Theorem mat1f

Description: There is a function 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	mat1f	|- ( ( R e. Ring /\ E e. V ) -> F : K --> B )

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	mat1f1o	|- ( ( R e. Ring /\ E e. V ) -> F : K -1-1-onto-> B )
7		f1of	|- ( F : K -1-1-onto-> B -> F : K --> B )
8	6 7	syl	|- ( ( R e. Ring /\ E e. V ) -> F : K --> B )