isfldidl - Mathbox for Jeff Madsen

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Theorem isfldidl 39078

Description: Obsolete theorem, use isfieldidl 21541 instead. Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 10-Jun-2010.) (New usage is discouraged.) (Proof modification is discouraged.)

**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		fldcrngo 39014	. . 3 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)
2		flddivrng 39009	. . . 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 38964	. . . 4 ⊢ (𝐾 ∈ DivRingOps → 𝑈 ≠ 𝑍)
9	2, 8	syl 18	. . 3 ⊢ (𝐾 ∈ Fld → 𝑈 ≠ 𝑍)
10	3, 4, 5, 6	divrngidl 39038	. . . 4 ⊢ (𝐾 ∈ DivRingOps → (Idl‘𝐾) = {{𝑍}, 𝑋})
11	2, 10	syl 18	. . 3 ⊢ (𝐾 ∈ Fld → (Idl‘𝐾) = {{𝑍}, 𝑋})
12	1, 9, 11	3jca 1146	. 2 ⊢ (𝐾 ∈ Fld → (𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
13		crngorngo 39010	. . . . . 6 ⊢ (𝐾 ∈ CRingOps → 𝐾 ∈ RingOps)
14	13	3ad2ant1 1151	. . . . 5 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ RingOps)
15		simp2 1155	. . . . 5 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝑈 ≠ 𝑍)
16	3	rneqi 5918	. . . . . . . . . . . . . . 15 ⊢ ran 𝐺 = ran (1^st ‘𝐾)
17	5, 16	eqtri 2783	. . . . . . . . . . . . . 14 ⊢ 𝑋 = ran (1^st ‘𝐾)
18	17, 4, 7	rngo1cl 38949	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ RingOps → 𝑈 ∈ 𝑋)
19	13, 18	syl 18	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ CRingOps → 𝑈 ∈ 𝑋)
20	19	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → 𝑈 ∈ 𝑋)
21		eldif 4049	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝑋 ∖ {𝑍}) ↔ (𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ {𝑍}))
22		snssi 4745	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝑋 → {𝑥} ⊆ 𝑋)
23	3, 5	igenss 39072	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ RingOps ∧ {𝑥} ⊆ 𝑋) → {𝑥} ⊆ (𝐾 IdlGen {𝑥}))
24	22, 23	sylan2 605	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → {𝑥} ⊆ (𝐾 IdlGen {𝑥}))
25		vex 3454	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑥 ∈ V
26	25	snss 4744	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝐾 IdlGen {𝑥}) ↔ {𝑥} ⊆ (𝐾 IdlGen {𝑥}))
27	26	biimpri 231	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({𝑥} ⊆ (𝐾 IdlGen {𝑥}) → 𝑥 ∈ (𝐾 IdlGen {𝑥}))
28		eleq2 2849	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 IdlGen {𝑥}) = {𝑍} → (𝑥 ∈ (𝐾 IdlGen {𝑥}) ↔ 𝑥 ∈ {𝑍}))
29	27, 28	syl5ibcom 248	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑥} ⊆ (𝐾 IdlGen {𝑥}) → ((𝐾 IdlGen {𝑥}) = {𝑍} → 𝑥 ∈ {𝑍}))
30	29	con3dimp 414	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑥} ⊆ (𝐾 IdlGen {𝑥}) ∧ ¬ 𝑥 ∈ {𝑍}) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
31	24, 30	sylan 592	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ RingOps ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ {𝑍}) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
32	31	anasss 472	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ RingOps ∧ (𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ {𝑍})) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
33	21, 32	sylan2b 606	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
34	33	adantlr 728	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
35		eldifi 4067	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝑋 ∖ {𝑍}) → 𝑥 ∈ 𝑋)
36	35	snssd 4746	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝑋 ∖ {𝑍}) → {𝑥} ⊆ 𝑋)
37	3, 5	igenidl 39073	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 ∈ RingOps ∧ {𝑥} ⊆ 𝑋) → (𝐾 IdlGen {𝑥}) ∈ (Idl‘𝐾))
38	36, 37	sylan2 605	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) ∈ (Idl‘𝐾))
39		eleq2 2849	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Idl‘𝐾) = {{𝑍}, 𝑋} → ((𝐾 IdlGen {𝑥}) ∈ (Idl‘𝐾) ↔ (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋}))
40	38, 39	syl5ibcom 248	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ((Idl‘𝐾) = {{𝑍}, 𝑋} → (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋}))
