Theorem iscrng 14307

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 5695	. . . 4 |- ( r = R -> (mulGrp ` r ) = (mulGrp ` R ) )
2		ringmgp.g	. . . 4 |- G = (mulGrp ` R )
3	1, 2	eqtr4di 2289	. . 3 |- ( r = R -> (mulGrp ` r ) = G )
4	3	eleq1d 2307	. 2 |- ( r = R -> ( (mulGrp ` r ) e. CMnd <-> G e. CMnd ) )
5		df-cring 14303	. 2 |- CRing = { r e. Ring | (mulGrp ` r ) e. CMnd }
6	4, 5	elrab2 2985	1 |- ( R e. CRing <-> ( R e. Ring /\ G e. CMnd ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   ` cfv 5377  CMndccmn 14087  mulGrpcmgp 14217   Ringcrg 14300   CRingccrg 14301

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-cring 14303

This theorem is used by:  crngmgp  14308  crngring  14312  iscrng2  14319  crngpropd  14344  iscrngd  14347  subrgcrng  14533