Theorem isfldidl2 38063

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 = (1st ‘𝐾)
isfldidl2.2	⊢ 𝐻 = (2nd ‘𝐾)
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

Step	Hyp	Ref	Expression
1		isfldidl2.1	. . 3 ⊢ 𝐺 = (1st ‘𝐾)
2		isfldidl2.2	. . 3 ⊢ 𝐻 = (2nd ‘𝐾)
3		isfldidl2.3	. . 3 ⊢ 𝑋 = ran 𝐺
4		isfldidl2.4	. . 3 ⊢ 𝑍 = (GId‘𝐺)
5		eqid 2729	. . 3 ⊢ (GId‘𝐻) = (GId‘𝐻)
6	1, 2, 3, 4, 5	isfldidl 38062	. 2 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
7		crngorngo 37994	. . . . . . 7 ⊢ (𝐾 ∈ CRingOps → 𝐾 ∈ RingOps)
8		eqcom 2736	. . . . . . . 8 ⊢ ((GId‘𝐻) = 𝑍 ↔ 𝑍 = (GId‘𝐻))
9	1, 2, 3, 4, 5	0rngo 38021	. . . . . . . 8 ⊢ (𝐾 ∈ RingOps → (𝑍 = (GId‘𝐻) ↔ 𝑋 = {𝑍}))
10	8, 9	bitrid 283	. . . . . . 7 ⊢ (𝐾 ∈ RingOps → ((GId‘𝐻) = 𝑍 ↔ 𝑋 = {𝑍}))
11	7, 10	syl 17	. . . . . 6 ⊢ (𝐾 ∈ CRingOps → ((GId‘𝐻) = 𝑍 ↔ 𝑋 = {𝑍}))
12	11	necon3bid 2969	. . . . 5 ⊢ (𝐾 ∈ CRingOps → ((GId‘𝐻) ≠ 𝑍 ↔ 𝑋 ≠ {𝑍}))
13	12	anbi1d 631	. . . 4 ⊢ (𝐾 ∈ CRingOps → (((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
14	13	pm5.32i 574	. . 3 ⊢ ((𝐾 ∈ CRingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})) ↔ (𝐾 ∈ CRingOps ∧ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
15		3anass 1094	. . 3 ⊢ ((𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
16		3anass 1094	. . 3 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ (𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
17	14, 15, 16	3bitr4i 303	. 2 ⊢ ((𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
18	6, 17	bitri 275	1 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  {csn 4589  {cpr 4591  ran crn 5639  ‘cfv 6511  1^st c1st 7966  2^nd c2nd 7967  GIdcgi 30419  RingOpscrngo 37888  Fldcfld 37985  CRingOpsccring 37987  Idlcidl 38001

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-1o 8434  df-en 8919  df-grpo 30422  df-gid 30423  df-ginv 30424  df-ablo 30474  df-ass 37837  df-exid 37839  df-mgmOLD 37843  df-sgrOLD 37855  df-mndo 37861  df-rngo 37889  df-drngo 37943  df-com2 37984  df-fld 37986  df-crngo 37988  df-idl 38004  df-igen 38054

This theorem is referenced by: (None)