Theorem matsca

Description: The matrix ring has the same scalars as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015) (Proof shortened by AV, 12-Nov-2024)

	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		scandxnmulrndx	|- ( Scalar ` ndx ) =/= ( .r ` ndx )
5	3 4	setsnid	|- ( Scalar ` G ) = ( Scalar ` ( G sSet <. ( .r ` ndx ) , ( R maMul <. N , N , N >. ) >. ) )
6		eqid	|- ( R maMul <. N , N , N >. ) = ( R maMul <. N , N , N >. )
7	1 2 6	matval	|- ( ( N e. Fin /\ R e. V ) -> A = ( G sSet <. ( .r ` ndx ) , ( R maMul <. N , N , N >. ) >. ) )
8	7	fveq2d	|- ( ( N e. Fin /\ R e. V ) -> ( Scalar ` A ) = ( Scalar ` ( G sSet <. ( .r ` ndx ) , ( R maMul <. N , N , N >. ) >. ) ) )
9	5 8	eqtr4id	|- ( ( N e. Fin /\ R e. V ) -> ( Scalar ` G ) = ( Scalar ` A ) )