Theorem drnginvrcld

Description: Closure of the multiplicative inverse in a division ring. ( reccld analog). (Contributed by SN, 14-Aug-2024)

	Ref	Expression
Hypotheses	invrcl.b	|- B = ( Base ` R )
	invrcl.z	|- .0. = ( 0g ` R )
	invrcl.i	|- I = ( invr ` R )
	drnginvrcld.r	|- ( ph -> R e. DivRing )
	drnginvrcld.x	|- ( ph -> X e. B )
	drnginvrcld.1	|- ( ph -> X =/= .0. )
Assertion	drnginvrcld	|- ( ph -> ( I ` X ) e. B )

Proof

Step	Hyp	Ref	Expression
1		invrcl.b	|- B = ( Base ` R )
2		invrcl.z	|- .0. = ( 0g ` R )
3		invrcl.i	|- I = ( invr ` R )
4		drnginvrcld.r	|- ( ph -> R e. DivRing )
5		drnginvrcld.x	|- ( ph -> X e. B )
6		drnginvrcld.1	|- ( ph -> X =/= .0. )
7	1 2 3	drnginvrcl	|- ( ( R e. DivRing /\ X e. B /\ X =/= .0. ) -> ( I ` X ) e. B )
8	4 5 6 7	syl3anc	|- ( ph -> ( I ` X ) e. B )