isfldidl - Mathbox for Jeff Madsen

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Theorem isfldidl 33867

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 33803	. . 3 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ CRingOps)
2		flddivrng 33798	. . . 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 33753	. . . 4 ⊢ (𝐾 ∈ DivRingOps → 𝑈 ≠ 𝑍)
9	2, 8	syl 17	. . 3 ⊢ (𝐾 ∈ Fld → 𝑈 ≠ 𝑍)
10	3, 4, 5, 6	divrngidl 33827	. . . 4 ⊢ (𝐾 ∈ DivRingOps → (Idl‘𝐾) = {{𝑍}, 𝑋})
11	2, 10	syl 17	. . 3 ⊢ (𝐾 ∈ Fld → (Idl‘𝐾) = {{𝑍}, 𝑋})
12	1, 9, 11	3jca 1242	. 2 ⊢ (𝐾 ∈ Fld → (𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))
13		crngorngo 33799	. . . . . 6 ⊢ (𝐾 ∈ CRingOps → 𝐾 ∈ RingOps)
14	13	3ad2ant1 1082	. . . . 5 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ RingOps)
15		simp2 1062	. . . . 5 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝑈 ≠ 𝑍)
16	3	rneqi 5352	. . . . . . . . . . . . . . 15 ⊢ ran 𝐺 = ran (1^st ‘𝐾)
17	5, 16	eqtri 2644	. . . . . . . . . . . . . 14 ⊢ 𝑋 = ran (1^st ‘𝐾)
18	17, 4, 7	rngo1cl 33738	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ RingOps → 𝑈 ∈ 𝑋)
19	13, 18	syl 17	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ CRingOps → 𝑈 ∈ 𝑋)
20	19	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → 𝑈 ∈ 𝑋)
21		eldif 3584	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝑋 ∖ {𝑍}) ↔ (𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ {𝑍}))
22		snssi 4339	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝑋 → {𝑥} ⊆ 𝑋)
23	3, 5	igenss 33861	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ RingOps ∧ {𝑥} ⊆ 𝑋) → {𝑥} ⊆ (𝐾 IdlGen {𝑥}))
24	22, 23	sylan2 491	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → {𝑥} ⊆ (𝐾 IdlGen {𝑥}))
25		vex 3203	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑥 ∈ V
26	25	snss 4316	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝐾 IdlGen {𝑥}) ↔ {𝑥} ⊆ (𝐾 IdlGen {𝑥}))
27	26	biimpri 218	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({𝑥} ⊆ (𝐾 IdlGen {𝑥}) → 𝑥 ∈ (𝐾 IdlGen {𝑥}))
28		eleq2 2690	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 IdlGen {𝑥}) = {𝑍} → (𝑥 ∈ (𝐾 IdlGen {𝑥}) ↔ 𝑥 ∈ {𝑍}))
29	27, 28	syl5ibcom 235	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑥} ⊆ (𝐾 IdlGen {𝑥}) → ((𝐾 IdlGen {𝑥}) = {𝑍} → 𝑥 ∈ {𝑍}))
30	29	con3dimp 457	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑥} ⊆ (𝐾 IdlGen {𝑥}) ∧ ¬ 𝑥 ∈ {𝑍}) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
31	24, 30	sylan 488	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ RingOps ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ {𝑍}) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
32	31	anasss 679	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ RingOps ∧ (𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ {𝑍})) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
33	21, 32	sylan2b 492	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
34	33	adantlr 751	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ¬ (𝐾 IdlGen {𝑥}) = {𝑍})
35		eldifi 3732	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (𝑋 ∖ {𝑍}) → 𝑥 ∈ 𝑋)
36	35	snssd 4340	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (𝑋 ∖ {𝑍}) → {𝑥} ⊆ 𝑋)
37	3, 5	igenidl 33862	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 ∈ RingOps ∧ {𝑥} ⊆ 𝑋) → (𝐾 IdlGen {𝑥}) ∈ (Idl‘𝐾))
38	36, 37	sylan2 491	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) ∈ (Idl‘𝐾))
39		eleq2 2690	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Idl‘𝐾) = {{𝑍}, 𝑋} → ((𝐾 IdlGen {𝑥}) ∈ (Idl‘𝐾) ↔ (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋}))
40	38, 39	syl5ibcom 235	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ((Idl‘𝐾) = {{𝑍}, 𝑋} → (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋}))
41	40	imp 445	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ RingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋})
