Theorem dflring3 34085

Description: Alternate definition of a local ring: local rings have a single maximal ideal. (Contributed by Thierry Arnoux, 2-Jun-2026.)

Assertion

Ref	Expression
dflring3	⊢ (𝑅 ∈ CRing → (𝑅 ∈ LRing ↔ (MaxIdeal‘𝑅) ≈ 1o))

Proof of Theorem dflring3

Dummy variables 𝑥 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		crngring 20487	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2	1	adantr 486	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → 𝑅 ∈ Ring)
3	2	adantr 486	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ Ring)
4		simpr 490	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (MaxIdeal‘𝑅))
5		eqid 2763	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
6		eqid 2763	. . . . . . . . 9 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
7		simpl 488	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → 𝑅 ∈ CRing)
8		simpr 490	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → 𝑅 ∈ LRing)
9	5, 6, 7, 8	dflringlem2 34083	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → ((Base‘𝑅) ∖ (Unit‘𝑅)) ∈ (LIdeal‘𝑅))
10	9	adantr 486	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → ((Base‘𝑅) ∖ (Unit‘𝑅)) ∈ (LIdeal‘𝑅))
11	5	mxidlidl 34044	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (LIdeal‘𝑅))
12	2, 11	sylan 592	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (LIdeal‘𝑅))
13		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
14	5, 13	lidlss 21505	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (LIdeal‘𝑅) → 𝑚 ⊆ (Base‘𝑅))
15	12, 14	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ⊆ (Base‘𝑅))
16	15	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅))) → 𝑚 ⊆ (Base‘𝑅))
17	16	sselda 3913	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅))) ∧ 𝑥 ∈ 𝑚) → 𝑥 ∈ (Base‘𝑅))
18		neldif 4073	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (Base‘𝑅) ∧ ¬ 𝑥 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))) → 𝑥 ∈ (Unit‘𝑅))
19	17, 18	sylan 592	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅))) ∧ 𝑥 ∈ 𝑚) ∧ ¬ 𝑥 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))) → 𝑥 ∈ (Unit‘𝑅))
20		simplr 781	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅))) ∧ 𝑥 ∈ 𝑚) ∧ ¬ 𝑥 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))) → 𝑥 ∈ 𝑚)
21	2	ad4antr 745	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅))) ∧ 𝑥 ∈ 𝑚) ∧ ¬ 𝑥 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))) → 𝑅 ∈ Ring)
22	12	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅))) ∧ 𝑥 ∈ 𝑚) ∧ ¬ 𝑥 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))) → 𝑚 ∈ (LIdeal‘𝑅))
23	5, 6, 19, 20, 21, 22	lidlunitel 34029	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅))) ∧ 𝑥 ∈ 𝑚) ∧ ¬ 𝑥 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))) → 𝑚 = (Base‘𝑅))
24		nssrex 3978	. . . . . . . . . 10 ⊢ (¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅)) ↔ ∃𝑥 ∈ 𝑚 ¬ 𝑥 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅)))
25	24	bilani 510	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅))) → ∃𝑥 ∈ 𝑚 ¬ 𝑥 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅)))
26	23, 25	r19.29a 3174	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅))) → 𝑚 = (Base‘𝑅))
27	2	ad2antrr 739	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅))) → 𝑅 ∈ Ring)
28		simplr 781	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅))) → 𝑚 ∈ (MaxIdeal‘𝑅))
29	5	mxidlnr 34045	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ≠ (Base‘𝑅))
30	27, 28, 29	syl2anc 596	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅))) → 𝑚 ≠ (Base‘𝑅))
31	30	neneqd 2964	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅))) → ¬ 𝑚 = (Base‘𝑅))
32	26, 31	condan 830	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅)))
33	5	mxidlmax 34046	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ (((Base‘𝑅) ∖ (Unit‘𝑅)) ∈ (LIdeal‘𝑅) ∧ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅)))) → (((Base‘𝑅) ∖ (Unit‘𝑅)) = 𝑚 ∨ ((Base‘𝑅) ∖ (Unit‘𝑅)) = (Base‘𝑅)))
34	3, 4, 10, 32, 33	syl22anc 852	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → (((Base‘𝑅) ∖ (Unit‘𝑅)) = 𝑚 ∨ ((Base‘𝑅) ∖ (Unit‘𝑅)) = (Base‘𝑅)))
35		eqid 2763	. . . . . . . . . . . 12 ⊢ (1r‘𝑅) = (1r‘𝑅)
36	5, 35, 1	ringidcld 20510	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → (1r‘𝑅) ∈ (Base‘𝑅))
37	36	adantr 486	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → (1r‘𝑅) ∈ (Base‘𝑅))
