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Theorem isfld2 34758
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 34752 . . 3 (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)
2 fldcrng 34757 . . 3 (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)
31, 2jca 504 . 2 (𝐾 ∈ Fld → (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps))
4 iscrngo 34749 . . . 4 (𝐾 ∈ CRingOps ↔ (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
54simprbi 489 . . 3 (𝐾 ∈ CRingOps → 𝐾 ∈ Com2)
6 elin 4052 . . . . 5 (𝐾 ∈ (DivRingOps ∩ Com2) ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2))
76biimpri 220 . . . 4 ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → 𝐾 ∈ (DivRingOps ∩ Com2))
8 df-fld 34745 . . . 4 Fld = (DivRingOps ∩ Com2)
97, 8syl6eleqr 2872 . . 3 ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → 𝐾 ∈ Fld)
105, 9sylan2 584 . 2 ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps) → 𝐾 ∈ Fld)
113, 10impbii 201 1 (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387  wcel 2051  cin 3823  RingOpscrngo 34647  DivRingOpscdrng 34701  Com2ccm2 34742  Fldcfld 34744  CRingOpsccring 34746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-sbc 3677  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-iota 6150  df-fun 6188  df-fv 6194  df-1st 7500  df-2nd 7501  df-drngo 34702  df-fld 34745  df-crngo 34747
This theorem is referenced by:  flddmn  34811  isfldidl  34821
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