42	41	an32s 846	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋})
43		ovex 6678	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 IdlGen {𝑥}) ∈ V
44	43	elpr 4198	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 IdlGen {𝑥}) ∈ {{𝑍}, 𝑋} ↔ ((𝐾 IdlGen {𝑥}) = {𝑍} ∨ (𝐾 IdlGen {𝑥}) = 𝑋))
45	42, 44	sylib 208	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ RingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ((𝐾 IdlGen {𝑥}) = {𝑍} ∨ (𝐾 IdlGen {𝑥}) = 𝑋))
46	45	ord 392	. . . . . . . . . . . . . 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 682	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) = 𝑋)
49	3, 4, 5	prnc 33866	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑥 ∈ 𝑋) → (𝐾 IdlGen {𝑥}) = {𝑧 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑧 = (𝑦𝐻𝑥)})
50	35, 49	sylan2 491	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) = {𝑧 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑧 = (𝑦𝐻𝑥)})
51	50	adantlr 751	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝐾 IdlGen {𝑥}) = {𝑧 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑧 = (𝑦𝐻𝑥)})
52	48, 51	eqtr3d 2658	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → 𝑋 = {𝑧 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑧 = (𝑦𝐻𝑥)})
53	20, 52	eleqtrd 2703	. . . . . . . . . 10 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → 𝑈 ∈ {𝑧 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑧 = (𝑦𝐻𝑥)})
54		eqeq1 2626	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑈 → (𝑧 = (𝑦𝐻𝑥) ↔ 𝑈 = (𝑦𝐻𝑥)))
55	54	rexbidv 3052	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑈 → (∃𝑦 ∈ 𝑋 𝑧 = (𝑦𝐻𝑥) ↔ ∃𝑦 ∈ 𝑋 𝑈 = (𝑦𝐻𝑥)))
56	55	elrab 3363	. . . . . . . . . 10 ⊢ (𝑈 ∈ {𝑧 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑧 = (𝑦𝐻𝑥)} ↔ (𝑈 ∈ 𝑋 ∧ ∃𝑦 ∈ 𝑋 𝑈 = (𝑦𝐻𝑥)))
57	53, 56	sylib 208	. . . . . . . . 9 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → (𝑈 ∈ 𝑋 ∧ ∃𝑦 ∈ 𝑋 𝑈 = (𝑦𝐻𝑥)))
58	57	simprd 479	. . . . . . . 8 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ∃𝑦 ∈ 𝑋 𝑈 = (𝑦𝐻𝑥))
59		eqcom 2629	. . . . . . . . 9 ⊢ ((𝑦𝐻𝑥) = 𝑈 ↔ 𝑈 = (𝑦𝐻𝑥))
60	59	rexbii 3041	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈 ↔ ∃𝑦 ∈ 𝑋 𝑈 = (𝑦𝐻𝑥))
61	58, 60	sylibr 224	. . . . . . 7 ⊢ (((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) ∧ 𝑥 ∈ (𝑋 ∖ {𝑍})) → ∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈)
62	61	ralrimiva 2966	. . . . . 6 ⊢ ((𝐾 ∈ CRingOps ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈)
63	62	3adant2 1080	. . . . 5 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈)
64	14, 15, 63	jca32 558	. . . 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 33758	. . . 4 ⊢ (𝐾 ∈ DivRingOps ↔ (𝐾 ∈ RingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ 𝑋 (𝑦𝐻𝑥) = 𝑈)))
66	64, 65	sylibr 224	. . 3 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ DivRingOps)
67		simp1 1061	. . 3 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ CRingOps)
68		isfld2 33804	. . 3 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps))
69	66, 67, 68	sylanbrc 698	. 2 ⊢ ((𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}) → 𝐾 ∈ Fld)
70	12, 69	impbii 199	1 ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 {crab 2916 ∖ cdif 3571 ⊆ wss 3574 {csn 4177 {cpr 4179 ran crn 5115 ‘cfv 5888 (class class class)co 6650 1^st c1st 7166 2^nd c2nd 7167 GIdcgi 27344 RingOpscrngo 33693 DivRingOpscdrng 33747 Fldcfld 33790 CRingOpsccring 33792 Idlcidl 33806 IdlGen cigen 33858

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-grpo 27347 df-gid 27348 df-ginv 27349 df-ablo 27399 df-ass 33642 df-exid 33644 df-mgmOLD 33648 df-sgrOLD 33660 df-mndo 33666 df-rngo 33694 df-drngo 33748 df-com2 33789 df-fld 33791 df-crngo 33793 df-idl 33809 df-igen 33859

This theorem is referenced by: isfldidl2 33868

W3C validator