38	6, 35	1unit 20619	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Unit‘𝑅))
39		elndif 4072	. . . . . . . . . . 11 ⊢ ((1r‘𝑅) ∈ (Unit‘𝑅) → ¬ (1r‘𝑅) ∈ ((Base‘𝑅) ∖ (Unit‘𝑅)))
40	2, 38, 39	3syl 19	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → ¬ (1r‘𝑅) ∈ ((Base‘𝑅) ∖ (Unit‘𝑅)))
41		nelne1 3056	. . . . . . . . . 10 ⊢ (((1r‘𝑅) ∈ (Base‘𝑅) ∧ ¬ (1r‘𝑅) ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))) → (Base‘𝑅) ≠ ((Base‘𝑅) ∖ (Unit‘𝑅)))
42	37, 40, 41	syl2anc 596	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → (Base‘𝑅) ≠ ((Base‘𝑅) ∖ (Unit‘𝑅)))
43	42	necomd 3014	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → ((Base‘𝑅) ∖ (Unit‘𝑅)) ≠ (Base‘𝑅))
44	43	adantr 486	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → ((Base‘𝑅) ∖ (Unit‘𝑅)) ≠ (Base‘𝑅))
45	44	neneqd 2964	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → ¬ ((Base‘𝑅) ∖ (Unit‘𝑅)) = (Base‘𝑅))
46	34, 45	olcnd 891	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → ((Base‘𝑅) ∖ (Unit‘𝑅)) = 𝑚)
47	46	eqcomd 2769	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 = ((Base‘𝑅) ∖ (Unit‘𝑅)))
48	5, 6, 7, 8	dflringlem3 34084	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → ((Base‘𝑅) ∖ (Unit‘𝑅)) ∈ (MaxIdeal‘𝑅))
49	47, 48	eqsnd 4789	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → (MaxIdeal‘𝑅) = {((Base‘𝑅) ∖ (Unit‘𝑅))})
50		ensn1g 9063	. . . 4 ⊢ (((Base‘𝑅) ∖ (Unit‘𝑅)) ∈ (LIdeal‘𝑅) → {((Base‘𝑅) ∖ (Unit‘𝑅))} ≈ 1o)
51	9, 50	syl 18	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → {((Base‘𝑅) ∖ (Unit‘𝑅))} ≈ 1o)
52	49, 51	eqbrtrd 5130	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → (MaxIdeal‘𝑅) ≈ 1o)
53		en1 9065	. . . . 5 ⊢ ((MaxIdeal‘𝑅) ≈ 1o ↔ ∃𝑚(MaxIdeal‘𝑅) = {𝑚})
54	53	bilani 510	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) ≈ 1o) → ∃𝑚(MaxIdeal‘𝑅) = {𝑚})
55	1	adantr 486	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) → 𝑅 ∈ Ring)
56		vsnid 4624	. . . . . . . 8 ⊢ 𝑚 ∈ {𝑚}
57		simpr 490	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) → (MaxIdeal‘𝑅) = {𝑚})
58	56, 57	eleqtrrid 2871	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) → 𝑚 ∈ (MaxIdeal‘𝑅))
59	5	mxidlnzr 34048	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ NzRing)
60	55, 58, 59	syl2anc 596	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) → 𝑅 ∈ NzRing)
61		simplll 787	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ ¬ 𝑥 ∈ 𝑚) → 𝑅 ∈ CRing)
62	58	ad2antrr 739	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ ¬ 𝑥 ∈ 𝑚) → 𝑚 ∈ (MaxIdeal‘𝑅))
63	57	ad2antrr 739	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ ¬ 𝑥 ∈ 𝑚) → (MaxIdeal‘𝑅) = {𝑚})
64		simplr 781	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ ¬ 𝑥 ∈ 𝑚) → 𝑥 ∈ (Base‘𝑅))
65		simpr 490	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ ¬ 𝑥 ∈ 𝑚) → ¬ 𝑥 ∈ 𝑚)
66	64, 65	eldifd 4053	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ ¬ 𝑥 ∈ 𝑚) → 𝑥 ∈ ((Base‘𝑅) ∖ 𝑚))
67	5, 6, 61, 62, 63, 66	dflringlem 34082	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ ¬ 𝑥 ∈ 𝑚) → 𝑥 ∈ (Unit‘𝑅))
68		simplll 787	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) → 𝑅 ∈ CRing)
69	58	ad2antrr 739	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) → 𝑚 ∈ (MaxIdeal‘𝑅))
70	57	ad2antrr 739	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) → (MaxIdeal‘𝑅) = {𝑚})
71		eqid 2763	. . . . . . . . . . 11 ⊢ (-g‘𝑅) = (-g‘𝑅)
72	1	ringgrpd 20484	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Grp)
73	72	ad3antrrr 743	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) → 𝑅 ∈ Grp)
74	36	ad3antrrr 743	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) → (1r‘𝑅) ∈ (Base‘𝑅))
75		simplr 781	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) → 𝑥 ∈ (Base‘𝑅))
76	5, 71, 73, 74, 75	grpsubcld 33658	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) → ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ (Base‘𝑅))
77	55	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) → 𝑅 ∈ Ring)
78	5, 35	mxidln1 34047	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → ¬ (1r‘𝑅) ∈ 𝑚)
79	77, 69, 78	syl2anc 596	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) → ¬ (1r‘𝑅) ∈ 𝑚)
80	73	adantr 486	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) ∧ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ 𝑚) → 𝑅 ∈ Grp)
81	74	adantr 486	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) ∧ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ 𝑚) → (1r‘𝑅) ∈ (Base‘𝑅))
