Theorem mat1rhmcl

Description: The value of the ring homomorphism F is a matrix with dimension 1. (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	mat1rhmcl	|- ( ( R e. Ring /\ E e. V /\ X e. K ) -> ( F ` X ) e. 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	2 1 4	mat1dimbas	|- ( ( R e. Ring /\ E e. V /\ X e. K ) -> { <. O , X >. } e. ( Base ` A ) )
7	1 2 3 4 5	mat1rhmval	|- ( ( R e. Ring /\ E e. V /\ X e. K ) -> ( F ` X ) = { <. O , X >. } )
8	3	a1i	|- ( ( R e. Ring /\ E e. V /\ X e. K ) -> B = ( Base ` A ) )
9	6 7 8	3eltr4d	|- ( ( R e. Ring /\ E e. V /\ X e. K ) -> ( F ` X ) e. B )