dflring4 - Mathbox for Thierry Arnoux

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Theorem dflring4 33797

Description: Alternate definition of a local ring: the set (𝐵 ∖ 𝑈) of non-units is an ideal. (Contributed by Thierry Arnoux, 2-Jun-2026.)

**Hypotheses**
Ref	Expression
dflring4.b	⊢ 𝐵 = (Base‘𝑅)
dflring4.u	⊢ 𝑈 = (Unit‘𝑅)

**Assertion**
Ref	Expression
dflring4	⊢ (𝑅 ∈ CRing → (𝑅 ∈ LRing ↔ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)))

Proof of Theorem dflring4 Dummy variables 𝑗 𝑥 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dflring4.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
2		dflring4.u	. . 3 ⊢ 𝑈 = (Unit‘𝑅)
3		simpl 487	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → 𝑅 ∈ CRing)
4		simpr 489	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → 𝑅 ∈ LRing)
5	1, 2, 3, 4	dflringlem2 33794	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅))
6		simpl 487	. . 3 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → 𝑅 ∈ CRing)
7	6	crngringd 20332	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → 𝑅 ∈ Ring)
8	7	adantr 485	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ Ring)
9		simpr 489	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (MaxIdeal‘𝑅))
10		simplr 780	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅))
11	1	mxidlidl 33755	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (LIdeal‘𝑅))
12	7, 11	sylan 591	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (LIdeal‘𝑅))
13		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
14	1, 13	lidlss 21345	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (LIdeal‘𝑅) → 𝑚 ⊆ 𝐵)
15	12, 14	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ⊆ 𝐵)
16	15	adantr 485	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → 𝑚 ⊆ 𝐵)
17	16	sselda 3937	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ 𝑚) → 𝑥 ∈ 𝐵)
18		neldif 4088	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝑈)) → 𝑥 ∈ 𝑈)
19	17, 18	sylan 591	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ 𝑚) ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝑈)) → 𝑥 ∈ 𝑈)
20		simplr 780	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ 𝑚) ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝑈)) → 𝑥 ∈ 𝑚)
21	8	ad3antrrr 742	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ 𝑚) ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝑈)) → 𝑅 ∈ Ring)
22	12	ad3antrrr 742	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ 𝑚) ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝑈)) → 𝑚 ∈ (LIdeal‘𝑅))
23	1, 2, 19, 20, 21, 22	lidlunitel 33740	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ 𝑚) ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝑈)) → 𝑚 = 𝐵)
24		nssrex 4002	. . . . . . . . . . 11 ⊢ (¬ 𝑚 ⊆ (𝐵 ∖ 𝑈) ↔ ∃𝑥 ∈ 𝑚 ¬ 𝑥 ∈ (𝐵 ∖ 𝑈))
25	24	bilani 509	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → ∃𝑥 ∈ 𝑚 ¬ 𝑥 ∈ (𝐵 ∖ 𝑈))
26	23, 25	r19.29a 3173	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → 𝑚 = 𝐵)
27	8	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → 𝑅 ∈ Ring)
28		simplr 780	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → 𝑚 ∈ (MaxIdeal‘𝑅))
29	1	mxidlnr 33756	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ≠ 𝐵)
30	27, 28, 29	syl2anc 595	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → 𝑚 ≠ 𝐵)
31	30	neneqd 2963	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → ¬ 𝑚 = 𝐵)
32	26, 31	condan 829	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ⊆ (𝐵 ∖ 𝑈))
