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Theorem isfldidl 38075
Description: Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
isfldidl.1 𝐺 = (1st𝐾)
isfldidl.2 𝐻 = (2nd𝐾)
isfldidl.3 𝑋 = ran 𝐺
isfldidl.4 𝑍 = (GId‘𝐺)
isfldidl.5 𝑈 = (GId‘𝐻)
Assertion
Ref Expression
isfldidl (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))

Proof of Theorem isfldidl
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fldcrngo 38011 . . 3 (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)
2 flddivrng 38006 . . . 4 (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)
3 isfldidl.1 . . . . 5 𝐺 = (1st𝐾)
4 isfldidl.2 . . . . 5 𝐻 = (2nd𝐾)
5 isfldidl.3 . . . . 5 𝑋 = ran 𝐺
6 isfldidl.4 . . . . 5 𝑍 = (GId‘𝐺)
7 isfldidl.5 . . . . 5 𝑈 = (GId‘𝐻)
83, 4, 5, 6, 7dvrunz 37961 . . . 4 (𝐾 ∈ DivRingOps → 𝑈𝑍)
92, 8syl 17 . . 3 (𝐾 ∈ Fld → 𝑈𝑍)
103, 4, 5, 6divrngidl 38035 . . . 4 (𝐾 ∈ DivRingOps → (Idl‘𝐾) = {{𝑍}, 𝑋})
112, 10syl 17 . . 3 (𝐾 ∈ Fld → (Idl‘𝐾) = {{𝑍}, 𝑋})
121, 9, 113jca 1129 . 2 (𝐾 ∈ Fld → (𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
13 crngorngo 38007 . . . . . 6 (𝐾 ∈ CRingOps → 𝐾 ∈ RingOps)
14133ad2ant1 1134 . . . . 5 ((𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ RingOps)
15 simp2 1138 . . . . 5 ((𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝑈𝑍)
163rneqi 5948 . . . . . . . . . . . . . . 15 ran 𝐺 = ran (1st𝐾)
175, 16eqtri 2765 . . . . . . . . . . . . . 14 𝑋 = ran (1st𝐾)
1817, 4, 7rngo1cl 37946 . . . . . . . . . . . . 13 (𝐾 ∈ RingOps → 𝑈𝑋)
1913, 18syl 17 . . . . . . . . . . . 12 (𝐾 ∈ CRingOps → 𝑈𝑋)
2019ad2antrr 726 . . . . . . . . . . 11 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → 𝑈𝑋)
21 eldif 3961 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝑋 ∖ {𝑍}) ↔ (𝑥𝑋 ∧ ¬ 𝑥 ∈ {𝑍}))
22 snssi 4808 . . . . . . . . . . . . . . . . . . 19 (𝑥𝑋 → {𝑥} ⊆ 𝑋)
233, 5igenss 38069 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ RingOps ∧ {𝑥} ⊆ 𝑋) → {𝑥} ⊆ (𝐾 IdlGen {𝑥}))
2422, 23sylan2 593 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ RingOps ∧ 𝑥𝑋) → {𝑥} ⊆ (𝐾 IdlGen {𝑥}))
25 vex 3484 . . . . . . . . . . . . . . . . . . . . . 22 𝑥 ∈ V
2625snss 4785 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝐾 IdlGen {𝑥}) ↔ {𝑥} ⊆ (𝐾 IdlGen {𝑥}))
2726biimpri 228 . . . . . . . . . . . . . . . . . . . 20 ({𝑥} ⊆ (𝐾 IdlGen {𝑥}) → 𝑥 ∈ (𝐾 IdlGen {𝑥}))
28 eleq2 2830 . . . . . . . . . . . . . . . . . . . 20 ((𝐾 IdlGen {𝑥}) = {𝑍} → (𝑥 ∈ (𝐾 IdlGen {𝑥}) ↔ 𝑥 ∈ {𝑍}))
2927, 28syl5ibcom 245 . . . . . . . . . . . . . . . . . . 19 ({𝑥} ⊆ (𝐾 IdlGen {𝑥}) → ((𝐾 IdlGen {𝑥}) = {𝑍} → 𝑥 ∈ {𝑍}))
3029con3dimp 408 . . . . . . . . . . . . . . . . . 18 (({𝑥} ⊆ (𝐾 IdlGen {𝑥}) ∧ ¬ 𝑥 ∈ {𝑍}) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
3124, 30sylan 580 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ RingOps ∧ 𝑥𝑋) ∧ ¬ 𝑥 ∈ {𝑍}) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
3231anasss 466 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ RingOps ∧ (𝑥𝑋 ∧ ¬ 𝑥 ∈ {𝑍})) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
3321, 32sylan2b 594 . . . . . . . . . . . . . . 15 ((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
3433adantlr 715 . . . . . . . . . . . . . 14 (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
35 eldifi 4131 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝑋 ∖ {𝑍}) → 𝑥𝑋)
3635snssd 4809 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (𝑋 ∖ {𝑍}) → {𝑥} ⊆ 𝑋)
373, 5igenidl 38070 . . . . . . . . . . . . . . . . . . . 20 ((𝐾 ∈ RingOps ∧ {𝑥} ⊆ 𝑋) → (𝐾 IdlGen {𝑥}) ∈ (Idl‘𝐾))
3836, 37sylan2 593 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) ∈ (Idl‘𝐾))
39 eleq2 2830 . . . . . . . . . . . . . . . . . . 19 ((Idl‘𝐾) = {{𝑍}, 𝑋} → ((𝐾 IdlGen {𝑥}) ∈ (Idl‘𝐾) ↔ (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋}))
