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Theorem isfldidl 32833
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 fldcrng 32769 . . 3 (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)
2 flddivrng 32764 . . . 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 32719 . . . 4 (𝐾 ∈ DivRingOps → 𝑈𝑍)
92, 8syl 17 . . 3 (𝐾 ∈ Fld → 𝑈𝑍)
103, 4, 5, 6divrngidl 32793 . . . 4 (𝐾 ∈ DivRingOps → (Idl‘𝐾) = {{𝑍}, 𝑋})
112, 10syl 17 . . 3 (𝐾 ∈ Fld → (Idl‘𝐾) = {{𝑍}, 𝑋})
121, 9, 113jca 1234 . 2 (𝐾 ∈ Fld → (𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
13 crngorngo 32765 . . . . . 6 (𝐾 ∈ CRingOps → 𝐾 ∈ RingOps)
14133ad2ant1 1074 . . . . 5 ((𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ RingOps)
15 simp2 1054 . . . . 5 ((𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝑈𝑍)
163rneqi 5260 . . . . . . . . . . . . . . 15 ran 𝐺 = ran (1st𝐾)
175, 16eqtri 2631 . . . . . . . . . . . . . 14 𝑋 = ran (1st𝐾)
1817, 4, 7rngo1cl 32704 . . . . . . . . . . . . 13 (𝐾 ∈ RingOps → 𝑈𝑋)
1913, 18syl 17 . . . . . . . . . . . 12 (𝐾 ∈ CRingOps → 𝑈𝑋)
2019ad2antrr 757 . . . . . . . . . . 11 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → 𝑈𝑋)
21 eldif 3549 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝑋 ∖ {𝑍}) ↔ (𝑥𝑋 ∧ ¬ 𝑥 ∈ {𝑍}))
22 snssi 4279 . . . . . . . . . . . . . . . . . . 19 (𝑥𝑋 → {𝑥} ⊆ 𝑋)
233, 5igenss 32827 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ RingOps ∧ {𝑥} ⊆ 𝑋) → {𝑥} ⊆ (𝐾 IdlGen {𝑥}))
2422, 23sylan2 489 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ RingOps ∧ 𝑥𝑋) → {𝑥} ⊆ (𝐾 IdlGen {𝑥}))
25 vex 3175 . . . . . . . . . . . . . . . . . . . . . 22 𝑥 ∈ V
2625snss 4258 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝐾 IdlGen {𝑥}) ↔ {𝑥} ⊆ (𝐾 IdlGen {𝑥}))
2726biimpri 216 . . . . . . . . . . . . . . . . . . . 20 ({𝑥} ⊆ (𝐾 IdlGen {𝑥}) → 𝑥 ∈ (𝐾 IdlGen {𝑥}))
28 eleq2 2676 . . . . . . . . . . . . . . . . . . . 20 ((𝐾 IdlGen {𝑥}) = {𝑍} → (𝑥 ∈ (𝐾 IdlGen {𝑥}) ↔ 𝑥 ∈ {𝑍}))
2927, 28syl5ibcom 233 . . . . . . . . . . . . . . . . . . 19 ({𝑥} ⊆ (𝐾 IdlGen {𝑥}) → ((𝐾 IdlGen {𝑥}) = {𝑍} → 𝑥 ∈ {𝑍}))
3029con3dimp 455 . . . . . . . . . . . . . . . . . 18 (({𝑥} ⊆ (𝐾 IdlGen {𝑥}) ∧ ¬ 𝑥 ∈ {𝑍}) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
3124, 30sylan 486 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ RingOps ∧ 𝑥𝑋) ∧ ¬ 𝑥 ∈ {𝑍}) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
3231anasss 676 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ RingOps ∧ (𝑥𝑋 ∧ ¬ 𝑥 ∈ {𝑍})) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
3321, 32sylan2b 490 . . . . . . . . . . . . . . 15 ((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
3433adantlr 746 . . . . . . . . . . . . . 14 (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
35 eldifi 3693 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝑋 ∖ {𝑍}) → 𝑥𝑋)
3635snssd 4280 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (𝑋 ∖ {𝑍}) → {𝑥} ⊆ 𝑋)
373, 5igenidl 32828 . . . . . . . . . . . . . . . . . . . 20 ((𝐾 ∈ RingOps ∧ {𝑥} ⊆ 𝑋) → (𝐾 IdlGen {𝑥}) ∈ (Idl‘𝐾))
3836, 37sylan2 489 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) ∈ (Idl‘𝐾))
39 eleq2 2676 . . . . . . . . . . . . . . . . . . 19 ((Idl‘𝐾) = {{𝑍}, 𝑋} → ((𝐾 IdlGen {𝑥}) ∈ (Idl‘𝐾) ↔ (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋}))
4038, 39syl5ibcom 233 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ((Idl‘𝐾) = {{𝑍}, 𝑋} → (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋}))
4140imp 443 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋})
4241an32s 841 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋})
