Theorem isfld2 33804

Description: The predicate "is a field". (Contributed by Jeff Madsen, 10-Jun-2010.)

Assertion

Ref	Expression
isfld2	|- ( K e. Fld <-> ( K e. DivRingOps /\ K e. CRingOps ) )

Proof of Theorem isfld2

Step	Hyp	Ref	Expression
1		flddivrng 33798	. . 3 |- ( K e. Fld -> K e. DivRingOps )
2		fldcrng 33803	. . 3 |- ( K e. Fld -> K e. CRingOps )
3	1, 2	jca 554	. 2 |- ( K e. Fld -> ( K e. DivRingOps /\ K e. CRingOps ) )
4		iscrngo 33795	. . . 4 |- ( K e. CRingOps <-> ( K e. RingOps /\ K e. Com2 ) )
5	4	simprbi 480	. . 3 |- ( K e. CRingOps -> K e. Com2 )
6		elin 3796	. . . . 5 |- ( K e. ( DivRingOps i^i Com2 ) <-> ( K e. DivRingOps /\ K e. Com2 ) )
7	6	biimpri 218	. . . 4 |- ( ( K e. DivRingOps /\ K e. Com2 ) -> K e. ( DivRingOps i^i Com2 ) )
8		df-fld 33791	. . . 4 |- Fld = ( DivRingOps i^i Com2 )
9	7, 8	syl6eleqr 2712	. . 3 |- ( ( K e. DivRingOps /\ K e. Com2 ) -> K e. Fld )
10	5, 9	sylan2 491	. 2 |- ( ( K e. DivRingOps /\ K e. CRingOps ) -> K e. Fld )
11	3, 10	impbii 199	1 |- ( K e. Fld <-> ( K e. DivRingOps /\ K e. CRingOps ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   i^i cin 3573   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:  flddmn  33857  isfldidl  33867