Metamath Proof Explorer


Theorem mat1rngiso

Description: There is a ring isomorphism from a ring to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019)

Ref Expression
Hypotheses mat1rhmval.k
|- K = ( Base ` R )
mat1rhmval.a
|- A = ( { E } Mat R )
mat1rhmval.b
|- B = ( Base ` A )
mat1rhmval.o
|- O = <. E , E >.
mat1rhmval.f
|- F = ( x e. K |-> { <. O , x >. } )
Assertion mat1rngiso
|- ( ( R e. Ring /\ E e. V ) -> F e. ( R RingIso A ) )

Proof

Step Hyp Ref Expression
1 mat1rhmval.k
 |-  K = ( Base ` R )
2 mat1rhmval.a
 |-  A = ( { E } Mat R )
3 mat1rhmval.b
 |-  B = ( Base ` A )
4 mat1rhmval.o
 |-  O = <. E , E >.
5 mat1rhmval.f
 |-  F = ( x e. K |-> { <. O , x >. } )
6 1 2 3 4 5 mat1rhm
 |-  ( ( R e. Ring /\ E e. V ) -> F e. ( R RingHom A ) )
7 1 2 3 4 5 mat1f1o
 |-  ( ( R e. Ring /\ E e. V ) -> F : K -1-1-onto-> B )
8 1 3 isrim
 |-  ( F e. ( R RingIso A ) <-> ( F e. ( R RingHom A ) /\ F : K -1-1-onto-> B ) )
9 6 7 8 sylanbrc
 |-  ( ( R e. Ring /\ E e. V ) -> F e. ( R RingIso A ) )