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Theorem fldcrng 33474
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 2621 . . . . 5 (1st𝐾) = (1st𝐾)
2 eqid 2621 . . . . 5 (2nd𝐾) = (2nd𝐾)
3 eqid 2621 . . . . 5 ran (1st𝐾) = ran (1st𝐾)
4 eqid 2621 . . . . 5 (GId‘(1st𝐾)) = (GId‘(1st𝐾))
51, 2, 3, 4drngoi 33421 . . . 4 (𝐾 ∈ DivRingOps → (𝐾 ∈ RingOps ∧ ((2nd𝐾) ↾ ((ran (1st𝐾) ∖ {(GId‘(1st𝐾))}) × (ran (1st𝐾) ∖ {(GId‘(1st𝐾))}))) ∈ GrpOp))
65simpld 475 . . 3 (𝐾 ∈ DivRingOps → 𝐾 ∈ RingOps)
76anim1i 591 . 2 ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
8 df-fld 33462 . . 3 Fld = (DivRingOps ∩ Com2)
98elin2 3785 . 2 (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2))
10 iscrngo 33466 . 2 (𝐾 ∈ CRingOps ↔ (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
117, 9, 103imtr4i 281 1 (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  cdif 3557  {csn 4155   × cxp 5082  ran crn 5085  cres 5086  cfv 5857  1st c1st 7126  2nd c2nd 7127  GrpOpcgr 27231  GIdcgi 27232  RingOpscrngo 33364  DivRingOpscdrng 33418  Com2ccm2 33459  Fldcfld 33461  CRingOpsccring 33463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-iota 5820  df-fun 5859  df-fv 5865  df-1st 7128  df-2nd 7129  df-drngo 33419  df-fld 33462  df-crngo 33464
This theorem is referenced by:  isfld2  33475  isfldidl  33538
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