /\ m e. B ) -> ( x e. N , y e. N |-> ( ( algSc ` P ) ` ( x m y ) ) ) e. _V )
12 eqid 
 |-  ( algSc ` P ) = ( algSc ` P )
13 1 2 3 4 12 mat2pmatfval 
 |-  ( ( N e. Fin /\ R e. Ring ) -> T = ( m e. B |-> ( x e. N , y e. N |-> ( ( algSc ` P ) ` ( x m y ) ) ) ) )
14 1 2 3 4 5 6 mat2pmatbas0 
 |-  ( ( N e. Fin /\ R e. Ring /\ m e. B ) -> ( T ` m ) e. H )
15 14 3expa 
 |-  ( ( ( N e. Fin /\ R e. Ring ) /\ m e. B ) -> ( T ` m ) e. H )
16 11 13 15 fmpt2d 
 |-  ( ( N e. Fin /\ R e. Ring ) -> T : B --> H )