Theorem dflring3 33753

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 20326	. . . . . . . . 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 2761	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
6		eqid 2761	. . . . . . . . 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 33751	. . . . . . . 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 33712	. . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . 15 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
14	5, 13	lidlss 21315	. . . . . . . . . . . . . 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 3936	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ ¬ 𝑚 ⊆ ((Base‘𝑅) ∖ (Unit‘𝑅))) ∧ 𝑥 ∈ 𝑚) → 𝑥 ∈ (Base‘𝑅))
18		neldif 4087	. . . . . . . . . . 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 33697	. . . . . . . . 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 4001	. . . . . . . . . 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 3171	. . . . . . . 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 33713	. . . . . . . . . 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 2961	. . . . . . . 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 33714	. . . . . . 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 2761	. . . . . . . . . . . 12 ⊢ (1r‘𝑅) = (1r‘𝑅)
36	5, 35, 1	ringidcld 20348	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → (1r‘𝑅) ∈ (Base‘𝑅))
37	36	adantr 485	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → (1r‘𝑅) ∈ (Base‘𝑅))
38	6, 35	1unit 20455	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Unit‘𝑅))
39		elndif 4086	. . . . . . . . . . 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 3053	. . . . . . . . . 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 3011	. . . . . . . 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 2961	. . . . . 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 2767	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 = ((Base‘𝑅) ∖ (Unit‘𝑅)))
48	5, 6, 7, 8	dflringlem3 33752	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → ((Base‘𝑅) ∖ (Unit‘𝑅)) ∈ (MaxIdeal‘𝑅))
49	47, 48	eqsnd 4795	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → (MaxIdeal‘𝑅) = {((Base‘𝑅) ∖ (Unit‘𝑅))})
50		ensn1g 9018	. . . 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 5132	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ LRing) → (MaxIdeal‘𝑅) ≈ 1o)
53		en1 9020	. . . . 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 4628	. . . . . . . 8 ⊢ 𝑚 ∈ {𝑚}
57		simpr 489	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) → (MaxIdeal‘𝑅) = {𝑚})
58	56, 57	eleqtrrid 2868	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) → 𝑚 ∈ (MaxIdeal‘𝑅))
59	5	mxidlnzr 33716	. . . . . . 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 3915	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (MaxIdeal‘𝑅) = {𝑚}) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ ¬ 𝑥 ∈ 𝑚) → 𝑥 ∈ ((Base‘𝑅) ∖ 𝑚))
67	5, 6, 61, 62, 63, 66	dflringlem 33750	. . . . . . . 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 2761	. . . . . . . . . . 11 ⊢ (-g‘𝑅) = (-g‘𝑅)
72	1	ringgrpd 20323	. . . . . . . . . . . 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 33327	. . . . . . . . . 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 33715	. . . . . . . . . . . 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 2761	. . . . . . . . . . . . . 14 ⊢ (+g‘𝑅) = (+g‘𝑅)
84	5, 83, 71	grpnpcan 19097	. . . . . . . . . . . . 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 21325	. . . . . . . . . . . . 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 2862	. . . . . . . . . . 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 3915	. . . . . . . . 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 33750	. . . . . . . 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 3155	. . . . . 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 1963	. . 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 33749	. . 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 1568  ∃wex 1807   ∈ wcel 2141   ≠ wne 2956  ∀wral 3077  ∃wrex 3087   ∖ cdif 3901   ⊆ wss 3904  {csn 4588   class class class wbr 5108  ‘cfv 6536  (class class class)co 7410  1_oc1o 8445   ≈ cen 8939  Basecbs 17268  +_gcplusg 17309  Grpcgrp 18999  -_gcsg 19001  1_rcur 20262  Ringcrg 20314  CRingccrg 20315  Unitcui 20436  NzRingcnzr 20594  LRingclring 20622  LIdealclidl 21309  MaxIdealcmxidl 33708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-ac2 10446  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-rpss 7720  df-om 7862  df-1st 7985  df-2nd 7986  df-tpos 8221  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-oadd 8456  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-dju 9886  df-card 9924  df-ac 10099  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-7 12307  df-8 12308  df-sets 17223  df-slot 17241  df-ndx 17253  df-base 17269  df-ress 17290  df-plusg 17322  df-mulr 17323  df-sca 17325  df-vsca 17326  df-ip 17327  df-0g 17493  df-mgm 18697  df-sgrp 18776  df-mnd 18792  df-grp 19002  df-minusg 19003  df-sbg 19004  df-subg 19188  df-cmn 19851  df-abl 19852  df-mgp 20216  df-rng 20230  df-ur 20263  df-ring 20316  df-cring 20317  df-oppr 20418  df-dvdsr 20438  df-unit 20439  df-invr 20469  df-dvr 20482  df-nzr 20595  df-lring 20623  df-subrg 20654  df-lmod 20962  df-lss 21032  df-lsp 21072  df-sra 21273  df-rgmod 21274  df-lidl 21311  df-rsp 21312  df-mxidl 33709

This theorem is referenced by:  dflring4  33754  fldlring  33755