Theorem drngmcl

Description: The product of two nonzero elements of a division ring is nonzero. (Contributed by NM, 7-Sep-2011) (Proof shortened by SN, 25-Jun-2025)

	Ref	Expression
Hypotheses	drngmcl.b	|- B = ( Base ` R )
	drngmcl.t	|- .x. = ( .r ` R )
	drngmcl.z	|- .0. = ( 0g ` R )
Assertion	drngmcl	|- ( ( R e. DivRing /\ X e. ( B \ { .0. } ) /\ Y e. ( B \ { .0. } ) ) -> ( X .x. Y ) e. ( B \ { .0. } ) )

Proof

Step	Hyp	Ref	Expression
1		drngmcl.b	|- B = ( Base ` R )
2		drngmcl.t	|- .x. = ( .r ` R )
3		drngmcl.z	|- .0. = ( 0g ` R )
4		drngring	|- ( R e. DivRing -> R e. Ring )
5		eldifi	|- ( X e. ( B \ { .0. } ) -> X e. B )
6		eldifi	|- ( Y e. ( B \ { .0. } ) -> Y e. B )
7	1 2	ringcl	|- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )
8	4 5 6 7	syl3an	|- ( ( R e. DivRing /\ X e. ( B \ { .0. } ) /\ Y e. ( B \ { .0. } ) ) -> ( X .x. Y ) e. B )
9		drngdomn	|- ( R e. DivRing -> R e. Domn )
10		eldifsn	|- ( X e. ( B \ { .0. } ) <-> ( X e. B /\ X =/= .0. ) )
11	10	biimpi	|- ( X e. ( B \ { .0. } ) -> ( X e. B /\ X =/= .0. ) )
12		eldifsn	|- ( Y e. ( B \ { .0. } ) <-> ( Y e. B /\ Y =/= .0. ) )
13	12	biimpi	|- ( Y e. ( B \ { .0. } ) -> ( Y e. B /\ Y =/= .0. ) )
14	1 2 3	domnmuln0	|- ( ( R e. Domn /\ ( X e. B /\ X =/= .0. ) /\ ( Y e. B /\ Y =/= .0. ) ) -> ( X .x. Y ) =/= .0. )
15	9 11 13 14	syl3an	|- ( ( R e. DivRing /\ X e. ( B \ { .0. } ) /\ Y e. ( B \ { .0. } ) ) -> ( X .x. Y ) =/= .0. )
16	8 15	eldifsnd	|- ( ( R e. DivRing /\ X e. ( B \ { .0. } ) /\ Y e. ( B \ { .0. } ) ) -> ( X .x. Y ) e. ( B \ { .0. } ) )