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Theorem isfldidl2 33998
Description: Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 6-Jan-2011.)
Hypotheses
Ref Expression
isfldidl2.1 𝐺 = (1st𝐾)
isfldidl2.2 𝐻 = (2nd𝐾)
isfldidl2.3 𝑋 = ran 𝐺
isfldidl2.4 𝑍 = (GId‘𝐺)
Assertion
Ref Expression
isfldidl2 (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))

Proof of Theorem isfldidl2
StepHypRef Expression
1 isfldidl2.1 . . 3 𝐺 = (1st𝐾)
2 isfldidl2.2 . . 3 𝐻 = (2nd𝐾)
3 isfldidl2.3 . . 3 𝑋 = ran 𝐺
4 isfldidl2.4 . . 3 𝑍 = (GId‘𝐺)
5 eqid 2651 . . 3 (GId‘𝐻) = (GId‘𝐻)
61, 2, 3, 4, 5isfldidl 33997 . 2 (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
7 crngorngo 33929 . . . . . . 7 (𝐾 ∈ CRingOps → 𝐾 ∈ RingOps)
8 eqcom 2658 . . . . . . . 8 ((GId‘𝐻) = 𝑍𝑍 = (GId‘𝐻))
91, 2, 3, 4, 50rngo 33956 . . . . . . . 8 (𝐾 ∈ RingOps → (𝑍 = (GId‘𝐻) ↔ 𝑋 = {𝑍}))
108, 9syl5bb 272 . . . . . . 7 (𝐾 ∈ RingOps → ((GId‘𝐻) = 𝑍𝑋 = {𝑍}))
117, 10syl 17 . . . . . 6 (𝐾 ∈ CRingOps → ((GId‘𝐻) = 𝑍𝑋 = {𝑍}))
1211necon3bid 2867 . . . . 5 (𝐾 ∈ CRingOps → ((GId‘𝐻) ≠ 𝑍𝑋 ≠ {𝑍}))
1312anbi1d 741 . . . 4 (𝐾 ∈ CRingOps → (((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
1413pm5.32i 670 . . 3 ((𝐾 ∈ CRingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})) ↔ (𝐾 ∈ CRingOps ∧ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
15 3anass 1059 . . 3 ((𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
16 3anass 1059 . . 3 ((𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
1714, 15, 163bitr4i 292 . 2 ((𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
186, 17bitri 264 1 (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  {csn 4210  {cpr 4212  ran crn 5144  cfv 5926  1st c1st 7208  2nd c2nd 7209  GIdcgi 27472  RingOpscrngo 33823  Fldcfld 33920  CRingOpsccring 33922  Idlcidl 33936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-grpo 27475  df-gid 27476  df-ginv 27477  df-ablo 27527  df-ass 33772  df-exid 33774  df-mgmOLD 33778  df-sgrOLD 33790  df-mndo 33796  df-rngo 33824  df-drngo 33878  df-com2 33919  df-fld 33921  df-crngo 33923  df-idl 33939  df-igen 33989
This theorem is referenced by: (None)
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