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Theorem isfldidl2 33038
 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 2610 . . 3 (GId‘𝐻) = (GId‘𝐻)
61, 2, 3, 4, 5isfldidl 33037 . 2 (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
7 crngorngo 32969 . . . . . . 7 (𝐾 ∈ CRingOps → 𝐾 ∈ RingOps)
8 eqcom 2617 . . . . . . . 8 ((GId‘𝐻) = 𝑍𝑍 = (GId‘𝐻))
91, 2, 3, 4, 50rngo 32996 . . . . . . . 8 (𝐾 ∈ RingOps → (𝑍 = (GId‘𝐻) ↔ 𝑋 = {𝑍}))
108, 9syl5bb 271 . . . . . . 7 (𝐾 ∈ RingOps → ((GId‘𝐻) = 𝑍𝑋 = {𝑍}))
117, 10syl 17 . . . . . 6 (𝐾 ∈ CRingOps → ((GId‘𝐻) = 𝑍𝑋 = {𝑍}))
1211necon3bid 2826 . . . . 5 (𝐾 ∈ CRingOps → ((GId‘𝐻) ≠ 𝑍𝑋 ≠ {𝑍}))
1312anbi1d 737 . . . 4 (𝐾 ∈ CRingOps → (((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
1413pm5.32i 667 . . 3 ((𝐾 ∈ CRingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})) ↔ (𝐾 ∈ CRingOps ∧ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
15 3anass 1035 . . 3 ((𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
16 3anass 1035 . . 3 ((𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
1714, 15, 163bitr4i 291 . 2 ((𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
186, 17bitri 263 1 (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {csn 4125  {cpr 4127  ran crn 5039  ‘cfv 5804  1st c1st 7057  2nd c2nd 7058  GIdcgi 26728  RingOpscrngo 32863  Fldcfld 32960  CRingOpsccring 32962  Idlcidl 32976 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-grpo 26731  df-gid 26732  df-ginv 26733  df-ablo 26783  df-ass 32812  df-exid 32814  df-mgmOLD 32818  df-sgrOLD 32830  df-mndo 32836  df-rngo 32864  df-drngo 32918  df-com2 32959  df-fld 32961  df-crngo 32963  df-idl 32979  df-igen 33029 This theorem is referenced by: (None)
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