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Theorem isfld2 37384
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 37378 . . 3 (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)
2 fldcrngo 37383 . . 3 (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)
31, 2jca 511 . 2 (𝐾 ∈ Fld → (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps))
4 iscrngo 37375 . . . 4 (𝐾 ∈ CRingOps ↔ (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
54simprbi 496 . . 3 (𝐾 ∈ CRingOps → 𝐾 ∈ Com2)
6 elin 3959 . . . . 5 (𝐾 ∈ (DivRingOps ∩ Com2) ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2))
76biimpri 227 . . . 4 ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → 𝐾 ∈ (DivRingOps ∩ Com2))
8 df-fld 37371 . . . 4 Fld = (DivRingOps ∩ Com2)
97, 8eleqtrrdi 2838 . . 3 ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → 𝐾 ∈ Fld)
105, 9sylan2 592 . 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 395  wcel 2098  cin 3942  RingOpscrngo 37273  DivRingOpscdrng 37327  Com2ccm2 37368  Fldcfld 37370  CRingOpsccring 37372
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-iota 6488  df-fun 6538  df-fv 6544  df-1st 7971  df-2nd 7972  df-drngo 37328  df-fld 37371  df-crngo 37373
This theorem is referenced by:  flddmn  37437  isfldidl  37447
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