Theorem qsdrng 33447

Description: An ideal 𝑀 is both left and right maximal if and only if the factor ring 𝑄 is a division ring. (Contributed by Thierry Arnoux, 13-Mar-2025.)

Hypotheses

Ref	Expression
qsdrng.0	⊢ 𝑂 = (oppr‘𝑅)
qsdrng.q	⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))
qsdrng.r	⊢ (𝜑 → 𝑅 ∈ NzRing)
qsdrng.2	⊢ (𝜑 → 𝑀 ∈ (2Ideal‘𝑅))

Assertion

Ref	Expression
qsdrng	⊢ (𝜑 → (𝑄 ∈ DivRing ↔ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))))

Proof of Theorem qsdrng

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

Step	Hyp	Ref	Expression
1		qsdrng.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ NzRing)
2		nzrring 20419	. . . . . 6 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
4	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → 𝑅 ∈ Ring)
5		qsdrng.2	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (2Ideal‘𝑅))
6	5	2idllidld 21179	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅))
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → 𝑀 ∈ (LIdeal‘𝑅))
8		drngnzr 20651	. . . . . . 7 ⊢ (𝑄 ∈ DivRing → 𝑄 ∈ NzRing)
9	8	ad2antlr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑀 = (Base‘𝑅)) → 𝑄 ∈ NzRing)
10		qsdrng.q	. . . . . . . . . . 11 ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))
11		eqid 2729	. . . . . . . . . . 11 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
12	10, 11	qusring 21200	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (2Ideal‘𝑅)) → 𝑄 ∈ Ring)
13	3, 5, 12	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ Ring)
14	13	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 = (Base‘𝑅)) → 𝑄 ∈ Ring)
15		oveq2 7361	. . . . . . . . . . . . . 14 ⊢ (𝑀 = (Base‘𝑅) → (𝑅 ~QG 𝑀) = (𝑅 ~QG (Base‘𝑅)))
16	15	oveq2d 7369	. . . . . . . . . . . . 13 ⊢ (𝑀 = (Base‘𝑅) → (𝑅 /s (𝑅 ~QG 𝑀)) = (𝑅 /s (𝑅 ~QG (Base‘𝑅))))
17	10, 16	eqtrid 2776	. . . . . . . . . . . 12 ⊢ (𝑀 = (Base‘𝑅) → 𝑄 = (𝑅 /s (𝑅 ~QG (Base‘𝑅))))
18	17	fveq2d 6830	. . . . . . . . . . 11 ⊢ (𝑀 = (Base‘𝑅) → (Base‘𝑄) = (Base‘(𝑅 /s (𝑅 ~QG (Base‘𝑅)))))
19	3	ringgrpd 20145	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ Grp)
20		eqid 2729	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
21		eqid 2729	. . . . . . . . . . . . 13 ⊢ (𝑅 /s (𝑅 ~QG (Base‘𝑅))) = (𝑅 /s (𝑅 ~QG (Base‘𝑅)))
22	20, 21	qustriv 33314	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Grp → (Base‘(𝑅 /s (𝑅 ~QG (Base‘𝑅)))) = {(Base‘𝑅)})
23	19, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘(𝑅 /s (𝑅 ~QG (Base‘𝑅)))) = {(Base‘𝑅)})
24	18, 23	sylan9eqr 2786	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 = (Base‘𝑅)) → (Base‘𝑄) = {(Base‘𝑅)})
25	24	fveq2d 6830	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = (♯‘{(Base‘𝑅)}))
26		fvex 6839	. . . . . . . . . 10 ⊢ (Base‘𝑅) ∈ V
27		hashsng 14294	. . . . . . . . . 10 ⊢ ((Base‘𝑅) ∈ V → (♯‘{(Base‘𝑅)}) = 1)
28	26, 27	ax-mp 5	. . . . . . . . 9 ⊢ (♯‘{(Base‘𝑅)}) = 1
29	25, 28	eqtrdi 2780	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = 1)
30		0ringnnzr 20428	. . . . . . . . 9 ⊢ (𝑄 ∈ Ring → ((♯‘(Base‘𝑄)) = 1 ↔ ¬ 𝑄 ∈ NzRing))
31	30	biimpa 476	. . . . . . . 8 ⊢ ((𝑄 ∈ Ring ∧ (♯‘(Base‘𝑄)) = 1) → ¬ 𝑄 ∈ NzRing)
32	14, 29, 31	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 = (Base‘𝑅)) → ¬ 𝑄 ∈ NzRing)
33	32	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑀 = (Base‘𝑅)) → ¬ 𝑄 ∈ NzRing)
34	9, 33	pm2.65da 816	. . . . 5 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → ¬ 𝑀 = (Base‘𝑅))
35	34	neqned 2932	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → 𝑀 ≠ (Base‘𝑅))
36		simplr 768	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑀 ⊆ 𝑗)
37		simpr 484	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → ¬ 𝑗 = 𝑀)
38	37	neqned 2932	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑗 ≠ 𝑀)
39	38	necomd 2980	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑀 ≠ 𝑗)
40		pssdifn0 4321	. . . . . . . . . . 11 ⊢ ((𝑀 ⊆ 𝑗 ∧ 𝑀 ≠ 𝑗) → (𝑗 ∖ 𝑀) ≠ ∅)
41	36, 39, 40	syl2anc 584	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → (𝑗 ∖ 𝑀) ≠ ∅)
42		n0 4306	. . . . . . . . . 10 ⊢ ((𝑗 ∖ 𝑀) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑗 ∖ 𝑀))
