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Theorem isfldidl2 38077
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 2736 . . 3 (GId‘𝐻) = (GId‘𝐻)
61, 2, 3, 4, 5isfldidl 38076 . 2 (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
7 crngorngo 38008 . . . . . . 7 (𝐾 ∈ CRingOps → 𝐾 ∈ RingOps)
8 eqcom 2743 . . . . . . . 8 ((GId‘𝐻) = 𝑍𝑍 = (GId‘𝐻))
91, 2, 3, 4, 50rngo 38035 . . . . . . . 8 (𝐾 ∈ RingOps → (𝑍 = (GId‘𝐻) ↔ 𝑋 = {𝑍}))
108, 9bitrid 283 . . . . . . 7 (𝐾 ∈ RingOps → ((GId‘𝐻) = 𝑍𝑋 = {𝑍}))
117, 10syl 17 . . . . . 6 (𝐾 ∈ CRingOps → ((GId‘𝐻) = 𝑍𝑋 = {𝑍}))
1211necon3bid 2984 . . . . 5 (𝐾 ∈ CRingOps → ((GId‘𝐻) ≠ 𝑍𝑋 ≠ {𝑍}))
1312anbi1d 631 . . . 4 (𝐾 ∈ CRingOps → (((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
1413pm5.32i 574 . . 3 ((𝐾 ∈ CRingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})) ↔ (𝐾 ∈ CRingOps ∧ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
15 3anass 1094 . . 3 ((𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
16 3anass 1094 . . 3 ((𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
1714, 15, 163bitr4i 303 . 2 ((𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
186, 17bitri 275 1 (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wne 2939  {csn 4625  {cpr 4627  ran crn 5685  cfv 6560  1st c1st 8013  2nd c2nd 8014  GIdcgi 30510  RingOpscrngo 37902  Fldcfld 37999  CRingOpsccring 38001  Idlcidl 38015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-1o 8507  df-en 8987  df-grpo 30513  df-gid 30514  df-ginv 30515  df-ablo 30565  df-ass 37851  df-exid 37853  df-mgmOLD 37857  df-sgrOLD 37869  df-mndo 37875  df-rngo 37903  df-drngo 37957  df-com2 37998  df-fld 38000  df-crngo 38002  df-idl 38018  df-igen 38068
This theorem is referenced by: (None)
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