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Metamath Proof Explorer

Theorem dmnnzd

Description: A domain has no zero-divisors (besides zero). (Contributed by Jeff Madsen, 19-Jun-2010)

Ref Expression
Hypotheses dmnnzd.1 
|- G = ( 1st ` R )
dmnnzd.2 
|- H = ( 2nd ` R )
dmnnzd.3 
|- X = ran G
dmnnzd.4 
|- Z = ( GId ` G )
Assertion dmnnzd 
|- ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ ( A H B ) = Z ) ) -> ( A = Z \/ B = Z ) )

Proof

Step Hyp Ref Expression
1 dmnnzd.1 
 |-  G = ( 1st ` R )
2 dmnnzd.2 
 |-  H = ( 2nd ` R )
3 dmnnzd.3 
 |-  X = ran G
4 dmnnzd.4 
 |-  Z = ( GId ` G )
5 eqid 
 |-  ( GId ` H ) = ( GId ` H )
6 1 2 3 4 5 isdmn3 
 |-  ( R e. Dmn <-> ( R e. CRingOps /\ ( GId ` H ) =/= Z /\ A. a e. X A. b e. X ( ( a H b ) = Z -> ( a = Z \/ b = Z ) ) ) )
7 6 simp3bi 
 |-  ( R e. Dmn -> A. a e. X A. b e. X ( ( a H b ) = Z -> ( a = Z \/ b = Z ) ) )
8 oveq1 
 |-  ( a = A -> ( a H b ) = ( A H b ) )
9 8 eqeq1d 
 |-  ( a = A -> ( ( a H b ) = Z <-> ( A H b ) = Z ) )
10 eqeq1 
 |-  ( a = A -> ( a = Z <-> A = Z ) )
11 10 orbi1d 
 |-  ( a = A -> ( ( a = Z \/ b = Z ) <-> ( A = Z \/ b = Z ) ) )
12 9 11 imbi12d 
 |-  ( a = A -> ( ( ( a H b ) = Z -> ( a = Z \/ b = Z ) ) <-> ( ( A H b ) = Z -> ( A = Z \/ b = Z ) ) ) )
13 oveq2 
 |-  ( b = B -> ( A H b ) = ( A H B ) )
14 13 eqeq1d 
 |-  ( b = B -> ( ( A H b ) = Z <-> ( A H B ) = Z ) )
15 eqeq1 
 |-  ( b = B -> ( b = Z <-> B = Z ) )
16 15 orbi2d 
 |-  ( b = B -> ( ( A = Z \/ b = Z ) <-> ( A = Z \/ B = Z ) ) )
17 14 16 imbi12d 
 |-  ( b = B -> ( ( ( A H b ) = Z -> ( A = Z \/ b = Z ) ) <-> ( ( A H B ) = Z -> ( A = Z \/ B = Z ) ) ) )
18 12 17 rspc2v 
 |-  ( ( A e. X /\ B e. X ) -> ( A. a e. X A. b e. X ( ( a H b ) = Z -> ( a = Z \/ b = Z ) ) -> ( ( A H B ) = Z -> ( A = Z \/ B = Z ) ) ) )
19 7 18 syl5com 
 |-  ( R e. Dmn -> ( ( A e. X /\ B e. X ) -> ( ( A H B ) = Z -> ( A = Z \/ B = Z ) ) ) )
20 19 expd 
 |-  ( R e. Dmn -> ( A e. X -> ( B e. X -> ( ( A H B ) = Z -> ( A = Z \/ B = Z ) ) ) ) )
21 20 3imp2 
 |-  ( ( R e. Dmn /\ ( A e. X /\ B e. X /\ ( A H B ) = Z ) ) -> ( A = Z \/ B = Z ) )