43	41, 42	sylib 218	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → ∃𝑥 𝑥 ∈ (𝑗 ∖ 𝑀))
44		qsdrng.0	. . . . . . . . . 10 ⊢ 𝑂 = (oppr‘𝑅)
45	1	ad5antr 734	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑅 ∈ NzRing)
46	5	ad5antr 734	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑀 ∈ (2Ideal‘𝑅))
47		simp-5r 785	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑄 ∈ DivRing)
48		simp-4r 783	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑗 ∈ (LIdeal‘𝑅))
49	36	adantr 480	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑀 ⊆ 𝑗)
50		simpr 484	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑥 ∈ (𝑗 ∖ 𝑀))
51	44, 10, 45, 46, 20, 47, 48, 49, 50	qsdrnglem2 33446	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑗 = (Base‘𝑅))
52	43, 51	exlimddv 1935	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑗 = (Base‘𝑅))
53	52	ex 412	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) → (¬ 𝑗 = 𝑀 → 𝑗 = (Base‘𝑅)))
54	53	orrd 863	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅)))
55	54	ex 412	. . . . 5 ⊢ (((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) → (𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
56	55	ralrimiva 3121	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
57	20	ismxidl 33412	. . . . 5 ⊢ (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))))
58	57	biimpar 477	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))) → 𝑀 ∈ (MaxIdeal‘𝑅))
59	4, 7, 35, 56, 58	syl13anc 1374	. . 3 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → 𝑀 ∈ (MaxIdeal‘𝑅))
60	44	opprring 20250	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring)
61	3, 60	syl 17	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ Ring)
62	61	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → 𝑂 ∈ Ring)
63	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → 𝑀 ∈ (2Ideal‘𝑅))
64	63, 44	2idlridld 21180	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → 𝑀 ∈ (LIdeal‘𝑂))
65		simplr 768	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑀 ⊆ 𝑗)
66		simpr 484	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → ¬ 𝑗 = 𝑀)
67	66	neqned 2932	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑗 ≠ 𝑀)
68	67	necomd 2980	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑀 ≠ 𝑗)
69	65, 68, 40	syl2anc 584	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → (𝑗 ∖ 𝑀) ≠ ∅)
70	69, 42	sylib 218	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → ∃𝑥 𝑥 ∈ (𝑗 ∖ 𝑀))
71		eqid 2729	. . . . . . . . . 10 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
72		eqid 2729	. . . . . . . . . 10 ⊢ (𝑂 /s (𝑂 ~QG 𝑀)) = (𝑂 /s (𝑂 ~QG 𝑀))
73	44	opprnzr 20425	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ NzRing → 𝑂 ∈ NzRing)
74	1, 73	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑂 ∈ NzRing)
75	74	ad5antr 734	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑂 ∈ NzRing)
76	44, 3	oppr2idl 33436	. . . . . . . . . . . 12 ⊢ (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂))
77	5, 76	eleqtrd 2830	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ (2Ideal‘𝑂))
78	77	ad5antr 734	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑀 ∈ (2Ideal‘𝑂))
79	44, 20	opprbas 20246	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑂)
80		eqid 2729	. . . . . . . . . . . . 13 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
81	80	opprdrng 20667	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ DivRing ↔ (oppr‘𝑄) ∈ DivRing)
82	20, 44, 10, 3, 5	opprqusdrng 33443	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((oppr‘𝑄) ∈ DivRing ↔ (𝑂 /s (𝑂 ~QG 𝑀)) ∈ DivRing))
83	82	biimpa 476	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (oppr‘𝑄) ∈ DivRing) → (𝑂 /s (𝑂 ~QG 𝑀)) ∈ DivRing)
84	81, 83	sylan2b 594	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → (𝑂 /s (𝑂 ~QG 𝑀)) ∈ DivRing)
85	84	ad4antr 732	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → (𝑂 /s (𝑂 ~QG 𝑀)) ∈ DivRing)
86		simp-4r 783	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑗 ∈ (LIdeal‘𝑂))
