Theorem iscrng 12979

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 5507	. . . 4 |- ( r = R -> (mulGrp ` r ) = (mulGrp ` R ) )
2		ringmgp.g	. . . 4 |- G = (mulGrp ` R )
3	1, 2	eqtr4di 2226	. . 3 |- ( r = R -> (mulGrp ` r ) = G )
4	3	eleq1d 2244	. 2 |- ( r = R -> ( (mulGrp ` r ) e. CMnd <-> G e. CMnd ) )
5		df-cring 12975	. 2 |- CRing = { r e. Ring | (mulGrp ` r ) e. CMnd }
6	4, 5	elrab2 2894	1 |- ( R e. CRing <-> ( R e. Ring /\ G e. CMnd ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2146   ` cfv 5208  CMndccmn 12884  mulGrpcmgp 12925   Ringcrg 12972   CRingccrg 12973

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-rab 2462  df-v 2737  df-un 3131  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-iota 5170  df-fv 5216  df-cring 12975

This theorem is referenced by:  crngmgp  12980  crngring  12984  iscrng2  12991  crngpropd  13010  iscrngd  13013