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Metamath Proof Explorer

Theorem iscrng2

Description: A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 15-Jun-2015)

Ref Expression
Hypotheses ringcl.b 
|- B = ( Base ` R )
ringcl.t 
|- .x. = ( .r ` R )
Assertion iscrng2 
|- ( R e. CRing <-> ( R e. Ring /\ A. x e. B A. y e. B ( x .x. y ) = ( y .x. x ) ) )

Proof

Step Hyp Ref Expression
1 ringcl.b 
 |-  B = ( Base ` R )
2 ringcl.t 
 |-  .x. = ( .r ` R )
3 eqid 
 |-  ( mulGrp ` R ) = ( mulGrp ` R )
4 3 iscrng 
 |-  ( R e. CRing <-> ( R e. Ring /\ ( mulGrp ` R ) e. CMnd ) )
5 3 ringmgp 
 |-  ( R e. Ring -> ( mulGrp ` R ) e. Mnd )
6 3 1 mgpbas 
 |-  B = ( Base ` ( mulGrp ` R ) )
7 3 2 mgpplusg 
 |-  .x. = ( +g ` ( mulGrp ` R ) )
8 6 7 iscmn 
 |-  ( ( mulGrp ` R ) e. CMnd <-> ( ( mulGrp ` R ) e. Mnd /\ A. x e. B A. y e. B ( x .x. y ) = ( y .x. x ) ) )
9 8 baib 
 |-  ( ( mulGrp ` R ) e. Mnd -> ( ( mulGrp ` R ) e. CMnd <-> A. x e. B A. y e. B ( x .x. y ) = ( y .x. x ) ) )
10 5 9 syl 
 |-  ( R e. Ring -> ( ( mulGrp ` R ) e. CMnd <-> A. x e. B A. y e. B ( x .x. y ) = ( y .x. x ) ) )
11 10 pm5.32i 
 |-  ( ( R e. Ring /\ ( mulGrp ` R ) e. CMnd ) <-> ( R e. Ring /\ A. x e. B A. y e. B ( x .x. y ) = ( y .x. x ) ) )
12 4 11 bitri 
 |-  ( R e. CRing <-> ( R e. Ring /\ A. x e. B A. y e. B ( x .x. y ) = ( y .x. x ) ) )