Theorem dvrcl

Description: Closure of division operation. (Contributed by Mario Carneiro, 2-Jul-2014)

	Ref	Expression
Hypotheses	dvrcl.b	|- B = ( Base ` R )
	dvrcl.o	|- U = ( Unit ` R )
	dvrcl.d	|- ./ = ( /r ` R )
Assertion	dvrcl	|- ( ( R e. Ring /\ X e. B /\ Y e. U ) -> ( X ./ Y ) e. B )

Proof

Step	Hyp	Ref	Expression
1		dvrcl.b	|- B = ( Base ` R )
2		dvrcl.o	|- U = ( Unit ` R )
3		dvrcl.d	|- ./ = ( /r ` R )
4		eqid	|- ( .r ` R ) = ( .r ` R )
5		eqid	|- ( invr ` R ) = ( invr ` R )
6	1 4 2 5 3	dvrval	|- ( ( X e. B /\ Y e. U ) -> ( X ./ Y ) = ( X ( .r ` R ) ( ( invr ` R ) ` Y ) ) )
7	6	3adant1	|- ( ( R e. Ring /\ X e. B /\ Y e. U ) -> ( X ./ Y ) = ( X ( .r ` R ) ( ( invr ` R ) ` Y ) ) )
8	2 5 1	ringinvcl	|- ( ( R e. Ring /\ Y e. U ) -> ( ( invr ` R ) ` Y ) e. B )
9	8	3adant2	|- ( ( R e. Ring /\ X e. B /\ Y e. U ) -> ( ( invr ` R ) ` Y ) e. B )
10	1 4	ringcl	|- ( ( R e. Ring /\ X e. B /\ ( ( invr ` R ) ` Y ) e. B ) -> ( X ( .r ` R ) ( ( invr ` R ) ` Y ) ) e. B )
11	9 10	syld3an3	|- ( ( R e. Ring /\ X e. B /\ Y e. U ) -> ( X ( .r ` R ) ( ( invr ` R ) ` Y ) ) e. B )
12	7 11	eqeltrd	|- ( ( R e. Ring /\ X e. B /\ Y e. U ) -> ( X ./ Y ) e. B )