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Metamath Proof Explorer

Theorem dvdsr02

Description: Only zero is divisible by zero. (Contributed by Stefan O'Rear, 29-Mar-2015)

Ref Expression
Hypotheses dvdsr0.b 
|- B = ( Base ` R )
dvdsr0.d 
|- .|| = ( ||r ` R )
dvdsr0.z 
|- .0. = ( 0g ` R )
Assertion dvdsr02 
|- ( ( R e. Ring /\ X e. B ) -> ( .0. .|| X <-> X = .0. ) )

Proof

Step Hyp Ref Expression
1 dvdsr0.b 
 |-  B = ( Base ` R )
2 dvdsr0.d 
 |-  .|| = ( ||r ` R )
3 dvdsr0.z 
 |-  .0. = ( 0g ` R )
4 1 3 ring0cl 
 |-  ( R e. Ring -> .0. e. B )
5 4 adantr 
 |-  ( ( R e. Ring /\ X e. B ) -> .0. e. B )
6 eqid 
 |-  ( .r ` R ) = ( .r ` R )
7 1 2 6 dvdsr2 
 |-  ( .0. e. B -> ( .0. .|| X <-> E. x e. B ( x ( .r ` R ) .0. ) = X ) )
8 5 7 syl 
 |-  ( ( R e. Ring /\ X e. B ) -> ( .0. .|| X <-> E. x e. B ( x ( .r ` R ) .0. ) = X ) )
9 1 6 3 ringrz 
 |-  ( ( R e. Ring /\ x e. B ) -> ( x ( .r ` R ) .0. ) = .0. )
10 9 eqeq1d 
 |-  ( ( R e. Ring /\ x e. B ) -> ( ( x ( .r ` R ) .0. ) = X <-> .0. = X ) )
11 eqcom 
 |-  ( .0. = X <-> X = .0. )
12 10 11 bitrdi 
 |-  ( ( R e. Ring /\ x e. B ) -> ( ( x ( .r ` R ) .0. ) = X <-> X = .0. ) )
13 12 rexbidva 
 |-  ( R e. Ring -> ( E. x e. B ( x ( .r ` R ) .0. ) = X <-> E. x e. B X = .0. ) )
14 ringgrp 
 |-  ( R e. Ring -> R e. Grp )
15 1 grpbn0 
 |-  ( R e. Grp -> B =/= (/) )
16 r19.9rzv 
 |-  ( B =/= (/) -> ( X = .0. <-> E. x e. B X = .0. ) )
17 14 15 16 3syl 
 |-  ( R e. Ring -> ( X = .0. <-> E. x e. B X = .0. ) )
18 13 17 bitr4d 
 |-  ( R e. Ring -> ( E. x e. B ( x ( .r ` R ) .0. ) = X <-> X = .0. ) )
19 18 adantr 
 |-  ( ( R e. Ring /\ X e. B ) -> ( E. x e. B ( x ( .r ` R ) .0. ) = X <-> X = .0. ) )
20 8 19 bitrd 
 |-  ( ( R e. Ring /\ X e. B ) -> ( .0. .|| X <-> X = .0. ) )