Theorem fldcrng 16152

Description: A field is a commutative ring.

Assertion

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

Proof of Theorem fldcrng

Step	Hyp	Ref	Expression
1		eqid 1884	. . . . 5 |- (1st` K) = (1st` K)
2		eqid 1884	. . . . 5 |- (2nd` K) = (2nd` K)
3		eqid 1884	. . . . 5 |- ran (1st` K) = ran (1st` K)
4		eqid 1884	. . . . 5 |- (Id` (1st` K)) = (Id` (1st` K))
5	1, 2, 3, 4	drngi 9493	. . . 4 |- (K e. DivRing -> (K e. Ring /\ ((2nd` K) |` ((ran (1st` K) \ {(Id` (1st` K))}) X. (ran (1st` K) \ {(Id` (1st` K))}))) e. Grp))
6	5	simplld 348	. . 3 |- (K e. DivRing -> K e. Ring)
7	6	anim1i 361	. 2 |- ((K e. DivRing /\ K e. Com2) -> (K e. Ring /\ K e. Com2))
8		df-fld 10398	. . . 4 |- Fld = (DivRing i^i Com2)
9	8	eleq2i 1961	. . 3 |- (K e. Fld <-> K e. (DivRing i^i Com2))
10		elin 2786	. . 3 |- (K e. (DivRing i^i Com2) <-> (K e. DivRing /\ K e. Com2))
11	9, 10	bitri 190	. 2 |- (K e. Fld <-> (K e. DivRing /\ K e. Com2))
12		iscring 16145	. 2 |- (K e. CRing <-> (K e. Ring /\ K e. Com2))
13	7, 11, 12	3imtr4i 236	1 |- (K e. Fld -> K e. CRing)

Syntax hints:   -> wi 3   /\ wa 240   e. wcel 1300   \ cdif 2590   i^i cin 2592  {csn 3044   X. cxp 3984  ran crn 3987   |` cres 3988  ` cfv 3998  1stc1st 5018  2ndc2nd 5019  Grpcgr 9311  Idcgi 9312  Ringcring 9463  DivRingcdrng 9491  Com2ccm2 10394  Fldcfld 10397  CRingccring 16143

This theorem is referenced by:  isfld2 16153  isfldidl 16216

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-1st 5020  df-2nd 5021  df-drng 9492  df-fld 10398  df-cring 16144

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