33	1	mxidlmax 33757	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ((𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅) ∧ 𝑚 ⊆ (𝐵 ∖ 𝑈))) → ((𝐵 ∖ 𝑈) = 𝑚 ∨ (𝐵 ∖ 𝑈) = 𝐵))
34	8, 9, 10, 32, 33	syl22anc 851	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → ((𝐵 ∖ 𝑈) = 𝑚 ∨ (𝐵 ∖ 𝑈) = 𝐵))
35		eqid 2763	. . . . . . . . . . . 12 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
36	1, 35, 7	ringidcld 20354	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (1_r‘𝑅) ∈ 𝐵)
37	2, 35	1unit 20461	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ 𝑈)
38	7, 37	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (1_r‘𝑅) ∈ 𝑈)
39		elndif 4087	. . . . . . . . . . . 12 ⊢ ((1_r‘𝑅) ∈ 𝑈 → ¬ (1_r‘𝑅) ∈ (𝐵 ∖ 𝑈))
40	38, 39	syl 18	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → ¬ (1_r‘𝑅) ∈ (𝐵 ∖ 𝑈))
41		nelne1 3055	. . . . . . . . . . 11 ⊢ (((1_r‘𝑅) ∈ 𝐵 ∧ ¬ (1_r‘𝑅) ∈ (𝐵 ∖ 𝑈)) → 𝐵 ≠ (𝐵 ∖ 𝑈))
42	36, 40, 41	syl2anc 595	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → 𝐵 ≠ (𝐵 ∖ 𝑈))
43	42	necomd 3013	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (𝐵 ∖ 𝑈) ≠ 𝐵)
44	43	adantr 485	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → (𝐵 ∖ 𝑈) ≠ 𝐵)
45	44	neneqd 2963	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → ¬ (𝐵 ∖ 𝑈) = 𝐵)
46	34, 45	olcnd 890	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → (𝐵 ∖ 𝑈) = 𝑚)
47	46	eqcomd 2769	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 = (𝐵 ∖ 𝑈))
48		simpr 489	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅))
49	1, 13	lidlss 21345	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (LIdeal‘𝑅) → 𝑗 ⊆ 𝐵)
50	49	ad3antlr 743	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → 𝑗 ⊆ 𝐵)
51		ssdif0 4321	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ⊆ 𝐵 ↔ (𝑗 ∖ 𝐵) = ∅)
52	50, 51	sylib 221	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → (𝑗 ∖ 𝐵) = ∅)
53	52	uneq1d 4121	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → ((𝑗 ∖ 𝐵) ∪ (𝑗 ∩ 𝑈)) = (∅ ∪ (𝑗 ∩ 𝑈)))
54		0un 4353	. . . . . . . . . . . . 13 ⊢ (∅ ∪ (𝑗 ∩ 𝑈)) = (𝑗 ∩ 𝑈)
55	53, 54	eqtr2di 2815	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → (𝑗 ∩ 𝑈) = ((𝑗 ∖ 𝐵) ∪ (𝑗 ∩ 𝑈)))
56		simplr 780	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → (𝐵 ∖ 𝑈) ⊆ 𝑗)
57		neqne 2966	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑗 = (𝐵 ∖ 𝑈) → 𝑗 ≠ (𝐵 ∖ 𝑈))
58	57	adantl 486	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → 𝑗 ≠ (𝐵 ∖ 𝑈))
59	58	necomd 3013	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → (𝐵 ∖ 𝑈) ≠ 𝑗)
60		difdif2 4249	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∖ (𝐵 ∖ 𝑈)) = ((𝑗 ∖ 𝐵) ∪ (𝑗 ∩ 𝑈))
61		pssdifn0 4323	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∖ 𝑈) ⊆ 𝑗 ∧ (𝐵 ∖ 𝑈) ≠ 𝑗) → (𝑗 ∖ (𝐵 ∖ 𝑈)) ≠ ∅)
62	60, 61	eqnetrrid 3033	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∖ 𝑈) ⊆ 𝑗 ∧ (𝐵 ∖ 𝑈) ≠ 𝑗) → ((𝑗 ∖ 𝐵) ∪ (𝑗 ∩ 𝑈)) ≠ ∅)
63	56, 59, 62	syl2anc 595	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → ((𝑗 ∖ 𝐵) ∪ (𝑗 ∩ 𝑈)) ≠ ∅)
64	55, 63	eqnetrd 3025	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → (𝑗 ∩ 𝑈) ≠ ∅)
65		simpr 489	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ (𝑗 ∩ 𝑈)) → 𝑥 ∈ (𝑗 ∩ 𝑈))
66	65	elin2d 4158	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ (𝑗 ∩ 𝑈)) → 𝑥 ∈ 𝑈)
67	65	elin1d 4157	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ (𝑗 ∩ 𝑈)) → 𝑥 ∈ 𝑗)
68	7	ad4antr 744	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ (𝑗 ∩ 𝑈)) → 𝑅 ∈ Ring)
69		simp-4r 795	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ (𝑗 ∩ 𝑈)) → 𝑗 ∈ (LIdeal‘𝑅))