4038, 39syl5ibcom 245 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ((Idl‘𝐾) = {{𝑍}, 𝑋} → (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋}))
4140imp 406 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋})
4241an32s 652 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋})
43 ovex 7464 . . . . . . . . . . . . . . . . 17 (𝐾 IdlGen {𝑥}) ∈ V
4443elpr 4650 . . . . . . . . . . . . . . . 16 ((𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋} ↔ ((𝐾 IdlGen {𝑥}) = {𝑍} ∨ (𝐾 IdlGen {𝑥}) = 𝑋))
4542, 44sylib 218 . . . . . . . . . . . . . . 15 (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ((𝐾 IdlGen {𝑥}) = {𝑍} ∨ (𝐾 IdlGen {𝑥}) = 𝑋))
4645ord 865 . . . . . . . . . . . . . 14 (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (¬ (𝐾 IdlGen {𝑥}) = {𝑍} → (𝐾 IdlGen {𝑥}) = 𝑋))
4734, 46mpd 15 . . . . . . . . . . . . 13 (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) = 𝑋)
4813, 47sylanl1 680 . . . . . . . . . . . 12 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) = 𝑋)
493, 4, 5prnc 38074 . . . . . . . . . . . . . 14 ((𝐾 ∈ CRingOps ∧ 𝑥𝑋) → (𝐾 IdlGen {𝑥}) = {𝑧𝑋 ∣ ∃𝑦𝑋 𝑧 = (𝑦𝐻𝑥)})
5035, 49sylan2 593 . . . . . . . . . . . . 13 ((𝐾 ∈ CRingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) = {𝑧𝑋 ∣ ∃𝑦𝑋 𝑧 = (𝑦𝐻𝑥)})
5150adantlr 715 . . . . . . . . . . . 12 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) = {𝑧𝑋 ∣ ∃𝑦𝑋 𝑧 = (𝑦𝐻𝑥)})
5248, 51eqtr3d 2779 . . . . . . . . . . 11 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → 𝑋 = {𝑧𝑋 ∣ ∃𝑦𝑋 𝑧 = (𝑦𝐻𝑥)})
5320, 52eleqtrd 2843 . . . . . . . . . 10 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → 𝑈 ∈ {𝑧𝑋 ∣ ∃𝑦𝑋 𝑧 = (𝑦𝐻𝑥)})
54 eqeq1 2741 . . . . . . . . . . . 12 (𝑧 = 𝑈 → (𝑧 = (𝑦𝐻𝑥) ↔ 𝑈 = (𝑦𝐻𝑥)))
5554rexbidv 3179 . . . . . . . . . . 11 (𝑧 = 𝑈 → (∃𝑦𝑋 𝑧 = (𝑦𝐻𝑥) ↔ ∃𝑦𝑋 𝑈 = (𝑦𝐻𝑥)))
5655elrab 3692 . . . . . . . . . 10 (𝑈 ∈ {𝑧𝑋 ∣ ∃𝑦𝑋 𝑧 = (𝑦𝐻𝑥)} ↔ (𝑈𝑋 ∧ ∃𝑦𝑋 𝑈 = (𝑦𝐻𝑥)))
5753, 56sylib 218 . . . . . . . . 9 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝑈𝑋 ∧ ∃𝑦𝑋 𝑈 = (𝑦𝐻𝑥)))
5857simprd 495 . . . . . . . 8 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ∃𝑦𝑋 𝑈 = (𝑦𝐻𝑥))
59 eqcom 2744 . . . . . . . . 9 ((𝑦𝐻𝑥) = 𝑈𝑈 = (𝑦𝐻𝑥))
6059rexbii 3094 . . . . . . . 8 (∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈 ↔ ∃𝑦𝑋 𝑈 = (𝑦𝐻𝑥))
6158, 60sylibr 234 . . . . . . 7 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)
6261ralrimiva 3146 . . . . . 6 ((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)
63623adant2 1132 . . . . 5 ((𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)
6414, 15, 63jca32 515 . . . 4 ((𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → (𝐾 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)))
653, 4, 6, 5, 7isdrngo3 37966 . . . 4 (𝐾 ∈ DivRingOps ↔ (𝐾 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)))
6664, 65sylibr 234 . . 3 ((𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ DivRingOps)
67 simp1 1137 . . 3 ((𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ CRingOps)
68 isfld2 38012 . . 3 (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps))
6966, 67, 68sylanbrc 583 . 2 ((𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ Fld)
7012, 69impbii 209 1 (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  {crab 3436  cdif 3948  wss 3951  {csn 4626  {cpr 4628  ran crn 5686  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  GIdcgi 30509  RingOpscrngo 37901  DivRingOpscdrng 37955  Fldcfld 37998  CRingOpsccring 38000  Idlcidl 38014   IdlGen cigen 38066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-1o 8506  df-en 8986  df-grpo 30512  df-gid 30513  df-ginv 30514  df-ablo 30564  df-ass 37850  df-exid 37852  df-mgmOLD 37856  df-sgrOLD 37868  df-mndo 37874  df-rngo 37902  df-drngo 37956  df-com2 37997  df-fld 37999  df-crngo 38001  df-idl 38017  df-igen 38067
This theorem is referenced by:  isfldidl2  38076
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