Metamath Proof Explorer

Theorem matsca

Description: The matrix ring has the same scalars as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015)

Ref Expression
Hypotheses matbas.a 
|- A = ( N Mat R )
matbas.g 
|- G = ( R freeLMod ( N X. N ) )
Assertion matsca 
|- ( ( N e. Fin /\ R e. V ) -> ( Scalar ` G ) = ( Scalar ` A ) )

Proof

Step Hyp Ref Expression
1 matbas.a 
 |-  A = ( N Mat R )
2 matbas.g 
 |-  G = ( R freeLMod ( N X. N ) )
3 scaid 
 |-  Scalar = Slot ( Scalar ` ndx )
4 3re 
 |-  3 e. RR
5 3lt5 
 |-  3 < 5
6 4 5 gtneii 
 |-  5 =/= 3
7 scandx 
 |-  ( Scalar ` ndx ) = 5
8 mulrndx 
 |-  ( .r ` ndx ) = 3
9 7 8 neeq12i 
 |-  ( ( Scalar ` ndx ) =/= ( .r ` ndx ) <-> 5 =/= 3 )
10 6 9 mpbir 
 |-  ( Scalar ` ndx ) =/= ( .r ` ndx )
11 3 10 setsnid 
 |-  ( Scalar ` G ) = ( Scalar ` ( G sSet <. ( .r ` ndx ) , ( R maMul <. N , N , N >. ) >. ) )
12 eqid 
 |-  ( R maMul <. N , N , N >. ) = ( R maMul <. N , N , N >. )
13 1 2 12 matval 
 |-  ( ( N e. Fin /\ R e. V ) -> A = ( G sSet <. ( .r ` ndx ) , ( R maMul <. N , N , N >. ) >. ) )
14 13 fveq2d 
 |-  ( ( N e. Fin /\ R e. V ) -> ( Scalar ` A ) = ( Scalar ` ( G sSet <. ( .r ` ndx ) , ( R maMul <. N , N , N >. ) >. ) ) )
15 11 14 eqtr4id 
 |-  ( ( N e. Fin /\ R e. V ) -> ( Scalar ` G ) = ( Scalar ` A ) )