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Theorem fldcrng 35151
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 2824 . . . . 5 (1st𝐾) = (1st𝐾)
2 eqid 2824 . . . . 5 (2nd𝐾) = (2nd𝐾)
3 eqid 2824 . . . . 5 ran (1st𝐾) = ran (1st𝐾)
4 eqid 2824 . . . . 5 (GId‘(1st𝐾)) = (GId‘(1st𝐾))
51, 2, 3, 4drngoi 35098 . . . 4 (𝐾 ∈ DivRingOps → (𝐾 ∈ RingOps ∧ ((2nd𝐾) ↾ ((ran (1st𝐾) ∖ {(GId‘(1st𝐾))}) × (ran (1st𝐾) ∖ {(GId‘(1st𝐾))}))) ∈ GrpOp))
65simpld 495 . . 3 (𝐾 ∈ DivRingOps → 𝐾 ∈ RingOps)
76anim1i 614 . 2 ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
8 df-fld 35139 . . 3 Fld = (DivRingOps ∩ Com2)
98elin2 4177 . 2 (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2))
10 iscrngo 35143 . 2 (𝐾 ∈ CRingOps ↔ (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
117, 9, 103imtr4i 293 1 (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  cdif 3936  {csn 4563   × cxp 5551  ran crn 5554  cres 5555  cfv 6351  1st c1st 7681  2nd c2nd 7682  GrpOpcgr 28181  GIdcgi 28182  RingOpscrngo 35041  DivRingOpscdrng 35095  Com2ccm2 35136  Fldcfld 35138  CRingOpsccring 35140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-iota 6311  df-fun 6353  df-fv 6359  df-1st 7683  df-2nd 7684  df-drngo 35096  df-fld 35139  df-crngo 35141
This theorem is referenced by:  isfld2  35152  isfldidl  35215
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