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Theorem isfld2 33475
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 33469 . . 3 (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)
2 fldcrng 33474 . . 3 (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)
31, 2jca 554 . 2 (𝐾 ∈ Fld → (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps))
4 iscrngo 33466 . . . 4 (𝐾 ∈ CRingOps ↔ (𝐾 ∈ RingOps ∧ 𝐾 ∈ Com2))
54simprbi 480 . . 3 (𝐾 ∈ CRingOps → 𝐾 ∈ Com2)
6 elin 3780 . . . . 5 (𝐾 ∈ (DivRingOps ∩ Com2) ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2))
76biimpri 218 . . . 4 ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → 𝐾 ∈ (DivRingOps ∩ Com2))
8 df-fld 33462 . . . 4 Fld = (DivRingOps ∩ Com2)
97, 8syl6eleqr 2709 . . 3 ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ Com2) → 𝐾 ∈ Fld)
105, 9sylan2 491 . 2 ((𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps) → 𝐾 ∈ Fld)
113, 10impbii 199 1 (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wcel 1987  cin 3559  RingOpscrngo 33364  DivRingOpscdrng 33418  Com2ccm2 33459  Fldcfld 33461  CRingOpsccring 33463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-iota 5820  df-fun 5859  df-fv 5865  df-1st 7128  df-2nd 7129  df-drngo 33419  df-fld 33462  df-crngo 33464
This theorem is referenced by:  flddmn  33528  isfldidl  33538
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