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Theorem fldcrng 34108
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 2802 . . . . 5 (1st𝐾) = (1st𝐾)
2 eqid 2802 . . . . 5 (2nd𝐾) = (2nd𝐾)
3 eqid 2802 . . . . 5 ran (1st𝐾) = ran (1st𝐾)
4 eqid 2802 . . . . 5 (GId‘(1st𝐾)) = (GId‘(1st𝐾))
51, 2, 3, 4drngoi 34055 . . . 4 (𝐾 ∈ DivRingOps → (𝐾 ∈ RingOps ∧ ((2nd𝐾) ↾ ((ran (1st𝐾) ∖ {(GId‘(1st𝐾))}) × (ran (1st𝐾) ∖ {(GId‘(1st𝐾))}))) ∈ GrpOp))
65simpld 484 . . 3 (𝐾 ∈ DivRingOps → 𝐾 ∈ RingOps)
76anim1i 604 . 2 ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
8 df-fld 34096 . . 3 Fld = (DivRingOps ∩ Com2)
98elin2 3994 . 2 (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2))
10 iscrngo 34100 . 2 (𝐾 ∈ CRingOps ↔ (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
117, 9, 103imtr4i 283 1 (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 2155  cdif 3760  {csn 4364   × cxp 5303  ran crn 5306  cres 5307  cfv 6095  1st c1st 7390  2nd c2nd 7391  GrpOpcgr 27666  GIdcgi 27667  RingOpscrngo 33998  DivRingOpscdrng 34052  Com2ccm2 34093  Fldcfld 34095  CRingOpsccring 34097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ral 3097  df-rex 3098  df-rab 3101  df-v 3389  df-sbc 3628  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-if 4274  df-sn 4365  df-pr 4367  df-op 4371  df-uni 4624  df-br 4838  df-opab 4900  df-mpt 4917  df-id 5213  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-iota 6058  df-fun 6097  df-fv 6103  df-1st 7392  df-2nd 7393  df-drngo 34053  df-fld 34096  df-crngo 34098
This theorem is referenced by:  isfld2  34109  isfldidl  34172
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