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Theorem isfld2 35443
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 35437 . . 3 (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)
2 fldcrng 35442 . . 3 (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)
31, 2jca 515 . 2 (𝐾 ∈ Fld → (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps))
4 iscrngo 35434 . . . 4 (𝐾 ∈ CRingOps ↔ (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
54simprbi 500 . . 3 (𝐾 ∈ CRingOps → 𝐾 ∈ Com2)
6 elin 3897 . . . . 5 (𝐾 ∈ (DivRingOps ∩ Com2) ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2))
76biimpri 231 . . . 4 ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → 𝐾 ∈ (DivRingOps ∩ Com2))
8 df-fld 35430 . . . 4 Fld = (DivRingOps ∩ Com2)
97, 8eleqtrrdi 2901 . . 3 ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → 𝐾 ∈ Fld)
105, 9sylan2 595 . 2 ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps) → 𝐾 ∈ Fld)
113, 10impbii 212 1 (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wcel 2111  cin 3880  RingOpscrngo 35332  DivRingOpscdrng 35386  Com2ccm2 35427  Fldcfld 35429  CRingOpsccring 35431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-iota 6283  df-fun 6326  df-fv 6332  df-1st 7671  df-2nd 7672  df-drngo 35387  df-fld 35430  df-crngo 35432
This theorem is referenced by:  flddmn  35496  isfldidl  35506
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