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Metamath Proof Explorer

Theorem matsca2

Description: The scalars of the matrix ring are the underlying ring. (Contributed by Stefan O'Rear, 5-Sep-2015)

Ref Expression
Hypothesis matsca2.a 
|- A = ( N Mat R )
Assertion matsca2 
|- ( ( N e. Fin /\ R e. V ) -> R = ( Scalar ` A ) )

Proof

Step Hyp Ref Expression
1 matsca2.a 
 |-  A = ( N Mat R )
2 xpfi 
 |-  ( ( N e. Fin /\ N e. Fin ) -> ( N X. N ) e. Fin )
3 2 anidms 
 |-  ( N e. Fin -> ( N X. N ) e. Fin )
4 eqid 
 |-  ( R freeLMod ( N X. N ) ) = ( R freeLMod ( N X. N ) )
5 4 frlmsca 
 |-  ( ( R e. V /\ ( N X. N ) e. Fin ) -> R = ( Scalar ` ( R freeLMod ( N X. N ) ) ) )
6 5 ancoms 
 |-  ( ( ( N X. N ) e. Fin /\ R e. V ) -> R = ( Scalar ` ( R freeLMod ( N X. N ) ) ) )
7 3 6 sylan 
 |-  ( ( N e. Fin /\ R e. V ) -> R = ( Scalar ` ( R freeLMod ( N X. N ) ) ) )
8 1 4 matsca 
 |-  ( ( N e. Fin /\ R e. V ) -> ( Scalar ` ( R freeLMod ( N X. N ) ) ) = ( Scalar ` A ) )
9 7 8 eqtrd 
 |-  ( ( N e. Fin /\ R e. V ) -> R = ( Scalar ` A ) )