dflring4 - Mathbox for Thierry Arnoux

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Theorem dflring4 33695

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 486	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → 𝑅 ∈ CRing)
4		simpr 488	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → 𝑅 ∈ LRing)
5	1, 2, 3, 4	dflringlem2 33692	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅))
6		simpl 486	. . 3 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → 𝑅 ∈ CRing)
7	6	crngringd 20297	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → 𝑅 ∈ Ring)
8	7	adantr 484	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ Ring)
9		simpr 488	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (MaxIdeal‘𝑅))
10		simplr 778	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅))
11	1	mxidlidl 33652	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (LIdeal‘𝑅))
12	7, 11	sylan 589	. . . . . . . . . . . . . . 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 21283	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (LIdeal‘𝑅) → 𝑚 ⊆ 𝐵)
15	12, 14	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ⊆ 𝐵)
16	15	adantr 484	. . . . . . . . . . . . 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 589	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ 𝑚) ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝑈)) → 𝑥 ∈ 𝑈)
20		simplr 778	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ 𝑚) ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝑈)) → 𝑥 ∈ 𝑚)
21	8	ad3antrrr 740	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ 𝑚) ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝑈)) → 𝑅 ∈ Ring)
22	12	ad3antrrr 740	. . . . . . . . . . 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 33610	. . . . . . . . . 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 508	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → ∃𝑥 ∈ 𝑚 ¬ 𝑥 ∈ (𝐵 ∖ 𝑈))
26	23, 25	r19.29a 3171	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → 𝑚 = 𝐵)
27	8	adantr 484	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → 𝑅 ∈ Ring)
28		simplr 778	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → 𝑚 ∈ (MaxIdeal‘𝑅))
29	1	mxidlnr 33653	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ≠ 𝐵)
30	27, 28, 29	syl2anc 593	. . . . . . . . . 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 827	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ⊆ (𝐵 ∖ 𝑈))
33	1	mxidlmax 33654	. . . . . . . 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 849	. . . . . . 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 20317	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (1_r‘𝑅) ∈ 𝐵)
37	2, 35	1unit 20424	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ 𝑈)
38	7, 37	syl 17	. . . . . . . . . . . 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 17	. . . . . . . . . . 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 593	. . . . . . . . . 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 484	. . . . . . . 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 888	. . . . . 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 488	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅))
49	1, 13	lidlss 21283	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (LIdeal‘𝑅) → 𝑗 ⊆ 𝐵)
50	49	ad3antlr 741	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → 𝑗 ⊆ 𝐵)
51		ssdif0 4320	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ⊆ 𝐵 ↔ (𝑗 ∖ 𝐵) = ∅)
52	50, 51	sylib 220	. . . . . . . . . . . . . 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 4351	. . . . . . . . . . . . 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 778	. . . . . . . . . . . . 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 485	. . . . . . . . . . . . . 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 4322	. . . . . . . . . . . . . 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 593	. . . . . . . . . . . 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 488	. . . . . . . . . . . . 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 742	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ (𝑗 ∩ 𝑈)) → 𝑅 ∈ Ring)
69		simp-4r 793	. . . . . . . . . . . 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 33610	. . . . . . . . . . 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 4307	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → 𝑗 = 𝐵)
72	71	ex 416	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) → (¬ 𝑗 = (𝐵 ∖ 𝑈) → 𝑗 = 𝐵))
73	72	orrd 874	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) → (𝑗 = (𝐵 ∖ 𝑈) ∨ 𝑗 = 𝐵))
74	73	ex 416	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) → ((𝐵 ∖ 𝑈) ⊆ 𝑗 → (𝑗 = (𝐵 ∖ 𝑈) ∨ 𝑗 = 𝐵)))
75	74	ralrimiva 3155	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → ∀𝑗 ∈ (LIdeal‘𝑅)((𝐵 ∖ 𝑈) ⊆ 𝑗 → (𝑗 = (𝐵 ∖ 𝑈) ∨ 𝑗 = 𝐵)))
76	1	ismxidl 33651	. . . . . . 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 481	. . . . . 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 1392	. . . . 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 6882	. . . . . . 7 ⊢ 𝐵 ∈ V
81	80	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → 𝐵 ∈ V)
82	81	difexd 5288	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (𝐵 ∖ 𝑈) ∈ V)
83		ensn1g 9004	. . . . 5 ⊢ ((𝐵 ∖ 𝑈) ∈ V → {(𝐵 ∖ 𝑈)} ≈ 1_o)
84	82, 83	syl 17	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → {(𝐵 ∖ 𝑈)} ≈ 1_o)
85	79, 84	eqbrtrd 5123	. . 3 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (MaxIdeal‘𝑅) ≈ 1_o)
86		dflring3 33694	. . . 4 ⊢ (𝑅 ∈ CRing → (𝑅 ∈ LRing ↔ (MaxIdeal‘𝑅) ≈ 1_o))
87	86	biimpar 481	. . 3 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) ≈ 1_o) → 𝑅 ∈ LRing)
88	6, 85, 87	syl2anc 593	. 2 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → 𝑅 ∈ LRing)
89	5, 88	impbida 810	1 ⊢ (𝑅 ∈ CRing → (𝑅 ∈ LRing ↔ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∨ wo 858 ∧ w3a 1099 = wceq 1561 ∈ wcel 2143 ≠ wne 2958 ∀wral 3077 ∃wrex 3087 Vcvv 3455 ∖ cdif 3902 ∪ cun 3903 ∩ cin 3904 ⊆ wss 3905 ∅c0 4286 {csn 4583 class class class wbr 5101 ‘cfv 6522 1_oc1o 8431 ≈ cen 8925 Basecbs 17246 1_rcur 20232 Ringcrg 20284 CRingccrg 20285 Unitcui 20405 LRingclring 20589 LIdealclidl 21277 MaxIdealcmxidl 33648

