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Theorem isfldidl 34221
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 34157 . . 3 (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)
2 flddivrng 34152 . . . 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 34107 . . . 4 (𝐾 ∈ DivRingOps → 𝑈𝑍)
92, 8syl 17 . . 3 (𝐾 ∈ Fld → 𝑈𝑍)
103, 4, 5, 6divrngidl 34181 . . . 4 (𝐾 ∈ DivRingOps → (Idl‘𝐾) = {{𝑍}, 𝑋})
112, 10syl 17 . . 3 (𝐾 ∈ Fld → (Idl‘𝐾) = {{𝑍}, 𝑋})
121, 9, 113jca 1158 . 2 (𝐾 ∈ Fld → (𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
13 crngorngo 34153 . . . . . 6 (𝐾 ∈ CRingOps → 𝐾 ∈ RingOps)
14133ad2ant1 1163 . . . . 5 ((𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ RingOps)
15 simp2 1167 . . . . 5 ((𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝑈𝑍)
163rneqi 5520 . . . . . . . . . . . . . . 15 ran 𝐺 = ran (1st𝐾)
175, 16eqtri 2787 . . . . . . . . . . . . . 14 𝑋 = ran (1st𝐾)
1817, 4, 7rngo1cl 34092 . . . . . . . . . . . . 13 (𝐾 ∈ RingOps → 𝑈𝑋)
1913, 18syl 17 . . . . . . . . . . . 12 (𝐾 ∈ CRingOps → 𝑈𝑋)
2019ad2antrr 717 . . . . . . . . . . 11 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → 𝑈𝑋)
21 eldif 3742 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝑋 ∖ {𝑍}) ↔ (𝑥𝑋 ∧ ¬ 𝑥 ∈ {𝑍}))
22 snssi 4493 . . . . . . . . . . . . . . . . . . 19 (𝑥𝑋 → {𝑥} ⊆ 𝑋)
233, 5igenss 34215 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ RingOps ∧ {𝑥} ⊆ 𝑋) → {𝑥} ⊆ (𝐾 IdlGen {𝑥}))
2422, 23sylan2 586 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ RingOps ∧ 𝑥𝑋) → {𝑥} ⊆ (𝐾 IdlGen {𝑥}))
25 vex 3353 . . . . . . . . . . . . . . . . . . . . . 22 𝑥 ∈ V
2625snss 4470 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝐾 IdlGen {𝑥}) ↔ {𝑥} ⊆ (𝐾 IdlGen {𝑥}))
2726biimpri 219 . . . . . . . . . . . . . . . . . . . 20 ({𝑥} ⊆ (𝐾 IdlGen {𝑥}) → 𝑥 ∈ (𝐾 IdlGen {𝑥}))
28 eleq2 2833 . . . . . . . . . . . . . . . . . . . 20 ((𝐾 IdlGen {𝑥}) = {𝑍} → (𝑥 ∈ (𝐾 IdlGen {𝑥}) ↔ 𝑥 ∈ {𝑍}))
2927, 28syl5ibcom 236 . . . . . . . . . . . . . . . . . . 19 ({𝑥} ⊆ (𝐾 IdlGen {𝑥}) → ((𝐾 IdlGen {𝑥}) = {𝑍} → 𝑥 ∈ {𝑍}))
3029con3dimp 397 . . . . . . . . . . . . . . . . . 18 (({𝑥} ⊆ (𝐾 IdlGen {𝑥}) ∧ ¬ 𝑥 ∈ {𝑍}) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
3124, 30sylan 575 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ RingOps ∧ 𝑥𝑋) ∧ ¬ 𝑥 ∈ {𝑍}) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
3231anasss 458 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ RingOps ∧ (𝑥𝑋 ∧ ¬ 𝑥 ∈ {𝑍})) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
3321, 32sylan2b 587 . . . . . . . . . . . . . . 15 ((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
3433adantlr 706 . . . . . . . . . . . . . 14 (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
35 eldifi 3894 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝑋 ∖ {𝑍}) → 𝑥𝑋)
3635snssd 4494 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (𝑋 ∖ {𝑍}) → {𝑥} ⊆ 𝑋)
373, 5igenidl 34216 . . . . . . . . . . . . . . . . . . . 20 ((𝐾 ∈ RingOps ∧ {𝑥} ⊆ 𝑋) → (𝐾 IdlGen {𝑥}) ∈ (Idl‘𝐾))
3836, 37sylan2 586 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) ∈ (Idl‘𝐾))
39 eleq2 2833 . . . . . . . . . . . . . . . . . . 19 ((Idl‘𝐾) = {{𝑍}, 𝑋} → ((𝐾 IdlGen {𝑥}) ∈ (Idl‘𝐾) ↔ (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋}))
4038, 39syl5ibcom 236 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ((Idl‘𝐾) = {{𝑍}, 𝑋} → (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋}))
4140imp 395 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋})
4241an32s 642 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋})
