Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isfld2 Structured version   Visualization version   GIF version

Theorem isfld2 36467
Description: The predicate "is a field". (Contributed by Jeff Madsen, 10-Jun-2010.)
Assertion
Ref Expression
isfld2 (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps))

Proof of Theorem isfld2
StepHypRef Expression
1 flddivrng 36461 . . 3 (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)
2 fldcrngo 36466 . . 3 (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)
31, 2jca 513 . 2 (𝐾 ∈ Fld → (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps))
4 iscrngo 36458 . . . 4 (𝐾 ∈ CRingOps ↔ (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
54simprbi 498 . . 3 (𝐾 ∈ CRingOps → 𝐾 ∈ Com2)
6 elin 3927 . . . . 5 (𝐾 ∈ (DivRingOps ∩ Com2) ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2))
76biimpri 227 . . . 4 ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → 𝐾 ∈ (DivRingOps ∩ Com2))
8 df-fld 36454 . . . 4 Fld = (DivRingOps ∩ Com2)
97, 8eleqtrrdi 2849 . . 3 ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → 𝐾 ∈ Fld)
105, 9sylan2 594 . 2 ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps) → 𝐾 ∈ Fld)
113, 10impbii 208 1 (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wcel 2107  cin 3910  RingOpscrngo 36356  DivRingOpscdrng 36410  Com2ccm2 36451  Fldcfld 36453  CRingOpsccring 36455
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 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-iota 6449  df-fun 6499  df-fv 6505  df-1st 7922  df-2nd 7923  df-drngo 36411  df-fld 36454  df-crngo 36456
This theorem is referenced by:  flddmn  36520  isfldidl  36530
  Copyright terms: Public domain W3C validator