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Metamath Proof Explorer

Theorem ringid

Description: The multiplication operation of a unital ring has (one or more) identity elements. (Contributed by Steve Rodriguez, 9-Sep-2007) (Revised by Mario Carneiro, 22-Dec-2013) (Revised by AV, 24-Aug-2021)

Ref Expression
Hypotheses ringid.b 
|- B = ( Base ` R )
ringid.t 
|- .x. = ( .r ` R )
Assertion ringid 
|- ( ( R e. Ring /\ X e. B ) -> E. u e. B ( ( u .x. X ) = X /\ ( X .x. u ) = X ) )

Proof

Step Hyp Ref Expression
1 ringid.b 
 |-  B = ( Base ` R )
2 ringid.t 
 |-  .x. = ( .r ` R )
3 eqid 
 |-  ( 1r ` R ) = ( 1r ` R )
4 1 3 ringidcl 
 |-  ( R e. Ring -> ( 1r ` R ) e. B )
5 4 adantr 
 |-  ( ( R e. Ring /\ X e. B ) -> ( 1r ` R ) e. B )
6 oveq1 
 |-  ( u = ( 1r ` R ) -> ( u .x. X ) = ( ( 1r ` R ) .x. X ) )
7 6 eqeq1d 
 |-  ( u = ( 1r ` R ) -> ( ( u .x. X ) = X <-> ( ( 1r ` R ) .x. X ) = X ) )
8 oveq2 
 |-  ( u = ( 1r ` R ) -> ( X .x. u ) = ( X .x. ( 1r ` R ) ) )
9 8 eqeq1d 
 |-  ( u = ( 1r ` R ) -> ( ( X .x. u ) = X <-> ( X .x. ( 1r ` R ) ) = X ) )
10 7 9 anbi12d 
 |-  ( u = ( 1r ` R ) -> ( ( ( u .x. X ) = X /\ ( X .x. u ) = X ) <-> ( ( ( 1r ` R ) .x. X ) = X /\ ( X .x. ( 1r ` R ) ) = X ) ) )
11 10 adantl 
 |-  ( ( ( R e. Ring /\ X e. B ) /\ u = ( 1r ` R ) ) -> ( ( ( u .x. X ) = X /\ ( X .x. u ) = X ) <-> ( ( ( 1r ` R ) .x. X ) = X /\ ( X .x. ( 1r ` R ) ) = X ) ) )
12 1 2 3 ringidmlem 
 |-  ( ( R e. Ring /\ X e. B ) -> ( ( ( 1r ` R ) .x. X ) = X /\ ( X .x. ( 1r ` R ) ) = X ) )
13 5 11 12 rspcedvd 
 |-  ( ( R e. Ring /\ X e. B ) -> E. u e. B ( ( u .x. X ) = X /\ ( X .x. u ) = X ) )