Metamath Proof Explorer

Theorem dvdsrcl2

Description: Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014)

Ref Expression
Hypotheses dvdsr.1 
|- B = ( Base ` R )
dvdsr.2 
|- .|| = ( ||r ` R )
Assertion dvdsrcl2 
|- ( ( R e. Ring /\ X .|| Y ) -> Y e. B )

Proof

Step Hyp Ref Expression
1 dvdsr.1 
 |-  B = ( Base ` R )
2 dvdsr.2 
 |-  .|| = ( ||r ` R )
3 eqid 
 |-  ( .r ` R ) = ( .r ` R )
4 1 2 3 dvdsr 
 |-  ( X .|| Y <-> ( X e. B /\ E. x e. B ( x ( .r ` R ) X ) = Y ) )
5 1 3 ringcl 
 |-  ( ( R e. Ring /\ x e. B /\ X e. B ) -> ( x ( .r ` R ) X ) e. B )
6 5 3expa 
 |-  ( ( ( R e. Ring /\ x e. B ) /\ X e. B ) -> ( x ( .r ` R ) X ) e. B )
7 6 an32s 
 |-  ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> ( x ( .r ` R ) X ) e. B )
8 eleq1 
 |-  ( ( x ( .r ` R ) X ) = Y -> ( ( x ( .r ` R ) X ) e. B <-> Y e. B ) )
9 7 8 syl5ibcom 
 |-  ( ( ( R e. Ring /\ X e. B ) /\ x e. B ) -> ( ( x ( .r ` R ) X ) = Y -> Y e. B ) )
10 9 rexlimdva 
 |-  ( ( R e. Ring /\ X e. B ) -> ( E. x e. B ( x ( .r ` R ) X ) = Y -> Y e. B ) )
11 10 impr 
 |-  ( ( R e. Ring /\ ( X e. B /\ E. x e. B ( x ( .r ` R ) X ) = Y ) ) -> Y e. B )
12 4 11 sylan2b 
 |-  ( ( R e. Ring /\ X .|| Y ) -> Y e. B )