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Metamath Proof Explorer

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. } ) )