Theorem isfldidl2 38029

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 2740	. . 3 ⊢ (GId‘𝐻) = (GId‘𝐻)
6	1, 2, 3, 4, 5	isfldidl 38028	. 2 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
7		crngorngo 37960	. . . . . . 7 ⊢ (𝐾 ∈ CRingOps → 𝐾 ∈ RingOps)
8		eqcom 2747	. . . . . . . 8 ⊢ ((GId‘𝐻) = 𝑍 ↔ 𝑍 = (GId‘𝐻))
9	1, 2, 3, 4, 5	0rngo 37987	. . . . . . . 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 2991	. . . . 5 ⊢ (𝐾 ∈ CRingOps → ((GId‘𝐻) ≠ 𝑍 ↔ 𝑋 ≠ {𝑍}))
13	12	anbi1d 630	. . . 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 1095	. . 3 ⊢ ((𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ↔ (𝐾 ∈ CRingOps ∧ ((GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})))
16		3anass 1095	. . 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 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  {csn 4648  {cpr 4650  ran crn 5701  ‘cfv 6573  1^st c1st 8028  2^nd c2nd 8029  GIdcgi 30522  RingOpscrngo 37854  Fldcfld 37951  CRingOpsccring 37953  Idlcidl 37967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-1o 8522  df-en 9004  df-grpo 30525  df-gid 30526  df-ginv 30527  df-ablo 30577  df-ass 37803  df-exid 37805  df-mgmOLD 37809  df-sgrOLD 37821  df-mndo 37827  df-rngo 37855  df-drngo 37909  df-com2 37950  df-fld 37952  df-crngo 37954  df-idl 37970  df-igen 38020

This theorem is referenced by: (None)