41	40	imp 412	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋})
42	41	an32s 665	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋})
43		ovex 7453	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 IdlGen {𝑥}) ∈ V
44	43	elpr 4608	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋} ↔ ((𝐾 IdlGen {𝑥}) = {𝑍} ∨ (𝐾 IdlGen {𝑥}) = 𝑋))
45	42, 44	sylib 221	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ((𝐾 IdlGen {𝑥}) = {𝑍} ∨ (𝐾 IdlGen {𝑥}) = 𝑋))
46	45	ord 878	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (¬ (𝐾 IdlGen {𝑥}) = {𝑍} → (𝐾 IdlGen {𝑥}) = 𝑋))
47	34, 46	mpd 16	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) = 𝑋)
48	13, 47	sylanl1 693	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) = 𝑋)
49	3, 4, 5	prnc 39077	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑥 ∈ 𝑋) → (𝐾 IdlGen {𝑥}) = {𝑧 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑧 = (𝑦𝐻𝑥)})
50	35, 49	sylan2 605	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) = {𝑧 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑧 = (𝑦𝐻𝑥)})
51	50	adantlr 728	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) = {𝑧 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑧 = (𝑦𝐻𝑥)})
52	48, 51	eqtr3d 2797	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → 𝑋 = {𝑧 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑧 = (𝑦𝐻𝑥)})
53	20, 52	eleqtrd 2862	. . . . . . . . . 10 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → 𝑈 ∈ {𝑧 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑧 = (𝑦𝐻𝑥)})
54		eqeq1 2764	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑈 → (𝑧 = (𝑦𝐻𝑥) ↔ 𝑈 = (𝑦𝐻𝑥)))
55	54	rexbidv 3186	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑈 → (∃𝑦 ∈ 𝑋 𝑧 = (𝑦𝐻𝑥) ↔ ∃𝑦 ∈ 𝑋 𝑈 = (𝑦𝐻𝑥)))
56	55	elrab 3644	. . . . . . . . . 10 ⊢ (𝑈 ∈ {𝑧 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑧 = (𝑦𝐻𝑥)} ↔ (𝑈 ∈ 𝑋 ∧ ∃𝑦 ∈ 𝑋 𝑈 = (𝑦𝐻𝑥)))
57	53, 56	sylib 221	. . . . . . . . 9 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝑈 ∈ 𝑋 ∧ ∃𝑦 ∈ 𝑋 𝑈 = (𝑦𝐻𝑥)))
58	57	simprd 501	. . . . . . . 8 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ∃𝑦 ∈ 𝑋 𝑈 = (𝑦𝐻𝑥))
59		eqcom 2767	. . . . . . . . 9 ⊢ ((𝑦𝐻𝑥) = 𝑈 ↔ 𝑈 = (𝑦𝐻𝑥))
60	59	rexbii 3109	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈 ↔ ∃𝑦 ∈ 𝑋 𝑈 = (𝑦𝐻𝑥))
61	58, 60	sylibr 237	. . . . . . 7 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈)
62	61	ralrimiva 3154	. . . . . 6 ⊢ ((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈)
63	62	3adant2 1149	. . . . 5 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈)
64	14, 15, 63	jca32 525	. . . 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 38969	. . . 4 ⊢ (𝐾 ∈ DivRingOps ↔ (𝐾 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈)))
66	64, 65	sylibr 237	. . 3 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ DivRingOps)
67		simp1 1154	. . 3 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ CRingOps)
68		isfld2 39015	. . 3 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps))
69	66, 67, 68	sylanbrc 595	. 2 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ Fld)
70	12, 69	impbii 212	1 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 {crab 3412 ⊆ wss 3895 ∖ cdif 4045 {csn 4583 {cpr 4585 ran crn 5651 ‘cfv 6538 (class class class)co 7420 1^st c1st 7999 2^nd c2nd 8000 GIdcgi 31093 RingOpscrngo 38904 DivRingOpscdrng 38958 Fldcfld 39001 CRingOpsccring 39003 Idlcidl 39017 IdlGen cigen 39069

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 8001 df-2nd 8002 df-1o 8477 df-en 8975 df-grpo 31096 df-gid 31097 df-ginv 31098 df-ablo 31148 df-ass 38853 df-exid 38855 df-mgmOLD 38859 df-sgrOLD 38871 df-mndo 38877 df-rngo 38905 df-drngo 38959 df-com2 39000 df-fld 39002 df-crngo 39004 df-idl 39020 df-igen 39070

This theorem is used by: isfldidl2 39079

W3C validator