dflring4 - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dflring4		Structured version Visualization version GIF version

Theorem dflring4 33590

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 483	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → 𝑅 ∈ CRing)
4		simpr 485	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → 𝑅 ∈ LRing)
5	1, 2, 3, 4	dflringlem2 33587	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅))
6		simpl 483	. . 3 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → 𝑅 ∈ CRing)
7	6	crngringd 20219	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → 𝑅 ∈ Ring)
8	7	adantr 481	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ Ring)
9		simpr 485	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (MaxIdeal‘𝑅))
10		simplr 774	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅))
11	1	mxidlidl 33547	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (LIdeal‘𝑅))
12	7, 11	sylan 586	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (LIdeal‘𝑅))
13		eqid 2739	. . . . . . . . . . . . . . . 16 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
14	1, 13	lidlss 21206	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → 𝑚 ⊆ 𝐵)
17	16	sselda 3915	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ 𝑚) → 𝑥 ∈ 𝐵)
18		neldif 4065	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝑈)) → 𝑥 ∈ 𝑈)
19	17, 18	sylan 586	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ 𝑚) ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝑈)) → 𝑥 ∈ 𝑈)
20		simplr 774	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ 𝑚) ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝑈)) → 𝑥 ∈ 𝑚)
21	8	ad3antrrr 736	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ 𝑚) ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝑈)) → 𝑅 ∈ Ring)
22	12	ad3antrrr 736	. . . . . . . . . . 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 33507	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ 𝑚) ∧ ¬ 𝑥 ∈ (𝐵 ∖ 𝑈)) → 𝑚 = 𝐵)
24		nssrex 3980	. . . . . . . . . . 11 ⊢ (¬ 𝑚 ⊆ (𝐵 ∖ 𝑈) ↔ ∃𝑥 ∈ 𝑚 ¬ 𝑥 ∈ (𝐵 ∖ 𝑈))
25	24	bilani 505	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → ∃𝑥 ∈ 𝑚 ¬ 𝑥 ∈ (𝐵 ∖ 𝑈))
26	23, 25	r19.29a 3147	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → 𝑚 = 𝐵)
27	8	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → 𝑅 ∈ Ring)
28		simplr 774	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → 𝑚 ∈ (MaxIdeal‘𝑅))
29	1	mxidlnr 33548	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ≠ 𝐵)
30	27, 28, 29	syl2anc 590	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → 𝑚 ≠ 𝐵)
31	30	neneqd 2939	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ (𝐵 ∖ 𝑈)) → ¬ 𝑚 = 𝐵)
32	26, 31	condan 823	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ⊆ (𝐵 ∖ 𝑈))
33	1	mxidlmax 33549	. . . . . . . 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 844	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → ((𝐵 ∖ 𝑈) = 𝑚 ∨ (𝐵 ∖ 𝑈) = 𝐵))
35		eqid 2739	. . . . . . . . . . . 12 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
36	1, 35, 7	ringidcld 20239	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (1_r‘𝑅) ∈ 𝐵)
37	2, 35	1unit 20346	. . . . . . . . . . . . 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 4064	. . . . . . . . . . . 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 3031	. . . . . . . . . . 11 ⊢ (((1_r‘𝑅) ∈ 𝐵 ∧ ¬ (1_r‘𝑅) ∈ (𝐵 ∖ 𝑈)) → 𝐵 ≠ (𝐵 ∖ 𝑈))
42	36, 40, 41	syl2anc 590	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → 𝐵 ≠ (𝐵 ∖ 𝑈))
43	42	necomd 2989	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (𝐵 ∖ 𝑈) ≠ 𝐵)
44	43	adantr 481	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → (𝐵 ∖ 𝑈) ≠ 𝐵)
45	44	neneqd 2939	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → ¬ (𝐵 ∖ 𝑈) = 𝐵)
46	34, 45	olcnd 883	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → (𝐵 ∖ 𝑈) = 𝑚)
47	46	eqcomd 2745	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 = (𝐵 ∖ 𝑈))
48		simpr 485	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅))
49	1, 13	lidlss 21206	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (LIdeal‘𝑅) → 𝑗 ⊆ 𝐵)
50	49	ad3antlr 737	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → 𝑗 ⊆ 𝐵)
51		ssdif0 4295	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ⊆ 𝐵 ↔ (𝑗 ∖ 𝐵) = ∅)
52	50, 51	sylib 219	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → (𝑗 ∖ 𝐵) = ∅)
53	52	uneq1d 4098	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → ((𝑗 ∖ 𝐵) ∪ (𝑗 ∩ 𝑈)) = (∅ ∪ (𝑗 ∩ 𝑈)))
54		0un 4325	. . . . . . . . . . . . 13 ⊢ (∅ ∪ (𝑗 ∩ 𝑈)) = (𝑗 ∩ 𝑈)
55	53, 54	eqtr2di 2791	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → (𝑗 ∩ 𝑈) = ((𝑗 ∖ 𝐵) ∪ (𝑗 ∩ 𝑈)))
