dflring4 - Mathbox for Thierry Arnoux

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Theorem dflring4 34086

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 488	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → 𝑅 ∈ CRing)
4		simpr 490	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → 𝑅 ∈ LRing)
5	1, 2, 3, 4	dflringlem2 34083	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅))
6		simpl 488	. . 3 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → 𝑅 ∈ CRing)
7	6	crngringd 20488	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → 𝑅 ∈ Ring)
8	7	adantr 486	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ Ring)
9		simpr 490	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (MaxIdeal‘𝑅))
10		simplr 781	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅))
11	1	mxidlidl 34044	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (LIdeal‘𝑅))
12	7, 11	sylan 592	. . . . . . . . . . . . . . 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 21505	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → 𝑚 ⊆ 𝐵)
17	16	sselda 3913	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ 𝑚) → 𝑥 ∈ 𝐵)
18		neldif 4073	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝑈)) → 𝑥 ∈ 𝑈)
19	17, 18	sylan 592	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ 𝑚) ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝑈)) → 𝑥 ∈ 𝑈)
20		simplr 781	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ 𝑚) ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝑈)) → 𝑥 ∈ 𝑚)
21	8	ad3antrrr 743	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ 𝑚) ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝑈)) → 𝑅 ∈ Ring)
22	12	ad3antrrr 743	. . . . . . . . . . 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 34029	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ 𝑚) ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝑈)) → 𝑚 = 𝐵)
24		nssrex 3978	. . . . . . . . . . 11 ⊢ (¬ 𝑚 ⊆ (𝐵 ∖ 𝑈) ↔ ∃𝑥 ∈ 𝑚 ¬ 𝑥 ∈ (𝐵 ∖ 𝑈))
25	24	bilani 510	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → ∃𝑥 ∈ 𝑚 ¬ 𝑥 ∈ (𝐵 ∖ 𝑈))
26	23, 25	r19.29a 3174	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → 𝑚 = 𝐵)
27	8	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → 𝑅 ∈ Ring)
28		simplr 781	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → 𝑚 ∈ (MaxIdeal‘𝑅))
29	1	mxidlnr 34045	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ≠ 𝐵)
30	27, 28, 29	syl2anc 596	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → 𝑚 ≠ 𝐵)
31	30	neneqd 2964	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → ¬ 𝑚 = 𝐵)
32	26, 31	condan 830	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ⊆ (𝐵 ∖ 𝑈))
33	1	mxidlmax 34046	. . . . . . . 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 852	. . . . . . 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 20510	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (1_r‘𝑅) ∈ 𝐵)
37	2, 35	1unit 20619	. . . . . . . . . . . . 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 4072	. . . . . . . . . . . 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 3056	. . . . . . . . . . 11 ⊢ (((1_r‘𝑅) ∈ 𝐵 ∧ ¬ (1_r‘𝑅) ∈ (𝐵 ∖ 𝑈)) → 𝐵 ≠ (𝐵 ∖ 𝑈))
42	36, 40, 41	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → 𝐵 ≠ (𝐵 ∖ 𝑈))
43	42	necomd 3014	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (𝐵 ∖ 𝑈) ≠ 𝐵)
44	43	adantr 486	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → (𝐵 ∖ 𝑈) ≠ 𝐵)
45	44	neneqd 2964	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → ¬ (𝐵 ∖ 𝑈) = 𝐵)
46	34, 45	olcnd 891	. . . . . 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 490	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅))
49	1, 13	lidlss 21505	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (LIdeal‘𝑅) → 𝑗 ⊆ 𝐵)
50	49	ad3antlr 744	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → 𝑗 ⊆ 𝐵)
51		ssdif0 4317	. . . . . . . . . . . . . . 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 4169	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → ((𝑗 ∖ 𝐵) ∪ (𝑗 ∩ 𝑈)) = (∅ ∪ (𝑗 ∩ 𝑈)))
54		0un 4349	. . . . . . . . . . . . 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 781	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → (𝐵 ∖ 𝑈) ⊆ 𝑗)
57		neqne 2967	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑗 = (𝐵 ∖ 𝑈) → 𝑗 ≠ (𝐵 ∖ 𝑈))
58	57	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → 𝑗 ≠ (𝐵 ∖ 𝑈))
59	58	necomd 3014	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → (𝐵 ∖ 𝑈) ≠ 𝑗)
60		difdif2 4244	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∖ (𝐵 ∖ 𝑈)) = ((𝑗 ∖ 𝐵) ∪ (𝑗 ∩ 𝑈))
61		pssdifn0 4319	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∖ 𝑈) ⊆ 𝑗 ∧ (𝐵 ∖ 𝑈) ≠ 𝑗) → (𝑗 ∖ (𝐵 ∖ 𝑈)) ≠ ∅)
62	60, 61	eqnetrrid 3034	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∖ 𝑈) ⊆ 𝑗 ∧ (𝐵 ∖ 𝑈) ≠ 𝑗) → ((𝑗 ∖ 𝐵) ∪ (𝑗 ∩ 𝑈)) ≠ ∅)
63	56, 59, 62	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → ((𝑗 ∖ 𝐵) ∪ (𝑗 ∩ 𝑈)) ≠ ∅)
64	55, 63	eqnetrd 3026	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → (𝑗 ∩ 𝑈) ≠ ∅)
65		simpr 490	. . . . . . . . . . . . 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 4105	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ (𝑗 ∩ 𝑈)) → 𝑥 ∈ 𝑈)
67	65	elin1d 4104	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ (𝑗 ∩ 𝑈)) → 𝑥 ∈ 𝑗)
68	7	ad4antr 745	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ (𝑗 ∩ 𝑈)) → 𝑅 ∈ Ring)
69		simp-4r 796	. . . . . . . . . . . 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 34029	. . . . . . . . . . 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 4304	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → 𝑗 = 𝐵)
72	71	ex 418	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) → (¬ 𝑗 = (𝐵 ∖ 𝑈) → 𝑗 = 𝐵))
73	72	orrd 877	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) → (𝑗 = (𝐵 ∖ 𝑈) ∨ 𝑗 = 𝐵))
74	73	ex 418	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) → ((𝐵 ∖ 𝑈) ⊆ 𝑗 → (𝑗 = (𝐵 ∖ 𝑈) ∨ 𝑗 = 𝐵)))
75	74	ralrimiva 3158	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → ∀𝑗 ∈ (LIdeal‘𝑅)((𝐵 ∖ 𝑈) ⊆ 𝑗 → (𝑗 = (𝐵 ∖ 𝑈) ∨ 𝑗 = 𝐵)))
76	1	ismxidl 34043	. . . . . . 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 483	. . . . . 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 4789	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (MaxIdeal‘𝑅) = {(𝐵 ∖ 𝑈)})
80	1	fvexi 6910	. . . . . . 7 ⊢ 𝐵 ∈ V
81	80	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → 𝐵 ∈ V)
82	81	difexd 5297	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (𝐵 ∖ 𝑈) ∈ V)
83		ensn1g 9063	. . . . 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 5130	. . 3 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (MaxIdeal‘𝑅) ≈ 1_o)
86		dflring3 34085	. . . 4 ⊢ (𝑅 ∈ CRing → (𝑅 ∈ LRing ↔ (MaxIdeal‘𝑅) ≈ 1_o))
87	86	biimpar 483	. . 3 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) ≈ 1_o) → 𝑅 ∈ LRing)
88	6, 85, 87	syl2anc 596	. 2 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → 𝑅 ∈ LRing)
89	5, 88	impbida 813	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 401 ∨ wo 861 ∧ w3a 1103 = wceq 1569 ∈ wcel 2146 ≠ wne 2959 ∀wral 3080 ∃wrex 3090 Vcvv 3454 ⊆ wss 3900 ∖ cdif 4048 ∩ cin 4092 ∪ cun 4152 ∅c0 4281 {csn 4588 class class class wbr 5106 ‘cfv 6547 1_oc1o 8483 ≈ cen 8984 Basecbs 17402 1_rcur 20422 Ringcrg 20474 CRingccrg 20475 Unitcui 20600 LRingclring 20805 LIdealclidl 21499 MaxIdealcmxidl 34040

