Metamath Proof Explorer

Theorem mat2pmatfval

Description: Value of the matrix transformation. (Contributed by AV, 31-Jul-2019)

Ref Expression
Hypotheses mat2pmatfval.t 
|- T = ( N matToPolyMat R )
mat2pmatfval.a 
|- A = ( N Mat R )
mat2pmatfval.b 
|- B = ( Base ` A )
mat2pmatfval.p 
|- P = ( Poly1 ` R )
mat2pmatfval.s 
|- S = ( algSc ` P )
Assertion mat2pmatfval 
|- ( ( N e. Fin /\ R e. V ) -> T = ( m e. B |-> ( x e. N , y e. N |-> ( S ` ( x m y ) ) ) ) )

Proof

Step Hyp Ref Expression
1 mat2pmatfval.t 
 |-  T = ( N matToPolyMat R )
2 mat2pmatfval.a 
 |-  A = ( N Mat R )
3 mat2pmatfval.b 
 |-  B = ( Base ` A )
4 mat2pmatfval.p 
 |-  P = ( Poly1 ` R )
5 mat2pmatfval.s 
 |-  S = ( algSc ` P )
6 df-mat2pmat 
 |-  matToPolyMat = ( n e. Fin , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) |-> ( x e. n , y e. n |-> ( ( algSc ` ( Poly1 ` r ) ) ` ( x m y ) ) ) ) )
7 6 a1i 
 |-  ( ( N e. Fin /\ R e. V ) -> matToPolyMat = ( n e. Fin , r e. _V |-> ( m e. ( Base ` ( n Mat r ) ) |-> ( x e. n , y e. n |-> ( ( algSc ` ( Poly1 ` r ) ) ` ( x m y ) ) ) ) ) )
8 oveq12 
 |-  ( ( n = N /\ r = R ) -> ( n Mat r ) = ( N Mat R ) )
9 8 fveq2d 
 |-  ( ( n = N /\ r = R ) -> ( Base ` ( n Mat r ) ) = ( Base ` ( N Mat R ) ) )
10 2 fveq2i 
 |-  ( Base ` A ) = ( Base ` ( N Mat R ) )
11 3 10 eqtr2i 
 |-  ( Base ` ( N Mat R ) ) = B
12 9 11 eqtrdi 
 |-  ( ( n = N /\ r = R ) -> ( Base ` ( n Mat r ) ) = B )
13 simpl 
 |-  ( ( n = N /\ r = R ) -> n = N )
14 2fveq3 
 |-  ( r = R -> ( algSc ` ( Poly1 ` r ) ) = ( algSc ` ( Poly1 ` R ) ) )
15 14 adantl 
 |-  ( ( n = N /\ r = R ) -> ( algSc ` ( Poly1 ` r ) ) = ( algSc ` ( Poly1 ` R ) ) )
16 4 fveq2i 
 |-  ( algSc ` P ) = ( algSc ` ( Poly1 ` R ) )
17 5 16 eqtr2i 
 |-  ( algSc ` ( Poly1 ` R ) ) = S
18 15 17 eqtrdi 
 |-  ( ( n = N /\ r = R ) -> ( algSc ` ( Poly1 ` r ) ) = S )
19 18 fveq1d 
 |-  ( ( n = N /\ r = R ) -> ( ( algSc ` ( Poly1 ` r ) ) ` ( x m y ) ) = ( S ` ( x m y ) ) )
20 13 13 19 mpoeq123dv 
 |-  ( ( n = N /\ r = R ) -> ( x e. n , y e. n |-> ( ( algSc ` ( Poly1 ` r ) ) ` ( x m y ) ) ) = ( x e. N , y e. N |-> ( S ` ( x m y ) ) ) )
21 12 20 mpteq12dv 
 |-  ( ( n = N /\ r = R ) -> ( m e. ( Base ` ( n Mat r ) ) |-> ( x e. n , y e. n |-> ( ( algSc ` ( Poly1 ` r ) ) ` ( x m y ) ) ) ) = ( m e. B |-> ( x e. N , y e. N |-> ( S ` ( x m y ) ) ) ) )
22 21 adantl 
 |-  ( ( ( N e. Fin /\ R e. V ) /\ ( n = N /\ r = R ) ) -> ( m e. ( Base ` ( n Mat r ) ) |-> ( x e. n , y e. n |-> ( ( algSc ` ( Poly1 ` r ) ) ` ( x m y ) ) ) ) = ( m e. B |-> ( x e. N , y e. N |-> ( S ` ( x m y ) ) ) ) )
23 simpl 
 |-  ( ( N e. Fin /\ R e. V ) -> N e. Fin )
24 elex 
 |-  ( R e. V -> R e. _V )
25 24 adantl 
 |-  ( ( N e. Fin /\ R e. V ) -> R e. _V )
26 3 fvexi 
 |-  B e. _V
27 mptexg 
 |-  ( B e. _V -> ( m e. B |-> ( x e. N , y e. N |-> ( S ` ( x m y ) ) ) ) e. _V )
28 26 27 mp1i 
 |-  ( ( N e. Fin /\ R e. V ) -> ( m e. B |-> ( x e. N , y e. N |-> ( S ` ( x m y ) ) ) ) e. _V )
29 7 22 23 25 28 ovmpod 
 |-  ( ( N e. Fin /\ R e. V ) -> ( N matToPolyMat R ) = ( m e. B |-> ( x e. N , y e. N |-> ( S ` ( x m y ) ) ) ) )
30 1 29 syl5eq 
 |-  ( ( N e. Fin /\ R e. V ) -> T = ( m e. B |-> ( x e. N , y e. N |-> ( S ` ( x m y ) ) ) ) )