Theorem isfldidl2 38108

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 2731	. . 3 ⊢ (GId‘𝐻) = (GId‘𝐻)
6	1, 2, 3, 4, 5	isfldidl 38107	. 2 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
7		crngorngo 38039	. . . . . . 7 ⊢ (𝐾 ∈ CRingOps → 𝐾 ∈ RingOps)
8		eqcom 2738	. . . . . . . 8 ⊢ ((GId‘𝐻) = 𝑍 ↔ 𝑍 = (GId‘𝐻))
9	1, 2, 3, 4, 5	0rngo 38066	. . . . . . . 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 2972	. . . . 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 1541   ∈ wcel 2111   ≠ wne 2928  {csn 4576  {cpr 4578  ran crn 5617  ‘cfv 6481  1^st c1st 7919  2^nd c2nd 7920  GIdcgi 30465  RingOpscrngo 37933  Fldcfld 38030  CRingOpsccring 38032  Idlcidl 38046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-1o 8385  df-en 8870  df-grpo 30468  df-gid 30469  df-ginv 30470  df-ablo 30520  df-ass 37882  df-exid 37884  df-mgmOLD 37888  df-sgrOLD 37900  df-mndo 37906  df-rngo 37934  df-drngo 37988  df-com2 38029  df-fld 38031  df-crngo 38033  df-idl 38049  df-igen 38099

This theorem is referenced by: (None)