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 ) ) |
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 ) ) |