Theorem fldcrng 33803

Description: A field is a commutative ring. (Contributed by Jeff Madsen, 8-Jun-2010.)

Assertion

Ref	Expression
fldcrng	|- ( K e. Fld -> K e. CRingOps )

Proof of Theorem fldcrng

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- ( 1st ` K ) = ( 1st ` K )
2		eqid 2622	. . . . 5 |- ( 2nd ` K ) = ( 2nd ` K )
3		eqid 2622	. . . . 5 |- ran ( 1st ` K ) = ran ( 1st ` K )
4		eqid 2622	. . . . 5 |- (GId ` ( 1st ` K ) ) = (GId ` ( 1st ` K ) )
5	1, 2, 3, 4	drngoi 33750	. . . 4 |- ( K e. DivRingOps -> ( K e. RingOps /\ ( ( 2nd ` K ) |` ( ( ran ( 1st ` K ) \ { (GId ` ( 1st ` K ) ) } ) X. ( ran ( 1st ` K ) \ { (GId ` ( 1st ` K ) ) } ) ) ) e. GrpOp ) )
6	5	simpld 475	. . 3 |- ( K e. DivRingOps -> K e. RingOps )
7	6	anim1i 592	. 2 |- ( ( K e. DivRingOps /\ K e. Com2 ) -> ( K e. RingOps /\ K e. Com2 ) )
8		df-fld 33791	. . 3 |- Fld = ( DivRingOps i^i Com2 )
9	8	elin2 3801	. 2 |- ( K e. Fld <-> ( K e. DivRingOps /\ K e. Com2 ) )
10		iscrngo 33795	. 2 |- ( K e. CRingOps <-> ( K e. RingOps /\ K e. Com2 ) )
11	7, 9, 10	3imtr4i 281	1 |- ( K e. Fld -> K e. CRingOps )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   \ cdif 3571   {csn 4177   X. cxp 5112   ran crn 5115   |` cres 5116   ` cfv 5888   1stc1st 7166   2ndc2nd 7167   GrpOpcgr 27343  GIdcgi 27344   RingOpscrngo 33693   DivRingOpscdrng 33747   Com2ccm2 33788   Fldcfld 33790  CRingOpsccring 33792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-2nd 7169  df-drngo 33748  df-fld 33791  df-crngo 33793

This theorem is referenced by:  isfld2  33804  isfldidl  33867