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7719 ax-ac2 10421 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 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 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-se 5602 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6289 df-ord 6350 df-on 6351 df-lim 6352 df-suc 6353 df-iota 6478 df-fun 6524 df-fn 6525 df-f 6526 df-f1 6527 df-fo 6528 df-f1o 6529 df-fv 6530 df-isom 6531 df-riota 7354 df-ov 7400 df-oprab 7401 df-mpo 7402 df-rpss 7707 df-om 7848 df-1st 7971 df-2nd 7972 df-tpos 8207 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8382 df-1o 8438 df-oadd 8442 df-er 8679 df-en 8929 df-dom 8930 df-sdom 8931 df-fin 8932 df-dju 9860 df-card 9898 df-ac 10073 df-pnf 11219 df-mnf 11220 df-xr 11221 df-ltxr 11222 df-le 11223 df-sub 11417 df-neg 11418 df-nn 12212 df-2 12281 df-3 12282 df-4 12283 df-5 12284 df-6 12285 df-7 12286 df-8 12287 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17247 df-ress 17268 df-plusg 17300 df-mulr 17301 df-sca 17303 df-vsca 17304 df-ip 17305 df-0g 17471 df-mgm 18675 df-sgrp 18754 df-mnd 18770 df-grp 18979 df-minusg 18980 df-sbg 18981 df-subg 19166 df-cmn 19823 df-abl 19824 df-mgp 20188 df-rng 20200 df-ur 20233 df-ring 20286 df-cring 20287 df-oppr 20387 df-dvdsr 20407 df-unit 20408 df-invr 20438 df-dvr 20451 df-nzr 20564 df-lring 20590 df-subrg 20621 df-lmod 20930 df-lss 21000 df-lsp 21040 df-sra 21241 df-rgmod 21242 df-lidl 21279 df-rsp 21280 df-mxidl 33649

This theorem is referenced by: (None)

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