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Theorem fldcrng 32973
 Description: A field is a commutative ring. (Contributed by Jeff Madsen, 8-Jun-2010.)
Assertion
Ref Expression
fldcrng (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)

Proof of Theorem fldcrng
StepHypRef Expression
1 eqid 2610 . . . . 5 (1st𝐾) = (1st𝐾)
2 eqid 2610 . . . . 5 (2nd𝐾) = (2nd𝐾)
3 eqid 2610 . . . . 5 ran (1st𝐾) = ran (1st𝐾)
4 eqid 2610 . . . . 5 (GId‘(1st𝐾)) = (GId‘(1st𝐾))
51, 2, 3, 4drngoi 32920 . . . 4 (𝐾 ∈ DivRingOps → (𝐾 ∈ RingOps ∧ ((2nd𝐾) ↾ ((ran (1st𝐾) ∖ {(GId‘(1st𝐾))}) × (ran (1st𝐾) ∖ {(GId‘(1st𝐾))}))) ∈ GrpOp))
65simpld 474 . . 3 (𝐾 ∈ DivRingOps → 𝐾 ∈ RingOps)
76anim1i 590 . 2 ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
8 df-fld 32961 . . 3 Fld = (DivRingOps ∩ Com2)
98elin2 3763 . 2 (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2))
10 iscrngo 32965 . 2 (𝐾 ∈ CRingOps ↔ (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
117, 9, 103imtr4i 280 1 (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977   ∖ cdif 3537  {csn 4125   × cxp 5036  ran crn 5039   ↾ cres 5040  ‘cfv 5804  1st c1st 7057  2nd c2nd 7058  GrpOpcgr 26727  GIdcgi 26728  RingOpscrngo 32863  DivRingOpscdrng 32917  Com2ccm2 32958  Fldcfld 32960  CRingOpsccring 32962 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-1st 7059  df-2nd 7060  df-drngo 32918  df-fld 32961  df-crngo 32963 This theorem is referenced by:  isfld2  32974  isfldidl  33037
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