Theorem iscrng 13880

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 5599	. . . 4 |- ( r = R -> (mulGrp ` r ) = (mulGrp ` R ) )
2		ringmgp.g	. . . 4 |- G = (mulGrp ` R )
3	1, 2	eqtr4di 2258	. . 3 |- ( r = R -> (mulGrp ` r ) = G )
4	3	eleq1d 2276	. 2 |- ( r = R -> ( (mulGrp ` r ) e. CMnd <-> G e. CMnd ) )
5		df-cring 13876	. 2 |- CRing = { r e. Ring | (mulGrp ` r ) e. CMnd }
6	4, 5	elrab2 2939	1 |- ( R e. CRing <-> ( R e. Ring /\ G e. CMnd ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   ` cfv 5290  CMndccmn 13735  mulGrpcmgp 13797   Ringcrg 13873   CRingccrg 13874

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-rab 2495  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-iota 5251  df-fv 5298  df-cring 13876

This theorem is referenced by:  crngmgp  13881  crngring  13885  iscrng2  13892  crngpropd  13916  iscrngd  13919  subrgcrng  14102