isfldidl - Mathbox for Jeff Madsen

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Theorem isfldidl 35361

Description: Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 10-Jun-2010.)

**Hypotheses**
Ref	Expression
isfldidl.1	⊢ 𝐺 = (1^st ‘𝐾)
isfldidl.2	⊢ 𝐻 = (2^nd ‘𝐾)
isfldidl.3	⊢ 𝑋 = ran 𝐺
isfldidl.4	⊢ 𝑍 = (GId‘𝐺)
isfldidl.5	⊢ 𝑈 = (GId‘𝐻)

**Assertion**
Ref	Expression
isfldidl	⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))

Proof of Theorem isfldidl Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fldcrng 35297	. . 3 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)
2		flddivrng 35292	. . . 4 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)
3		isfldidl.1	. . . . 5 ⊢ 𝐺 = (1^st ‘𝐾)
4		isfldidl.2	. . . . 5 ⊢ 𝐻 = (2^nd ‘𝐾)
5		isfldidl.3	. . . . 5 ⊢ 𝑋 = ran 𝐺
6		isfldidl.4	. . . . 5 ⊢ 𝑍 = (GId‘𝐺)
7		isfldidl.5	. . . . 5 ⊢ 𝑈 = (GId‘𝐻)
8	3, 4, 5, 6, 7	dvrunz 35247	. . . 4 ⊢ (𝐾 ∈ DivRingOps → 𝑈 ≠ 𝑍)
9	2, 8	syl 17	. . 3 ⊢ (𝐾 ∈ Fld → 𝑈 ≠ 𝑍)
10	3, 4, 5, 6	divrngidl 35321	. . . 4 ⊢ (𝐾 ∈ DivRingOps → (Idl‘𝐾) = {{𝑍}, 𝑋})
11	2, 10	syl 17	. . 3 ⊢ (𝐾 ∈ Fld → (Idl‘𝐾) = {{𝑍}, 𝑋})
12	1, 9, 11	3jca 1124	. 2 ⊢ (𝐾 ∈ Fld → (𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
13		crngorngo 35293	. . . . . 6 ⊢ (𝐾 ∈ CRingOps → 𝐾 ∈ RingOps)
14	13	3ad2ant1 1129	. . . . 5 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ RingOps)
15		simp2 1133	. . . . 5 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝑈 ≠ 𝑍)
16	3	rneqi 5807	. . . . . . . . . . . . . . 15 ⊢ ran 𝐺 = ran (1^st ‘𝐾)
17	5, 16	eqtri 2844	. . . . . . . . . . . . . 14 ⊢ 𝑋 = ran (1^st ‘𝐾)
18	17, 4, 7	rngo1cl 35232	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ RingOps → 𝑈 ∈ 𝑋)
19	13, 18	syl 17	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ CRingOps → 𝑈 ∈ 𝑋)
20	19	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → 𝑈 ∈ 𝑋)
21		eldif 3946	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝑋 ∖ {𝑍}) ↔ (𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ {𝑍}))
22		snssi 4741	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝑋 → {𝑥} ⊆ 𝑋)
23	3, 5	igenss 35355	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ RingOps ∧ {𝑥} ⊆ 𝑋) → {𝑥} ⊆ (𝐾 IdlGen {𝑥}))
24	22, 23	sylan2 594	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → {𝑥} ⊆ (𝐾 IdlGen {𝑥}))
25		vex 3497	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑥 ∈ V
26	25	snss 4718	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝐾 IdlGen {𝑥}) ↔ {𝑥} ⊆ (𝐾 IdlGen {𝑥}))
27	26	biimpri 230	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({𝑥} ⊆ (𝐾 IdlGen {𝑥}) → 𝑥 ∈ (𝐾 IdlGen {𝑥}))
28		eleq2 2901	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 IdlGen {𝑥}) = {𝑍} → (𝑥 ∈ (𝐾 IdlGen {𝑥}) ↔ 𝑥 ∈ {𝑍}))
29	27, 28	syl5ibcom 247	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑥} ⊆ (𝐾 IdlGen {𝑥}) → ((𝐾 IdlGen {𝑥}) = {𝑍} → 𝑥 ∈ {𝑍}))
30	29	con3dimp 411	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑥} ⊆ (𝐾 IdlGen {𝑥}) ∧ ¬ 𝑥 ∈ {𝑍}) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
31	24, 30	sylan 582	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ RingOps ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ {𝑍}) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
32	31	anasss 469	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ RingOps ∧ (𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ {𝑍})) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
33	21, 32	sylan2b 595	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
34	33	adantlr 713	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
35		eldifi 4103	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝑋 ∖ {𝑍}) → 𝑥 ∈ 𝑋)
36	35	snssd 4742	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝑋 ∖ {𝑍}) → {𝑥} ⊆ 𝑋)
37	3, 5	igenidl 35356	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 ∈ RingOps ∧ {𝑥} ⊆ 𝑋) → (𝐾 IdlGen {𝑥}) ∈ (Idl‘𝐾))
38	36, 37	sylan2 594	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) ∈ (Idl‘𝐾))
39		eleq2 2901	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Idl‘𝐾) = {{𝑍}, 𝑋} → ((𝐾 IdlGen {𝑥}) ∈ (Idl‘𝐾) ↔ (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋}))
