Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fldcrngo Structured version   Visualization version   GIF version

Theorem fldcrngo 36395
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 2738 . . . . 5 (1st𝐾) = (1st𝐾)
2 eqid 2738 . . . . 5 (2nd𝐾) = (2nd𝐾)
3 eqid 2738 . . . . 5 ran (1st𝐾) = ran (1st𝐾)
4 eqid 2738 . . . . 5 (GId‘(1st𝐾)) = (GId‘(1st𝐾))
51, 2, 3, 4drngoi 36342 . . . 4 (𝐾 ∈ DivRingOps → (𝐾 ∈ RingOps ∧ ((2nd𝐾) ↾ ((ran (1st𝐾) ∖ {(GId‘(1st𝐾))}) × (ran (1st𝐾) ∖ {(GId‘(1st𝐾))}))) ∈ GrpOp))
65simpld 496 . . 3 (𝐾 ∈ DivRingOps → 𝐾 ∈ RingOps)
76anim1i 616 . 2 ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
8 df-fld 36383 . . 3 Fld = (DivRingOps ∩ Com2)
98elin2 4156 . 2 (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2))
10 iscrngo 36387 . 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 397  wcel 2107  cdif 3906  {csn 4585   × cxp 5630  ran crn 5633  cres 5634  cfv 6494  1st c1st 7912  2nd c2nd 7913  GrpOpcgr 29260  GIdcgi 29261  RingOpscrngo 36285  DivRingOpscdrng 36339  Com2ccm2 36380  Fldcfld 36382  CRingOpsccring 36384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383  ax-un 7665
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-iota 6446  df-fun 6496  df-fv 6502  df-1st 7914  df-2nd 7915  df-drngo 36340  df-fld 36383  df-crngo 36385
This theorem is referenced by:  isfld2  36396  isfldidl  36459
  Copyright terms: Public domain W3C validator