43 ovex 6874 . . . . . . . . . . . . . . . . 17 (𝐾 IdlGen {𝑥}) ∈ V
4443elpr 4357 . . . . . . . . . . . . . . . 16 ((𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋} ↔ ((𝐾 IdlGen {𝑥}) = {𝑍} ∨ (𝐾 IdlGen {𝑥}) = 𝑋))
4542, 44sylib 209 . . . . . . . . . . . . . . 15 (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ((𝐾 IdlGen {𝑥}) = {𝑍} ∨ (𝐾 IdlGen {𝑥}) = 𝑋))
4645ord 890 . . . . . . . . . . . . . 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 670 . . . . . . . . . . . 12 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) = 𝑋)
493, 4, 5prnc 34220 . . . . . . . . . . . . . 14 ((𝐾 ∈ CRingOps ∧ 𝑥𝑋) → (𝐾 IdlGen {𝑥}) = {𝑧𝑋 ∣ ∃𝑦𝑋 𝑧 = (𝑦𝐻𝑥)})
5035, 49sylan2 586 . . . . . . . . . . . . 13 ((𝐾 ∈ CRingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) = {𝑧𝑋 ∣ ∃𝑦𝑋 𝑧 = (𝑦𝐻𝑥)})
5150adantlr 706 . . . . . . . . . . . 12 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) = {𝑧𝑋 ∣ ∃𝑦𝑋 𝑧 = (𝑦𝐻𝑥)})
5248, 51eqtr3d 2801 . . . . . . . . . . 11 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → 𝑋 = {𝑧𝑋 ∣ ∃𝑦𝑋 𝑧 = (𝑦𝐻𝑥)})
5320, 52eleqtrd 2846 . . . . . . . . . 10 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → 𝑈 ∈ {𝑧𝑋 ∣ ∃𝑦𝑋 𝑧 = (𝑦𝐻𝑥)})
54 eqeq1 2769 . . . . . . . . . . . 12 (𝑧 = 𝑈 → (𝑧 = (𝑦𝐻𝑥) ↔ 𝑈 = (𝑦𝐻𝑥)))
5554rexbidv 3199 . . . . . . . . . . 11 (𝑧 = 𝑈 → (∃𝑦𝑋 𝑧 = (𝑦𝐻𝑥) ↔ ∃𝑦𝑋 𝑈 = (𝑦𝐻𝑥)))
5655elrab 3519 . . . . . . . . . 10 (𝑈 ∈ {𝑧𝑋 ∣ ∃𝑦𝑋 𝑧 = (𝑦𝐻𝑥)} ↔ (𝑈𝑋 ∧ ∃𝑦𝑋 𝑈 = (𝑦𝐻𝑥)))
5753, 56sylib 209 . . . . . . . . 9 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝑈𝑋 ∧ ∃𝑦𝑋 𝑈 = (𝑦𝐻𝑥)))
5857simprd 489 . . . . . . . 8 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ∃𝑦𝑋 𝑈 = (𝑦𝐻𝑥))
59 eqcom 2772 . . . . . . . . 9 ((𝑦𝐻𝑥) = 𝑈𝑈 = (𝑦𝐻𝑥))
6059rexbii 3188 . . . . . . . 8 (∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈 ↔ ∃𝑦𝑋 𝑈 = (𝑦𝐻𝑥))
6158, 60sylibr 225 . . . . . . 7 (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)
6261ralrimiva 3113 . . . . . 6 ((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)
63623adant2 1161 . . . . 5 ((𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)
6414, 15, 63jca32 511 . . . 4 ((𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → (𝐾 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)))
653, 4, 6, 5, 7isdrngo3 34112 . . . 4 (𝐾 ∈ DivRingOps ↔ (𝐾 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)))
6664, 65sylibr 225 . . 3 ((𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ DivRingOps)
67 simp1 1166 . . 3 ((𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ CRingOps)
68 isfld2 34158 . . 3 (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps))
6966, 67, 68sylanbrc 578 . 2 ((𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ Fld)
7012, 69impbii 200 1 (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑈𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  {crab 3059  cdif 3729  wss 3732  {csn 4334  {cpr 4336  ran crn 5278  cfv 6068  (class class class)co 6842  1st c1st 7364  2nd c2nd 7365  GIdcgi 27736  RingOpscrngo 34047  DivRingOpscdrng 34101  Fldcfld 34144  CRingOpsccring 34146  Idlcidl 34160   IdlGen cigen 34212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-1o 7764  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-grpo 27739  df-gid 27740  df-ginv 27741  df-ablo 27791  df-ass 33996  df-exid 33998  df-mgmOLD 34002  df-sgrOLD 34014  df-mndo 34020  df-rngo 34048  df-drngo 34102  df-com2 34143  df-fld 34145  df-crngo 34147  df-idl 34163  df-igen 34213
This theorem is referenced by:  isfldidl2  34222
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