87	65	adantr 480	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑀 ⊆ 𝑗)
88		simpr 484	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑥 ∈ (𝑗 ∖ 𝑀))
89	71, 72, 75, 78, 79, 85, 86, 87, 88	qsdrnglem2 33446	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑗 = (Base‘𝑅))
90	70, 89	exlimddv 1935	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑗 = (Base‘𝑅))
91	90	ex 412	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) → (¬ 𝑗 = 𝑀 → 𝑗 = (Base‘𝑅)))
92	91	orrd 863	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅)))
93	92	ex 412	. . . . 5 ⊢ (((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) → (𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
94	93	ralrimiva 3121	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → ∀𝑗 ∈ (LIdeal‘𝑂)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
95	79	ismxidl 33412	. . . . 5 ⊢ (𝑂 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑂) ↔ (𝑀 ∈ (LIdeal‘𝑂) ∧ 𝑀 ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘𝑂)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))))
96	95	biimpar 477	. . . 4 ⊢ ((𝑂 ∈ Ring ∧ (𝑀 ∈ (LIdeal‘𝑂) ∧ 𝑀 ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘𝑂)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))) → 𝑀 ∈ (MaxIdeal‘𝑂))
97	62, 64, 35, 94, 96	syl13anc 1374	. . 3 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → 𝑀 ∈ (MaxIdeal‘𝑂))
98	59, 97	jca 511	. 2 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂)))
99	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))) → 𝑅 ∈ NzRing)
100		simprl 770	. . 3 ⊢ ((𝜑 ∧ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))) → 𝑀 ∈ (MaxIdeal‘𝑅))
101		simprr 772	. . 3 ⊢ ((𝜑 ∧ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))) → 𝑀 ∈ (MaxIdeal‘𝑂))
102	44, 10, 99, 100, 101	qsdrngi 33445	. 2 ⊢ ((𝜑 ∧ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))) → 𝑄 ∈ DivRing)
103	98, 102	impbida 800	1 ⊢ (𝜑 → (𝑄 ∈ DivRing ↔ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  Vcvv 3438   ∖ cdif 3902   ⊆ wss 3905  ∅c0 4286  {csn 4579  ‘cfv 6486  (class class class)co 7353  1c1 11029  ♯chash 14255  Basecbs 17138   /_s cqus 17427  Grpcgrp 18830   ~_QG cqg 19019  Ringcrg 20136  opp_rcoppr 20239  NzRingcnzr 20415  DivRingcdr 20632  LIdealclidl 21131  2Idealc2idl 21174  MaxIdealcmxidl 33409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-iin 4947  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  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 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-of 7617  df-om 7807  df-1st 7931  df-2nd 7932  df-supp 8101  df-tpos 8166  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-er 8632  df-ec 8634  df-qs 8638  df-map 8762  df-ixp 8832  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-fsupp 9271  df-sup 9351  df-inf 9352  df-oi 9421  df-dju 9816  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-3 12210  df-4 12211  df-5 12212  df-6 12213  df-7 12214  df-8 12215  df-9 12216  df-n0 12403  df-xnn0 12476  df-z 12490  df-dec 12610  df-uz 12754  df-fz 13429  df-fzo 13576  df-seq 13927  df-hash 14256  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-plusg 17192  df-mulr 17193  df-sca 17195  df-vsca 17196  df-ip 17197  df-tset 17198  df-ple 17199  df-ds 17201  df-hom 17203  df-cco 17204  df-0g 17363  df-gsum 17364  df-prds 17369  df-pws 17371  df-imas 17430  df-qus 17431  df-mre 17506  df-mrc 17507  df-acs 17509  df-mgm 18532  df-sgrp 18611  df-mnd 18627  df-mhm 18675  df-submnd 18676  df-grp 18833  df-minusg 18834  df-sbg 18835  df-mulg 18965  df-subg 19020  df-nsg 19021  df-eqg 19022  df-ghm 19110  df-cntz 19214  df-oppg 19243  df-lsm 19533  df-cmn 19679  df-abl 19680  df-mgp 20044  df-rng 20056  df-ur 20085  df-ring 20138  df-oppr 20240  df-dvdsr 20260  df-unit 20261  df-invr 20291  df-nzr 20416  df-subrg 20473  df-drng 20634  df-lmod 20783  df-lss 20853  df-lsp 20893  df-lmhm 20944  df-lbs 20997  df-sra 21095  df-rgmod 21096  df-lidl 21133  df-rsp 21134  df-2idl 21175  df-dsmm 21657  df-frlm 21672  df-uvc 21708  df-mxidl 33410

This theorem is referenced by:  qsfld  33448