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 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7761 ax-ac2 10554 ax-cnex 11269 ax-resscn 11270 ax-1cn 11271 ax-icn 11272 ax-addcl 11273 ax-addrcl 11274 ax-mulcl 11275 ax-mulrcl 11276 ax-mulcom 11277 ax-addass 11278 ax-mulass 11279 ax-distr 11280 ax-i2m1 11281 ax-1ne0 11282 ax-1rid 11283 ax-rnegex 11284 ax-rrecex 11285 ax-cnre 11286 ax-pre-lttri 11287 ax-pre-lttrn 11288 ax-pre-ltadd 11289 ax-pre-mulgt0 11290

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6311 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-isom 6556 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-rpss 7749 df-om 7888 df-1st 8011 df-2nd 8012 df-tpos 8257 df-frecs 8313 df-wrecs 8344 df-recs 8393 df-rdg 8432 df-1o 8490 df-oadd 8494 df-er 8731 df-en 8988 df-dom 8989 df-sdom 8990 df-fin 8991 df-dju 9995 df-card 10033 df-ac 10208 df-pnf 11358 df-mnf 11359 df-xr 11360 df-ltxr 11361 df-le 11362 df-sub 11556 df-neg 11557 df-nn 12349 df-2 12418 df-3 12419 df-4 12420 df-5 12421 df-6 12422 df-7 12423 df-8 12424 df-sets 17357 df-slot 17375 df-ndx 17387 df-base 17403 df-ress 17424 df-plusg 17456 df-mulr 17457 df-sca 17459 df-vsca 17460 df-ip 17461 df-0g 17627 df-mgm 18831 df-sgrp 18923 df-mnd 18939 df-grp 19162 df-minusg 19163 df-sbg 19164 df-subg 19348 df-cmn 20011 df-abl 20012 df-mgp 20376 df-rng 20390 df-ur 20423 df-ring 20476 df-cring 20477 df-oppr 20582 df-dvdsr 20602 df-unit 20603 df-invr 20633 df-dvr 20646 df-nzr 20778 df-lring 20806 df-subrg 20837 df-lmod 21152 df-lss 21222 df-lsp 21262 df-sra 21463 df-rgmod 21464 df-lidl 21501 df-rsp 21502 df-mxidl 34041

This theorem is used by: (None)

W3C validator