82	75	adantr 486	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) ∧ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ 𝑚) → 𝑥 ∈ (Base‘𝑅))
83		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (+g‘𝑅) = (+g‘𝑅)
84	5, 83, 71	grpnpcan 19257	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Grp ∧ (1r‘𝑅) ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (((1r‘𝑅)(-g‘𝑅)𝑥)(+g‘𝑅)𝑥) = (1r‘𝑅))
85	80, 81, 82, 84	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) ∧ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ 𝑚) → (((1r‘𝑅)(-g‘𝑅)𝑥)(+g‘𝑅)𝑥) = (1r‘𝑅))
86	77	adantr 486	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) ∧ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ 𝑚) → 𝑅 ∈ Ring)
87	69	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) ∧ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ 𝑚) → 𝑚 ∈ (MaxIdeal‘𝑅))
88	86, 87, 11	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) ∧ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ 𝑚) → 𝑚 ∈ (LIdeal‘𝑅))
89		simpr 490	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) ∧ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ 𝑚) → ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ 𝑚)
90		simplr 781	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) ∧ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ 𝑚) → 𝑥 ∈ 𝑚)
91	13, 83	lidlacl 21515	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑚 ∈ (LIdeal‘𝑅)) ∧ (((1r‘𝑅)(-g‘𝑅)𝑥) ∈ 𝑚 ∧ 𝑥 ∈ 𝑚)) → (((1r‘𝑅)(-g‘𝑅)𝑥)(+g‘𝑅)𝑥) ∈ 𝑚)
92	86, 88, 89, 90, 91	syl22anc 852	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) ∧ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ 𝑚) → (((1r‘𝑅)(-g‘𝑅)𝑥)(+g‘𝑅)𝑥) ∈ 𝑚)
93	85, 92	eqeltrrd 2865	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) ∧ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ 𝑚) → (1r‘𝑅) ∈ 𝑚)
94	79, 93	mtand 828	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) → ¬ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ 𝑚)
95	76, 94	eldifd 4053	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) → ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ ((Base‘𝑅) ∖ 𝑚))
96	5, 6, 68, 69, 70, 95	dflringlem 34082	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) → ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ (Unit‘𝑅))
97		exmidd 909	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 ∈ 𝑚 ∨ ¬ 𝑥 ∈ 𝑚))
98	97	orcomd 885	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) → (¬ 𝑥 ∈ 𝑚 ∨ 𝑥 ∈ 𝑚))
99	67, 96, 98	orim12da 980	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 ∈ (Unit‘𝑅) ∨ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ (Unit‘𝑅)))
100	99	ralrimiva 3158	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) → ∀𝑥 ∈ (Base‘𝑅)(𝑥 ∈ (Unit‘𝑅) ∨ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ (Unit‘𝑅)))
101	60, 100	jca 521	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) → (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)(𝑥 ∈ (Unit‘𝑅) ∨ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ (Unit‘𝑅))))
102	101	adantlr 728	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) ≈ 1o) ∧ (MaxIdeal‘𝑅) = {𝑚}) → (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)(𝑥 ∈ (Unit‘𝑅) ∨ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ (Unit‘𝑅))))
103	54, 102	exlimddv 1969	. . 3 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) ≈ 1o) → (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)(𝑥 ∈ (Unit‘𝑅) ∨ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ (Unit‘𝑅))))
104	5, 6, 35, 71	dflring2 34081	. . 3 ⊢ (𝑅 ∈ LRing ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)(𝑥 ∈ (Unit‘𝑅) ∨ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ (Unit‘𝑅))))
105	103, 104	sylibr 237	. 2 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) ≈ 1o) → 𝑅 ∈ LRing)
106	52, 105	impbida 813	1 ⊢ (𝑅 ∈ CRing → (𝑅 ∈ LRing ↔ (MaxIdeal‘𝑅) ≈ 1o))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569  ∃wex 1811   ∈ wcel 2146   ≠ wne 2959  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∖ cdif 4048  {csn 4588   class class class wbr 5106  ‘cfv 6547  (class class class)co 7430  1_oc1o 8483   ≈ cen 8984  Basecbs 17402  +_gcplusg 17443  Grpcgrp 19159  -_gcsg 19161  1_rcur 20422  Ringcrg 20474  CRingccrg 20475  Unitcui 20600  NzRingcnzr 20777  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:  dflring4  34086  fldlring  34087