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Theorem isfldidl2 35234
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 2826 . . 3 (GId‘𝐻) = (GId‘𝐻)
61, 2, 3, 4, 5isfldidl 35233 . 2 (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
7 crngorngo 35165 . . . . . . 7 (𝐾 ∈ CRingOps → 𝐾 ∈ RingOps)
8 eqcom 2833 . . . . . . . 8 ((GId‘𝐻) = 𝑍𝑍 = (GId‘𝐻))
91, 2, 3, 4, 50rngo 35192 . . . . . . . 8 (𝐾 ∈ RingOps → (𝑍 = (GId‘𝐻) ↔ 𝑋 = {𝑍}))
108, 9syl5bb 284 . . . . . . 7 (𝐾 ∈ RingOps → ((GId‘𝐻) = 𝑍𝑋 = {𝑍}))
117, 10syl 17 . . . . . 6 (𝐾 ∈ CRingOps → ((GId‘𝐻) = 𝑍𝑋 = {𝑍}))
1211necon3bid 3065 . . . . 5 (𝐾 ∈ CRingOps → ((GId‘𝐻) ≠ 𝑍𝑋 ≠ {𝑍}))
1312anbi1d 629 . . . 4 (𝐾 ∈ CRingOps → (((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
1413pm5.32i 575 . . 3 ((𝐾 ∈ CRingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})) ↔ (𝐾 ∈ CRingOps ∧ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
15 3anass 1089 . . 3 ((𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
16 3anass 1089 . . 3 ((𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
1714, 15, 163bitr4i 304 . 2 ((𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
186, 17bitri 276 1 (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021  {csn 4564  {cpr 4566  ran crn 5555  cfv 6354  1st c1st 7683  2nd c2nd 7684  GIdcgi 28200  RingOpscrngo 35059  Fldcfld 35156  CRingOpsccring 35158  Idlcidl 35172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-1o 8098  df-er 8284  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-grpo 28203  df-gid 28204  df-ginv 28205  df-ablo 28255  df-ass 35008  df-exid 35010  df-mgmOLD 35014  df-sgrOLD 35026  df-mndo 35032  df-rngo 35060  df-drngo 35114  df-com2 35155  df-fld 35157  df-crngo 35159  df-idl 35175  df-igen 35225
This theorem is referenced by: (None)
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