40	38, 39	syl5ibcom 247	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ((Idl‘𝐾) = {{𝑍}, 𝑋} → (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋}))
41	40	imp 409	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋})
42	41	an32s 650	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋})
43		ovex 7189	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 IdlGen {𝑥}) ∈ V
44	43	elpr 4590	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋} ↔ ((𝐾 IdlGen {𝑥}) = {𝑍} ∨ (𝐾 IdlGen {𝑥}) = 𝑋))
45	42, 44	sylib 220	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ((𝐾 IdlGen {𝑥}) = {𝑍} ∨ (𝐾 IdlGen {𝑥}) = 𝑋))
46	45	ord 860	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (¬ (𝐾 IdlGen {𝑥}) = {𝑍} → (𝐾 IdlGen {𝑥}) = 𝑋))
47	34, 46	mpd 15	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) = 𝑋)
48	13, 47	sylanl1 678	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) = 𝑋)
49	3, 4, 5	prnc 35360	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑥 ∈ 𝑋) → (𝐾 IdlGen {𝑥}) = {𝑧 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑧 = (𝑦𝐻𝑥)})
50	35, 49	sylan2 594	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) = {𝑧 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑧 = (𝑦𝐻𝑥)})
51	50	adantlr 713	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) = {𝑧 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑧 = (𝑦𝐻𝑥)})
52	48, 51	eqtr3d 2858	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → 𝑋 = {𝑧 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑧 = (𝑦𝐻𝑥)})
53	20, 52	eleqtrd 2915	. . . . . . . . . 10 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → 𝑈 ∈ {𝑧 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑧 = (𝑦𝐻𝑥)})
54		eqeq1 2825	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑈 → (𝑧 = (𝑦𝐻𝑥) ↔ 𝑈 = (𝑦𝐻𝑥)))
55	54	rexbidv 3297	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑈 → (∃𝑦 ∈ 𝑋 𝑧 = (𝑦𝐻𝑥) ↔ ∃𝑦 ∈ 𝑋 𝑈 = (𝑦𝐻𝑥)))
56	55	elrab 3680	. . . . . . . . . 10 ⊢ (𝑈 ∈ {𝑧 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑧 = (𝑦𝐻𝑥)} ↔ (𝑈 ∈ 𝑋 ∧ ∃𝑦 ∈ 𝑋 𝑈 = (𝑦𝐻𝑥)))
57	53, 56	sylib 220	. . . . . . . . 9 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝑈 ∈ 𝑋 ∧ ∃𝑦 ∈ 𝑋 𝑈 = (𝑦𝐻𝑥)))
58	57	simprd 498	. . . . . . . 8 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ∃𝑦 ∈ 𝑋 𝑈 = (𝑦𝐻𝑥))
59		eqcom 2828	. . . . . . . . 9 ⊢ ((𝑦𝐻𝑥) = 𝑈 ↔ 𝑈 = (𝑦𝐻𝑥))
60	59	rexbii 3247	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈 ↔ ∃𝑦 ∈ 𝑋 𝑈 = (𝑦𝐻𝑥))
61	58, 60	sylibr 236	. . . . . . 7 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈)
62	61	ralrimiva 3182	. . . . . 6 ⊢ ((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈)
63	62	3adant2 1127	. . . . 5 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈)
64	14, 15, 63	jca32 518	. . . 4 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → (𝐾 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈)))
65	3, 4, 6, 5, 7	isdrngo3 35252	. . . 4 ⊢ (𝐾 ∈ DivRingOps ↔ (𝐾 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈)))
66	64, 65	sylibr 236	. . 3 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ DivRingOps)
67		simp1 1132	. . 3 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ CRingOps)
68		isfld2 35298	. . 3 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps))
69	66, 67, 68	sylanbrc 585	. 2 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ Fld)
70	12, 69	impbii 211	1 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 {crab 3142 ∖ cdif 3933 ⊆ wss 3936 {csn 4567 {cpr 4569 ran crn 5556 ‘cfv 6355 (class class class)co 7156 1^st c1st 7687 2^nd c2nd 7688 GIdcgi 28267 RingOpscrngo 35187 DivRingOpscdrng 35241 Fldcfld 35284 CRingOpsccring 35286 Idlcidl 35300 IdlGen cigen 35352

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-1o 8102 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-grpo 28270 df-gid 28271 df-ginv 28272 df-ablo 28322 df-ass 35136 df-exid 35138 df-mgmOLD 35142 df-sgrOLD 35154 df-mndo 35160 df-rngo 35188 df-drngo 35242 df-com2 35283 df-fld 35285 df-crngo 35287 df-idl 35303 df-igen 35353

This theorem is referenced by: isfldidl2 35362

W3C validator