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Theorem fldcrngo 37991
Description: A field is a commutative ring. (Contributed by Jeff Madsen, 8-Jun-2010.)
Assertion
Ref Expression
fldcrngo (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)

Proof of Theorem fldcrngo
StepHypRef Expression
1 eqid 2735 . . . . 5 (1st𝐾) = (1st𝐾)
2 eqid 2735 . . . . 5 (2nd𝐾) = (2nd𝐾)
3 eqid 2735 . . . . 5 ran (1st𝐾) = ran (1st𝐾)
4 eqid 2735 . . . . 5 (GId‘(1st𝐾)) = (GId‘(1st𝐾))
51, 2, 3, 4drngoi 37938 . . . 4 (𝐾 ∈ DivRingOps → (𝐾 ∈ RingOps ∧ ((2nd𝐾) ↾ ((ran (1st𝐾) ∖ {(GId‘(1st𝐾))}) × (ran (1st𝐾) ∖ {(GId‘(1st𝐾))}))) ∈ GrpOp))
65simpld 494 . . 3 (𝐾 ∈ DivRingOps → 𝐾 ∈ RingOps)
76anim1i 615 . 2 ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
8 df-fld 37979 . . 3 Fld = (DivRingOps ∩ Com2)
98elin2 4213 . 2 (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2))
10 iscrngo 37983 . 2 (𝐾 ∈ CRingOps ↔ (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
117, 9, 103imtr4i 292 1 (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  cdif 3960  {csn 4631   × cxp 5687  ran crn 5690  cres 5691  cfv 6563  1st c1st 8011  2nd c2nd 8012  GrpOpcgr 30518  GIdcgi 30519  RingOpscrngo 37881  DivRingOpscdrng 37935  Com2ccm2 37976  Fldcfld 37978  CRingOpsccring 37980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013  df-2nd 8014  df-drngo 37936  df-fld 37979  df-crngo 37981
This theorem is referenced by:  isfld2  37992  isfldidl  38055
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