70	1, 2, 66, 67, 68, 69	lidlunitel 33740	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ (𝑗 ∩ 𝑈)) → 𝑗 = 𝐵)
71	64, 70	n0limd 4308	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → 𝑗 = 𝐵)
72	71	ex 417	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) → (¬ 𝑗 = (𝐵 ∖ 𝑈) → 𝑗 = 𝐵))
73	72	orrd 876	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) → (𝑗 = (𝐵 ∖ 𝑈) ∨ 𝑗 = 𝐵))
74	73	ex 417	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) → ((𝐵 ∖ 𝑈) ⊆ 𝑗 → (𝑗 = (𝐵 ∖ 𝑈) ∨ 𝑗 = 𝐵)))
75	74	ralrimiva 3157	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → ∀𝑗 ∈ (LIdeal‘𝑅)((𝐵 ∖ 𝑈) ⊆ 𝑗 → (𝑗 = (𝐵 ∖ 𝑈) ∨ 𝑗 = 𝐵)))
76	1	ismxidl 33754	. . . . . . 7 ⊢ (𝑅 ∈ Ring → ((𝐵 ∖ 𝑈) ∈ (MaxIdeal‘𝑅) ↔ ((𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅) ∧ (𝐵 ∖ 𝑈) ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)((𝐵 ∖ 𝑈) ⊆ 𝑗 → (𝑗 = (𝐵 ∖ 𝑈) ∨ 𝑗 = 𝐵)))))
77	76	biimpar 482	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ ((𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅) ∧ (𝐵 ∖ 𝑈) ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)((𝐵 ∖ 𝑈) ⊆ 𝑗 → (𝑗 = (𝐵 ∖ 𝑈) ∨ 𝑗 = 𝐵)))) → (𝐵 ∖ 𝑈) ∈ (MaxIdeal‘𝑅))
78	7, 48, 43, 75, 77	syl13anc 1399	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (𝐵 ∖ 𝑈) ∈ (MaxIdeal‘𝑅))
79	47, 78	eqsnd 4796	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (MaxIdeal‘𝑅) = {(𝐵 ∖ 𝑈)})
80	1	fvexi 6895	. . . . . . 7 ⊢ 𝐵 ∈ V
81	80	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → 𝐵 ∈ V)
82	81	difexd 5302	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (𝐵 ∖ 𝑈) ∈ V)
83		ensn1g 9015	. . . . 5 ⊢ ((𝐵 ∖ 𝑈) ∈ V → {(𝐵 ∖ 𝑈)} ≈ 1_o)
84	82, 83	syl 18	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → {(𝐵 ∖ 𝑈)} ≈ 1_o)
85	79, 84	eqbrtrd 5133	. . 3 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (MaxIdeal‘𝑅) ≈ 1_o)
86		dflring3 33796	. . . 4 ⊢ (𝑅 ∈ CRing → (𝑅 ∈ LRing ↔ (MaxIdeal‘𝑅) ≈ 1_o))
87	86	biimpar 482	. . 3 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) ≈ 1_o) → 𝑅 ∈ LRing)
88	6, 85, 87	syl2anc 595	. 2 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → 𝑅 ∈ LRing)
89	5, 88	impbida 812	1 ⊢ (𝑅 ∈ CRing → (𝑅 ∈ LRing ↔ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 Vcvv 3455 ∖ cdif 3902 ∪ cun 3903 ∩ cin 3904 ⊆ wss 3905 ∅c0 4286 {csn 4589 class class class wbr 5109 ‘cfv 6536 1_oc1o 8442 ≈ cen 8936 Basecbs 17273 1_rcur 20267 Ringcrg 20319 CRingccrg 20320 Unitcui 20442 LRingclring 20646 LIdealclidl 21339 MaxIdealcmxidl 33751

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-ac2 10451 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-rpss 7720 df-om 7859 df-1st 7982 df-2nd 7983 df-tpos 8218 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-oadd 8453 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-dju 9892 df-card 9930 df-ac 10105 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-nn 12238 df-2 12307 df-3 12308 df-4 12309 df-5 12310 df-6 12311 df-7 12312 df-8 12313 df-sets 17228 df-slot 17246 df-ndx 17258 df-base 17274 df-ress 17295 df-plusg 17327 df-mulr 17328 df-sca 17330 df-vsca 17331 df-ip 17332 df-0g 17498 df-mgm 18702 df-sgrp 18781 df-mnd 18797 df-grp 19007 df-minusg 19008 df-sbg 19009 df-subg 19193 df-cmn 19856 df-abl 19857 df-mgp 20221 df-rng 20235 df-ur 20268 df-ring 20321 df-cring 20322 df-oppr 20424 df-dvdsr 20444 df-unit 20445 df-invr 20475 df-dvr 20488 df-nzr 20619 df-lring 20647 df-subrg 20678 df-lmod 20992 df-lss 21062 df-lsp 21102 df-sra 21303 df-rgmod 21304 df-lidl 21341 df-rsp 21342 df-mxidl 33752

This theorem is used by: (None)

W3C validator