Theorem iscrng 13186

Description: A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)

Hypothesis

Ref	Expression
ringmgp.g	|- G = (mulGrp ` R )

Assertion

Ref	Expression
iscrng	|- ( R e. CRing <-> ( R e. Ring /\ G e. CMnd ) )

Proof of Theorem iscrng

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5516	. . . 4 |- ( r = R -> (mulGrp ` r ) = (mulGrp ` R ) )
2		ringmgp.g	. . . 4 |- G = (mulGrp ` R )
3	1, 2	eqtr4di 2228	. . 3 |- ( r = R -> (mulGrp ` r ) = G )
4	3	eleq1d 2246	. 2 |- ( r = R -> ( (mulGrp ` r ) e. CMnd <-> G e. CMnd ) )
5		df-cring 13182	. 2 |- CRing = { r e. Ring | (mulGrp ` r ) e. CMnd }
6	4, 5	elrab2 2897	1 |- ( R e. CRing <-> ( R e. Ring /\ G e. CMnd ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   ` cfv 5217  CMndccmn 13088  mulGrpcmgp 13130   Ringcrg 13179   CRingccrg 13180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-rab 2464  df-v 2740  df-un 3134  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-iota 5179  df-fv 5225  df-cring 13182

This theorem is referenced by:  crngmgp  13187  crngring  13191  iscrng2  13198  crngpropd  13218  iscrngd  13221  subrgcrng  13346