43 ovex 6555 . . . . . . . . . . . . . . . . 17 (𝐾 IdlGen {𝑥}) ∈ V
4443elpr 4145 . . . . . . . . . . . . . . . 16 ((𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋} ↔ ((𝐾 IdlGen {𝑥}) = {𝑍} ∨ (𝐾 IdlGen {𝑥}) = 𝑋))
4542, 44sylib 206 . . . . . . . . . . . . . . 15 (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ((𝐾 IdlGen {𝑥}) = {𝑍} ∨ (𝐾 IdlGen {𝑥}) = 𝑋))
4645ord 390 . . . . . . . . . . . . . 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 679 . . . . . . . . . . . 12 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) = 𝑋)
493, 4, 5prnc 32832 . . . . . . . . . . . . . 14 ((𝐾 ∈ CRingOps ∧ 𝑥𝑋) → (𝐾 IdlGen {𝑥}) = {𝑧𝑋 ∣ ∃𝑦𝑋 𝑧 = (𝑦𝐻𝑥)})
5035, 49sylan2 489 . . . . . . . . . . . . 13 ((𝐾 ∈ CRingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) = {𝑧𝑋 ∣ ∃𝑦𝑋 𝑧 = (𝑦𝐻𝑥)})
5150adantlr 746 . . . . . . . . . . . 12 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) = {𝑧𝑋 ∣ ∃𝑦𝑋 𝑧 = (𝑦𝐻𝑥)})
5248, 51eqtr3d 2645 . . . . . . . . . . 11 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → 𝑋 = {𝑧𝑋 ∣ ∃𝑦𝑋 𝑧 = (𝑦𝐻𝑥)})
5320, 52eleqtrd 2689 . . . . . . . . . 10 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → 𝑈 ∈ {𝑧𝑋 ∣ ∃𝑦𝑋 𝑧 = (𝑦𝐻𝑥)})
54 eqeq1 2613 . . . . . . . . . . . 12 (𝑧 = 𝑈 → (𝑧 = (𝑦𝐻𝑥) ↔ 𝑈 = (𝑦𝐻𝑥)))
5554rexbidv 3033 . . . . . . . . . . 11 (𝑧 = 𝑈 → (∃𝑦𝑋 𝑧 = (𝑦𝐻𝑥) ↔ ∃𝑦𝑋 𝑈 = (𝑦𝐻𝑥)))
5655elrab 3330 . . . . . . . . . 10 (𝑈 ∈ {𝑧𝑋 ∣ ∃𝑦𝑋 𝑧 = (𝑦𝐻𝑥)} ↔ (𝑈𝑋 ∧ ∃𝑦𝑋 𝑈 = (𝑦𝐻𝑥)))
5753, 56sylib 206 . . . . . . . . 9 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝑈𝑋 ∧ ∃𝑦𝑋 𝑈 = (𝑦𝐻𝑥)))
5857simprd 477 . . . . . . . 8 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ∃𝑦𝑋 𝑈 = (𝑦𝐻𝑥))
59 eqcom 2616 . . . . . . . . 9 ((𝑦𝐻𝑥) = 𝑈𝑈 = (𝑦𝐻𝑥))
6059rexbii 3022 . . . . . . . 8 (∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈 ↔ ∃𝑦𝑋 𝑈 = (𝑦𝐻𝑥))
6158, 60sylibr 222 . . . . . . 7 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)
6261ralrimiva 2948 . . . . . 6 ((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)
63623adant2 1072 . . . . 5 ((𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)
6414, 15, 63jca32 555 . . . 4 ((𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → (𝐾 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)))
653, 4, 6, 5, 7isdrngo3 32724 . . . 4 (𝐾 ∈ DivRingOps ↔ (𝐾 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)))
6664, 65sylibr 222 . . 3 ((𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ DivRingOps)
67 simp1 1053 . . 3 ((𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ CRingOps)
68 isfld2 32770 . . 3 (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps))
6966, 67, 68sylanbrc 694 . 2 ((𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ Fld)
7012, 69impbii 197 1 (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 194  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  {crab 2899  cdif 3536  wss 3539  {csn 4124  {cpr 4126  ran crn 5029  cfv 5790  (class class class)co 6527  1st c1st 7034  2nd c2nd 7035  GIdcgi 26494  RingOpscrngo 32659  DivRingOpscdrng 32713  Fldcfld 32756  CRingOpsccring 32758  Idlcidl 32772   IdlGen cigen 32824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-1o 7424  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-grpo 26497  df-gid 26498  df-ginv 26499  df-ablo 26552  df-ass 32608  df-exid 32610  df-mgmOLD 32614  df-sgrOLD 32626  df-mndo 32632  df-rngo 32660  df-drngo 32714  df-com2 32755  df-fld 32757  df-crngo 32759  df-idl 32775  df-igen 32825
This theorem is referenced by:  isfldidl2  32834
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