Theorem dflring3 33732

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 20327	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2	1	adantr 485	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → 𝑅 ∈ Ring)
3	2	adantr 485	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ Ring)
4		simpr 489	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (MaxIdeal‘𝑅))
5		eqid 2769	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
6		eqid 2769	. . . . . . . . 9 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
7		simpl 487	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → 𝑅 ∈ CRing)
8		simpr 489	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → 𝑅 ∈ LRing)
9	5, 6, 7, 8	dflringlem2 33730	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → ((Base‘𝑅) ∖ (Unit‘𝑅)) ∈ (LIdeal‘𝑅))
10	9	adantr 485	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → ((Base‘𝑅) ∖ (Unit‘𝑅)) ∈ (LIdeal‘𝑅))
11	5	mxidlidl 33691	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (LIdeal‘𝑅))
12	2, 11	sylan 591	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (LIdeal‘𝑅))
13		eqid 2769	. . . . . . . . . . . . . . 15 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
14	5, 13	lidlss 21314	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅))) → 𝑚 ⊆ (Base‘𝑅))
17	16	sselda 3945	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅))) ∧ 𝑥 ∈ 𝑚) → 𝑥 ∈ (Base‘𝑅))
18		neldif 4096	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (Base‘𝑅) ∧ ¬ 𝑥 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))) → 𝑥 ∈ (Unit‘𝑅))
19	17, 18	sylan 591	. . . . . . . . . 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 780	. . . . . . . . . 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 744	. . . . . . . . . 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 742	. . . . . . . . . 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 33675	. . . . . . . . 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 4010	. . . . . . . . . 10 ⊢ (¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅)) ↔ ∃𝑥 ∈ 𝑚 ¬ 𝑥 ∈ ((Base‘𝑅) ∖ (Unit‘𝑅)))
25	24	bilani 509	. . . . . . . . 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 3179	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅))) → 𝑚 = (Base‘𝑅))
27	2	ad2antrr 738	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅))) → 𝑅 ∈ Ring)
28		simplr 780	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅))) → 𝑚 ∈ (MaxIdeal‘𝑅))
29	5	mxidlnr 33692	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ≠ (Base‘𝑅))
30	27, 28, 29	syl2anc 595	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅))) → 𝑚 ≠ (Base‘𝑅))
31	30	neneqd 2969	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅))) → ¬ 𝑚 = (Base‘𝑅))
32	26, 31	condan 829	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅)))
33	5	mxidlmax 33693	. . . . . . 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 851	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → (((Base‘𝑅) ∖ (Unit‘𝑅)) = 𝑚 ∨ ((Base‘𝑅) ∖ (Unit‘𝑅)) = (Base‘𝑅)))
35		eqid 2769	. . . . . . . . . . . 12 ⊢ (1r‘𝑅) = (1r‘𝑅)
36	5, 35, 1	ringidcld 20349	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → (1r‘𝑅) ∈ (Base‘𝑅))
37	36	adantr 485	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → (1r‘𝑅) ∈ (Base‘𝑅))
38	6, 35	1unit 20456	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Unit‘𝑅))
39		elndif 4095	. . . . . . . . . . 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 3061	. . . . . . . . . 10 ⊢ (((1r‘𝑅) ∈ (Base‘𝑅) ∧ ¬ (1r‘𝑅) ∈ ((Base‘𝑅) ∖ (Unit‘𝑅))) → (Base‘𝑅) ≠ ((Base‘𝑅) ∖ (Unit‘𝑅)))
42	37, 40, 41	syl2anc 595	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → (Base‘𝑅) ≠ ((Base‘𝑅) ∖ (Unit‘𝑅)))
43	42	necomd 3019	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → ((Base‘𝑅) ∖ (Unit‘𝑅)) ≠ (Base‘𝑅))
44	43	adantr 485	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → ((Base‘𝑅) ∖ (Unit‘𝑅)) ≠ (Base‘𝑅))
45	44	neneqd 2969	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → ¬ ((Base‘𝑅) ∖ (Unit‘𝑅)) = (Base‘𝑅))
46	34, 45	olcnd 890	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → ((Base‘𝑅) ∖ (Unit‘𝑅)) = 𝑚)
47	46	eqcomd 2775	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 = ((Base‘𝑅) ∖ (Unit‘𝑅)))
48	5, 6, 7, 8	dflringlem3 33731	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → ((Base‘𝑅) ∖ (Unit‘𝑅)) ∈ (MaxIdeal‘𝑅))
49	47, 48	eqsnd 4800	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → (MaxIdeal‘𝑅) = {((Base‘𝑅) ∖ (Unit‘𝑅))})
50		ensn1g 9019	. . . 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 5137	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → (MaxIdeal‘𝑅) ≈ 1o)
53		en1 9021	. . . . 5 ⊢ ((MaxIdeal‘𝑅) ≈ 1o ↔ ∃𝑚(MaxIdeal‘𝑅) = {𝑚})
54	53	bilani 509	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) ≈ 1o) → ∃𝑚(MaxIdeal‘𝑅) = {𝑚})
55	1	adantr 485	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) → 𝑅 ∈ Ring)
56		vsnid 4634	. . . . . . . 8 ⊢ 𝑚 ∈ {𝑚}
57		simpr 489	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) → (MaxIdeal‘𝑅) = {𝑚})
58	56, 57	eleqtrrid 2876	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) → 𝑚 ∈ (MaxIdeal‘𝑅))
59	5	mxidlnzr 33695	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ NzRing)
60	55, 58, 59	syl2anc 595	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) → 𝑅 ∈ NzRing)
61		simplll 786	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ ¬ 𝑥 ∈ 𝑚) → 𝑅 ∈ CRing)
62	58	ad2antrr 738	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ ¬ 𝑥 ∈ 𝑚) → 𝑚 ∈ (MaxIdeal‘𝑅))
63	57	ad2antrr 738	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ ¬ 𝑥 ∈ 𝑚) → (MaxIdeal‘𝑅) = {𝑚})
64		simplr 780	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ ¬ 𝑥 ∈ 𝑚) → 𝑥 ∈ (Base‘𝑅))
