Theorem iscrng 13559

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 5558	. . . 4 |- ( r = R -> (mulGrp ` r ) = (mulGrp ` R ) )
2		ringmgp.g	. . . 4 |- G = (mulGrp ` R )
3	1, 2	eqtr4di 2247	. . 3 |- ( r = R -> (mulGrp ` r ) = G )
4	3	eleq1d 2265	. 2 |- ( r = R -> ( (mulGrp ` r ) e. CMnd <-> G e. CMnd ) )
5		df-cring 13555	. 2 |- CRing = { r e. Ring | (mulGrp ` r ) e. CMnd }
6	4, 5	elrab2 2923	1 |- ( R e. CRing <-> ( R e. Ring /\ G e. CMnd ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   ` cfv 5258  CMndccmn 13414  mulGrpcmgp 13476   Ringcrg 13552   CRingccrg 13553

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-rab 2484  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266  df-cring 13555

This theorem is referenced by:  crngmgp  13560  crngring  13564  iscrng2  13571  crngpropd  13595  iscrngd  13598  subrgcrng  13781