Theorem dflring3 33855

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 20375	. . . . . . . . 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 2765	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
6		eqid 2765	. . . . . . . . 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 33853	. . . . . . . 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 33814	. . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . 15 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
14	5, 13	lidlss 21390	. . . . . . . . . . . . . 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 3938	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅))) ∧ 𝑥 ∈ 𝑚) → 𝑥 ∈ (Base‘𝑅))
18		neldif 4088	. . . . . . . . . . 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 33799	. . . . . . . . 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 4003	. . . . . . . . . 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 3175	. . . . . . . 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 33815	. . . . . . . . . 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 2965	. . . . . . . 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 33816	. . . . . . 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 2765	. . . . . . . . . . . 12 ⊢ (1r‘𝑅) = (1r‘𝑅)
36	5, 35, 1	ringidcld 20398	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → (1r‘𝑅) ∈ (Base‘𝑅))
37	36	adantr 486	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → (1r‘𝑅) ∈ (Base‘𝑅))
38	6, 35	1unit 20506	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Unit‘𝑅))
39		elndif 4087	. . . . . . . . . . 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 3057	. . . . . . . . . 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 3015	. . . . . . . 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 2965	. . . . . 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 2771	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 = ((Base‘𝑅) ∖ (Unit‘𝑅)))
48	5, 6, 7, 8	dflringlem3 33854	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → ((Base‘𝑅) ∖ (Unit‘𝑅)) ∈ (MaxIdeal‘𝑅))
49	47, 48	eqsnd 4798	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → (MaxIdeal‘𝑅) = {((Base‘𝑅) ∖ (Unit‘𝑅))})
50		ensn1g 9025	. . . 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 5135	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → (MaxIdeal‘𝑅) ≈ 1o)
53		en1 9027	. . . . 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 4631	. . . . . . . 8 ⊢ 𝑚 ∈ {𝑚}
57		simpr 490	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) → (MaxIdeal‘𝑅) = {𝑚})
58	56, 57	eleqtrrid 2872	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) → 𝑚 ∈ (MaxIdeal‘𝑅))
59	5	mxidlnzr 33818	. . . . . . 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 3917	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ ¬ 𝑥 ∈ 𝑚) → 𝑥 ∈ ((Base‘𝑅) ∖ 𝑚))
67	5, 6, 61, 62, 63, 66	dflringlem 33852	. . . . . . . 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 2765	. . . . . . . . . . 11 ⊢ (-g‘𝑅) = (-g‘𝑅)
72	1	ringgrpd 20372	. . . . . . . . . . . 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 33429	. . . . . . . . . 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 33817	. . . . . . . . . . . 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 2765	. . . . . . . . . . . . . 14 ⊢ (+g‘𝑅) = (+g‘𝑅)
84	5, 83, 71	grpnpcan 19146	. . . . . . . . . . . . 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 21400	. . . . . . . . . . . . 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 2866	. . . . . . . . . . 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 3917	. . . . . . . . 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 33852	. . . . . . . 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 3159	. . . . . 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 1968	. . 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 33851	. . 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 1570  ∃wex 1812   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091   ∖ cdif 3903   ⊆ wss 3906  {csn 4591   class class class wbr 5111  ‘cfv 6540  (class class class)co 7419  1_oc1o 8452   ≈ cen 8946  Basecbs 17295  +_gcplusg 17336  Grpcgrp 19048  -_gcsg 19050  1_rcur 20311  Ringcrg 20363  CRingccrg 20364  Unitcui 20487  NzRingcnzr 20663  LRingclring 20691  LIdealclidl 21384  MaxIdealcmxidl 33810

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-ac2 10462  ax-cnex 11175  ax-resscn 11176  ax-1cn 11177  ax-icn 11178  ax-addcl 11179  ax-addrcl 11180  ax-mulcl 11181  ax-mulrcl 11182  ax-mulcom 11183  ax-addass 11184  ax-mulass 11185  ax-distr 11186  ax-i2m1 11187  ax-1ne0 11188  ax-1rid 11189  ax-rnegex 11190  ax-rrecex 11191  ax-cnre 11192  ax-pre-lttri 11193  ax-pre-lttrn 11194  ax-pre-ltadd 11195  ax-pre-mulgt0 11196

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-rpss 7730  df-om 7869  df-1st 7992  df-2nd 7993  df-tpos 8228  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-oadd 8463  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-dju 9903  df-card 9941  df-ac 10116  df-pnf 11264  df-mnf 11265  df-xr 11266  df-ltxr 11267  df-le 11268  df-sub 11462  df-neg 11463  df-nn 12253  df-2 12322  df-3 12323  df-4 12324  df-5 12325  df-6 12326  df-7 12327  df-8 12328  df-sets 17250  df-slot 17268  df-ndx 17280  df-base 17296  df-ress 17317  df-plusg 17349  df-mulr 17350  df-sca 17352  df-vsca 17353  df-ip 17354  df-0g 17520  df-mgm 18724  df-sgrp 18813  df-mnd 18829  df-grp 19051  df-minusg 19052  df-sbg 19053  df-subg 19237  df-cmn 19900  df-abl 19901  df-mgp 20265  df-rng 20279  df-ur 20312  df-ring 20365  df-cring 20366  df-oppr 20469  df-dvdsr 20489  df-unit 20490  df-invr 20520  df-dvr 20533  df-nzr 20664  df-lring 20692  df-subrg 20723  df-lmod 21037  df-lss 21107  df-lsp 21147  df-sra 21348  df-rgmod 21349  df-lidl 21386  df-rsp 21387  df-mxidl 33811

This theorem is used by:  dflring4  33856  fldlring  33857