65		simpr 489	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ ¬ 𝑥 ∈ 𝑚) → ¬ 𝑥 ∈ 𝑚)
66	64, 65	eldifd 3924	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ ¬ 𝑥 ∈ 𝑚) → 𝑥 ∈ ((Base‘𝑅) ∖ 𝑚))
67	5, 6, 61, 62, 63, 66	dflringlem 33729	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ ¬ 𝑥 ∈ 𝑚) → 𝑥 ∈ (Unit‘𝑅))
68		simplll 786	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) → 𝑅 ∈ CRing)
69	58	ad2antrr 738	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) → 𝑚 ∈ (MaxIdeal‘𝑅))
70	57	ad2antrr 738	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) → (MaxIdeal‘𝑅) = {𝑚})
71		eqid 2769	. . . . . . . . . . 11 ⊢ (-g‘𝑅) = (-g‘𝑅)
72	1	ringgrpd 20324	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Grp)
73	72	ad3antrrr 742	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) → 𝑅 ∈ Grp)
74	36	ad3antrrr 742	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) → (1r‘𝑅) ∈ (Base‘𝑅))
75		simplr 780	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) → 𝑥 ∈ (Base‘𝑅))
76	5, 71, 73, 74, 75	grpsubcld 33302	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) → ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ (Base‘𝑅))
77	55	ad2antrr 738	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) → 𝑅 ∈ Ring)
78	5, 35	mxidln1 33694	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → ¬ (1r‘𝑅) ∈ 𝑚)
79	77, 69, 78	syl2anc 595	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) → ¬ (1r‘𝑅) ∈ 𝑚)
80	73	adantr 485	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) ∧ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ 𝑚) → 𝑅 ∈ Grp)
81	74	adantr 485	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) ∧ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ 𝑚) → (1r‘𝑅) ∈ (Base‘𝑅))
82	75	adantr 485	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) ∧ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ 𝑚) → 𝑥 ∈ (Base‘𝑅))
83		eqid 2769	. . . . . . . . . . . . . 14 ⊢ (+g‘𝑅) = (+g‘𝑅)
84	5, 83, 71	grpnpcan 19098	. . . . . . . . . . . . 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 1396	. . . . . . . . . . . 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 485	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) ∧ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ 𝑚) → 𝑅 ∈ Ring)
87	69	adantr 485	. . . . . . . . . . . . . 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 595	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) ∧ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ 𝑚) → 𝑚 ∈ (LIdeal‘𝑅))
89		simpr 489	. . . . . . . . . . . . 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 780	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) ∧ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ 𝑚) → 𝑥 ∈ 𝑚)
91	13, 83	lidlacl 21324	. . . . . . . . . . . . 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 851	. . . . . . . . . . . 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 2870	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) ∧ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ 𝑚) → (1r‘𝑅) ∈ 𝑚)
94	79, 93	mtand 827	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) → ¬ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ 𝑚)
95	76, 94	eldifd 3924	. . . . . . . . 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 33729	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑥 ∈ 𝑚) → ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ (Unit‘𝑅))
97		exmidd 908	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 ∈ 𝑚 ∨ ¬ 𝑥 ∈ 𝑚))
98	97	orcomd 884	. . . . . . . 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 3163	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) → ∀𝑥 ∈ (Base‘𝑅)(𝑥 ∈ (Unit‘𝑅) ∨ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ (Unit‘𝑅)))
101	60, 100	jca 520	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) → (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)(𝑥 ∈ (Unit‘𝑅) ∨ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ (Unit‘𝑅))))
102	101	adantlr 727	. . . 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 1962	. . 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 33728	. . 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 812	1 ⊢ (𝑅 ∈ CRing → (𝑅 ∈ LRing ↔ (MaxIdeal‘𝑅) ≈ 1o))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1567  ∃wex 1806   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095   ∖ cdif 3910   ⊆ wss 3913  {csn 4594   class class class wbr 5113  ‘cfv 6537  (class class class)co 7411  1_oc1o 8446   ≈ cen 8940  Basecbs 17269  +_gcplusg 17310  Grpcgrp 19000  -_gcsg 19002  1_rcur 20263  Ringcrg 20315  CRingccrg 20316  Unitcui 20437  NzRingcnzr 20595  LRingclring 20623  LIdealclidl 21308  MaxIdealcmxidl 33687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-ac2 10447  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-rpss 7721  df-om 7863  df-1st 7986  df-2nd 7987  df-tpos 8222  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-oadd 8457  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-dju 9887  df-card 9925  df-ac 10100  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-sets 17224  df-slot 17242  df-ndx 17254  df-base 17270  df-ress 17291  df-plusg 17323  df-mulr 17324  df-sca 17326  df-vsca 17327  df-ip 17328  df-0g 17494  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-grp 19003  df-minusg 19004  df-sbg 19005  df-subg 19189  df-cmn 19852  df-abl 19853  df-mgp 20217  df-rng 20231  df-ur 20264  df-ring 20317  df-cring 20318  df-oppr 20419  df-dvdsr 20439  df-unit 20440  df-invr 20470  df-dvr 20483  df-nzr 20596  df-lring 20624  df-subrg 20655  df-lmod 20961  df-lss 21031  df-lsp 21071  df-sra 21272  df-rgmod 21273  df-lidl 21310  df-rsp 21311  df-mxidl 33688

This theorem is referenced by:  dflring4  33733  fldlring  33734