56		simplr 774	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → (𝐵 ∖ 𝑈) ⊆ 𝑗)
57		neqne 2942	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑗 = (𝐵 ∖ 𝑈) → 𝑗 ≠ (𝐵 ∖ 𝑈))
58	57	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → 𝑗 ≠ (𝐵 ∖ 𝑈))
59	58	necomd 2989	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → (𝐵 ∖ 𝑈) ≠ 𝑗)
60		difdif2 4225	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∖ (𝐵 ∖ 𝑈)) = ((𝑗 ∖ 𝐵) ∪ (𝑗 ∩ 𝑈))
61		pssdifn0 4297	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∖ 𝑈) ⊆ 𝑗 ∧ (𝐵 ∖ 𝑈) ≠ 𝑗) → (𝑗 ∖ (𝐵 ∖ 𝑈)) ≠ ∅)
62	60, 61	eqnetrrid 3009	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∖ 𝑈) ⊆ 𝑗 ∧ (𝐵 ∖ 𝑈) ≠ 𝑗) → ((𝑗 ∖ 𝐵) ∪ (𝑗 ∩ 𝑈)) ≠ ∅)
63	56, 59, 62	syl2anc 590	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → ((𝑗 ∖ 𝐵) ∪ (𝑗 ∩ 𝑈)) ≠ ∅)
64	55, 63	eqnetrd 3001	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → (𝑗 ∩ 𝑈) ≠ ∅)
65		simpr 485	. . . . . . . . . . . . 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 4135	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ (𝑗 ∩ 𝑈)) → 𝑥 ∈ 𝑈)
67	65	elin1d 4134	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ (𝑗 ∩ 𝑈)) → 𝑥 ∈ 𝑗)
68	7	ad4antr 738	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) ∧ 𝑥 ∈ (𝑗 ∩ 𝑈)) → 𝑅 ∈ Ring)
69		simp-4r 789	. . . . . . . . . . . 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 33507	. . . . . . . . . . 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 32560	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) ∧ ¬ 𝑗 = (𝐵 ∖ 𝑈)) → 𝑗 = 𝐵)
72	71	ex 413	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) → (¬ 𝑗 = (𝐵 ∖ 𝑈) → 𝑗 = 𝐵))
73	72	orrd 869	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝐵 ∖ 𝑈) ⊆ 𝑗) → (𝑗 = (𝐵 ∖ 𝑈) ∨ 𝑗 = 𝐵))
74	73	ex 413	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) → ((𝐵 ∖ 𝑈) ⊆ 𝑗 → (𝑗 = (𝐵 ∖ 𝑈) ∨ 𝑗 = 𝐵)))
75	74	ralrimiva 3131	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → ∀𝑗 ∈ (LIdeal‘𝑅)((𝐵 ∖ 𝑈) ⊆ 𝑗 → (𝑗 = (𝐵 ∖ 𝑈) ∨ 𝑗 = 𝐵)))
76	1	ismxidl 33546	. . . . . . 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 478	. . . . . 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 1380	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (𝐵 ∖ 𝑈) ∈ (MaxIdeal‘𝑅))
79	47, 78	eqsnd 4762	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (MaxIdeal‘𝑅) = {(𝐵 ∖ 𝑈)})
80	1	fvexi 6842	. . . . . . 7 ⊢ 𝐵 ∈ V
81	80	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → 𝐵 ∈ V)
82	81	difexd 5260	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (𝐵 ∖ 𝑈) ∈ V)
83		ensn1g 8960	. . . . 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 5095	. . 3 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → (MaxIdeal‘𝑅) ≈ 1_o)
86		dflring3 33589	. . . 4 ⊢ (𝑅 ∈ CRing → (𝑅 ∈ LRing ↔ (MaxIdeal‘𝑅) ≈ 1_o))
87	86	biimpar 478	. . 3 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) ≈ 1_o) → 𝑅 ∈ LRing)
88	6, 85, 87	syl2anc 590	. 2 ⊢ ((𝑅 ∈ CRing ∧ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)) → 𝑅 ∈ LRing)
89	5, 88	impbida 806	1 ⊢ (𝑅 ∈ CRing → (𝑅 ∈ LRing ↔ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 853 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∀wral 3053 ∃wrex 3063 Vcvv 3431 ∖ cdif 3880 ∪ cun 3881 ∩ cin 3882 ⊆ wss 3883 ∅c0 4262 {csn 4556 class class class wbr 5073 ‘cfv 6486 1_oc1o 8389 ≈ cen 8881 Basecbs 17171 1_rcur 20154 Ringcrg 20206 CRingccrg 20207 Unitcui 20327 LRingclring 20511 LIdealclidl 21200 MaxIdealcmxidl 33543

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-ac2 10377 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4840 df-int 4879 df-iun 4924 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-rpss 7667 df-om 7808 df-1st 7932 df-2nd 7933 df-tpos 8167 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-oadd 8400 df-er 8634 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-dju 9817 df-card 9855 df-ac 10030 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-nn 12167 df-2 12236 df-3 12237 df-4 12238 df-5 12239 df-6 12240 df-7 12241 df-8 12242 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17172 df-ress 17193 df-plusg 17225 df-mulr 17226 df-sca 17228 df-vsca 17229 df-ip 17230 df-0g 17396 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18904 df-minusg 18905 df-sbg 18906 df-subg 19091 df-cmn 19749 df-abl 19750 df-mgp 20114 df-rng 20126 df-ur 20155 df-ring 20208 df-cring 20209 df-oppr 20309 df-dvdsr 20329 df-unit 20330 df-invr 20360 df-dvr 20373 df-nzr 20486 df-lring 20512 df-subrg 20543 df-lmod 20853 df-lss 20923 df-lsp 20963 df-sra 21164 df-rgmod 21165 df-lidl 21202 df-rsp 21203 df-mxidl 33544

This theorem is referenced by: (None)

W3C validator