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Theorem isfldidl2 34199
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 2760 . . 3 (GId‘𝐻) = (GId‘𝐻)
61, 2, 3, 4, 5isfldidl 34198 . 2 (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
7 crngorngo 34130 . . . . . . 7 (𝐾 ∈ CRingOps → 𝐾 ∈ RingOps)
8 eqcom 2767 . . . . . . . 8 ((GId‘𝐻) = 𝑍𝑍 = (GId‘𝐻))
91, 2, 3, 4, 50rngo 34157 . . . . . . . 8 (𝐾 ∈ RingOps → (𝑍 = (GId‘𝐻) ↔ 𝑋 = {𝑍}))
108, 9syl5bb 272 . . . . . . 7 (𝐾 ∈ RingOps → ((GId‘𝐻) = 𝑍𝑋 = {𝑍}))
117, 10syl 17 . . . . . 6 (𝐾 ∈ CRingOps → ((GId‘𝐻) = 𝑍𝑋 = {𝑍}))
1211necon3bid 2976 . . . . 5 (𝐾 ∈ CRingOps → ((GId‘𝐻) ≠ 𝑍𝑋 ≠ {𝑍}))
1312anbi1d 743 . . . 4 (𝐾 ∈ CRingOps → (((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
1413pm5.32i 672 . . 3 ((𝐾 ∈ CRingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})) ↔ (𝐾 ∈ CRingOps ∧ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
15 3anass 1081 . . 3 ((𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
16 3anass 1081 . . 3 ((𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
1714, 15, 163bitr4i 292 . 2 ((𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
186, 17bitri 264 1 (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  {csn 4321  {cpr 4323  ran crn 5267  cfv 6049  1st c1st 7332  2nd c2nd 7333  GIdcgi 27674  RingOpscrngo 34024  Fldcfld 34121  CRingOpsccring 34123  Idlcidl 34137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-1o 7730  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-grpo 27677  df-gid 27678  df-ginv 27679  df-ablo 27729  df-ass 33973  df-exid 33975  df-mgmOLD 33979  df-sgrOLD 33991  df-mndo 33997  df-rngo 34025  df-drngo 34079  df-com2 34120  df-fld 34122  df-crngo 34124  df-idl 34140  df-igen 34190
This theorem is referenced by: (None)
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