dflring4 - Mathbox for Thierry Arnoux

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Theorem dflring4 33911

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 33908	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅))
6		simpl 488	. . 3 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → 𝑅 ∈ CRing)
7	6	crngringd 20388	. . . . . . . . 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 33869	. . . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . 16 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
14	1, 13	lidlss 21402	. . . . . . . . . . . . . . 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 3931	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ 𝑚) → 𝑥 ∈ 𝐵)
18		neldif 4081	. . . . . . . . . . . 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 33854	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ 𝑚) ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝑈)) → 𝑚 = 𝐵)
24		nssrex 3996	. . . . . . . . . . 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 3170	. . . . . . . . 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 33870	. . . . . . . . . . 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 2960	. . . . . . . . 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 33871	. . . . . . . 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 2760	. . . . . . . . . . . 12 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
36	1, 35, 7	ringidcld 20410	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (1_r‘𝑅) ∈ 𝐵)
37	2, 35	1unit 20518	. . . . . . . . . . . . 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 4080	. . . . . . . . . . . 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 3052	. . . . . . . . . . 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 3010	. . . . . . . . 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 2960	. . . . . . 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 2766	. . . . 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 21402	. . . . . . . . . . . . . . . 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 4314	. . . . . . . . . . . . . . 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 4114	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → ((𝑗 ∖ 𝐵) ∪ (𝑗 ∩ 𝑈)) = (∅ ∪ (𝑗 ∩ 𝑈)))
54		0un 4346	. . . . . . . . . . . . 13 ⊢ (∅ ∪ (𝑗 ∩ 𝑈)) = (𝑗 ∩ 𝑈)
55	53, 54	eqtr2di 2812	. . . . . . . . . . . 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 2963	. . . . . . . . . . . . . . 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 3010	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → (𝐵 ∖ 𝑈) ≠ 𝑗)
60		difdif2 4242	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∖ (𝐵 ∖ 𝑈)) = ((𝑗 ∖ 𝐵) ∪ (𝑗 ∩ 𝑈))
61		pssdifn0 4316	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∖ 𝑈) ⊆ 𝑗 ∧ (𝐵 ∖ 𝑈) ≠ 𝑗) → (𝑗 ∖ (𝐵 ∖ 𝑈)) ≠ ∅)
62	60, 61	eqnetrrid 3030	. . . . . . . . . . . . 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 3022	. . . . . . . . . . 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 4151	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ (𝑗 ∩ 𝑈)) → 𝑥 ∈ 𝑈)
67	65	elin1d 4150	. . . . . . . . . . . 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 33854	. . . . . . . . . . 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 4301	. . . . . . . . . 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 3154	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → ∀𝑗 ∈ (LIdeal‘𝑅)((𝐵 ∖ 𝑈) ⊆ 𝑗 → (𝑗 = (𝐵 ∖ 𝑈) ∨ 𝑗 = 𝐵)))
76	1	ismxidl 33868	. . . . . . 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 4791	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (MaxIdeal‘𝑅) = {(𝐵 ∖ 𝑈)})
80	1	fvexi 6893	. . . . . . 7 ⊢ 𝐵 ∈ V
81	80	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → 𝐵 ∈ V)
82	81	difexd 5296	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (𝐵 ∖ 𝑈) ∈ V)
83		ensn1g 9031	. . . . 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 5127	. . 3 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (MaxIdeal‘𝑅) ≈ 1_o)
86		dflring3 33910	. . . 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 1570 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ∖ cdif 3896 ∪ cun 3897 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 {csn 4584 class class class wbr 5103 ‘cfv 6533 1_oc1o 8451 ≈ cen 8952 Basecbs 17304 1_rcur 20323 Ringcrg 20375 CRingccrg 20376 Unitcui 20499 LRingclring 20703 LIdealclidl 21396 MaxIdealcmxidl 33865

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-ac2 10468 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 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 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-rpss 7725 df-om 7864 df-1st 7987 df-2nd 7988 df-tpos 8225 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8458 df-oadd 8462 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-dju 9909 df-card 9947 df-ac 10122 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-sets 17259 df-slot 17277 df-ndx 17289 df-base 17305 df-ress 17326 df-plusg 17358 df-mulr 17359 df-sca 17361 df-vsca 17362 df-ip 17363 df-0g 17529 df-mgm 18733 df-sgrp 18824 df-mnd 18840 df-grp 19063 df-minusg 19064 df-sbg 19065 df-subg 19249 df-cmn 19912 df-abl 19913 df-mgp 20277 df-rng 20291 df-ur 20324 df-ring 20377 df-cring 20378 df-oppr 20481 df-dvdsr 20501 df-unit 20502 df-invr 20532 df-dvr 20545 df-nzr 20676 df-lring 20704 df-subrg 20735 df-lmod 21049 df-lss 21119 df-lsp 21159 df-sra 21360 df-rgmod 21361 df-lidl 21398 df-rsp 21399 df-mxidl 33866

This theorem is used by: (None)

W3C validator