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Theorem fldcrngo 38011
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 2737 . . . . 5 (1st𝐾) = (1st𝐾)
2 eqid 2737 . . . . 5 (2nd𝐾) = (2nd𝐾)
3 eqid 2737 . . . . 5 ran (1st𝐾) = ran (1st𝐾)
4 eqid 2737 . . . . 5 (GId‘(1st𝐾)) = (GId‘(1st𝐾))
51, 2, 3, 4drngoi 37958 . . . 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 37999 . . 3 Fld = (DivRingOps ∩ Com2)
98elin2 4203 . 2 (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2))
10 iscrngo 38003 . 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 2108  cdif 3948  {csn 4626   × cxp 5683  ran crn 5686  cres 5687  cfv 6561  1st c1st 8012  2nd c2nd 8013  GrpOpcgr 30508  GIdcgi 30509  RingOpscrngo 37901  DivRingOpscdrng 37955  Com2ccm2 37996  Fldcfld 37998  CRingOpsccring 38000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-1st 8014  df-2nd 8015  df-drngo 37956  df-fld 37999  df-crngo 38001
This theorem is referenced by:  isfld2  38012  isfldidl  38075
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