Theorem qsdrng 33907

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 20682	. . . . . 6 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
3	1, 2	syl 18	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
4	3	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → 𝑅 ∈ Ring)
5		qsdrng.2	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (2Ideal‘𝑅))
6	5	2idllidld 21462	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅))
7	6	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → 𝑀 ∈ (LIdeal‘𝑅))
8		drngnzr 20917	. . . . . . 7 ⊢ (𝑄 ∈ DivRing → 𝑄 ∈ NzRing)
9	8	ad2antlr 740	. . . . . 6 ⊢ (((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑀 = (Base‘𝑅)) → 𝑄 ∈ NzRing)
10		qsdrng.q	. . . . . . . . . . 11 ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))
11		eqid 2762	. . . . . . . . . . 11 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
12	10, 11	qusring 21483	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (2Ideal‘𝑅)) → 𝑄 ∈ Ring)
13	3, 5, 12	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ Ring)
14	13	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 = (Base‘𝑅)) → 𝑄 ∈ Ring)
15		oveq2 7425	. . . . . . . . . . . . . 14 ⊢ (𝑀 = (Base‘𝑅) → (𝑅 ~QG 𝑀) = (𝑅 ~QG (Base‘𝑅)))
16	15	oveq2d 7433	. . . . . . . . . . . . 13 ⊢ (𝑀 = (Base‘𝑅) → (𝑅 /s (𝑅 ~QG 𝑀)) = (𝑅 /s (𝑅 ~QG (Base‘𝑅))))
17	10, 16	eqtrid 2809	. . . . . . . . . . . 12 ⊢ (𝑀 = (Base‘𝑅) → 𝑄 = (𝑅 /s (𝑅 ~QG (Base‘𝑅))))
18	17	fveq2d 6886	. . . . . . . . . . 11 ⊢ (𝑀 = (Base‘𝑅) → (Base‘𝑄) = (Base‘(𝑅 /s (𝑅 ~QG (Base‘𝑅)))))
19	3	ringgrpd 20387	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ Grp)
20		eqid 2762	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
21		eqid 2762	. . . . . . . . . . . . 13 ⊢ (𝑅 /s (𝑅 ~QG (Base‘𝑅))) = (𝑅 /s (𝑅 ~QG (Base‘𝑅)))
22	20, 21	qustriv 19315	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Grp → (Base‘(𝑅 /s (𝑅 ~QG (Base‘𝑅)))) = {(Base‘𝑅)})
23	19, 22	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘(𝑅 /s (𝑅 ~QG (Base‘𝑅)))) = {(Base‘𝑅)})
24	18, 23	sylan9eqr 2819	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 = (Base‘𝑅)) → (Base‘𝑄) = {(Base‘𝑅)})
25	24	fveq2d 6886	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = (♯‘{(Base‘𝑅)}))
26		fvex 6895	. . . . . . . . . 10 ⊢ (Base‘𝑅) ∈ V
27		hashsng 14437	. . . . . . . . . 10 ⊢ ((Base‘𝑅) ∈ V → (♯‘{(Base‘𝑅)}) = 1)
28	26, 27	ax-mp 5	. . . . . . . . 9 ⊢ (♯‘{(Base‘𝑅)}) = 1
29	25, 28	eqtrdi 2813	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = 1)
30		0ringnnzr 20692	. . . . . . . . 9 ⊢ (𝑄 ∈ Ring → ((♯‘(Base‘𝑄)) = 1 ↔ ¬ 𝑄 ∈ NzRing))
31	30	biimpa 482	. . . . . . . 8 ⊢ ((𝑄 ∈ Ring ∧ (♯‘(Base‘𝑄)) = 1) → ¬ 𝑄 ∈ NzRing)
32	14, 29, 31	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 = (Base‘𝑅)) → ¬ 𝑄 ∈ NzRing)
33	32	adantlr 728	. . . . . 6 ⊢ (((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑀 = (Base‘𝑅)) → ¬ 𝑄 ∈ NzRing)
34	9, 33	pm2.65da 829	. . . . 5 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → ¬ 𝑀 = (Base‘𝑅))
35	34	neqned 2964	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → 𝑀 ≠ (Base‘𝑅))
36		simplr 781	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑀 ⊆ 𝑗)
37		simpr 490	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → ¬ 𝑗 = 𝑀)
38	37	neqned 2964	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑗 ≠ 𝑀)
39	38	necomd 3012	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑀 ≠ 𝑗)
40		pssdifn0 4319	. . . . . . . . . . 11 ⊢ ((𝑀 ⊆ 𝑗 ∧ 𝑀 ≠ 𝑗) → (𝑗 ∖ 𝑀) ≠ ∅)
41	36, 39, 40	syl2anc 596	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → (𝑗 ∖ 𝑀) ≠ ∅)
42		n0 4303	. . . . . . . . . 10 ⊢ ((𝑗 ∖ 𝑀) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑗 ∖ 𝑀))
43	41, 42	sylib 221	. . . . . . . . 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 747	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑅 ∈ NzRing)
46	5	ad5antr 747	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑀 ∈ (2Ideal‘𝑅))
47		simp-5r 798	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑄 ∈ DivRing)
48		simp-4r 796	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑗 ∈ (LIdeal‘𝑅))
49	36	adantr 486	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑀 ⊆ 𝑗)
50		simpr 490	. . . . . . . . . 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 33906	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑗 = (Base‘𝑅))
52	43, 51	exlimddv 1968	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑗 = (Base‘𝑅))
53	52	ex 418	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) → (¬ 𝑗 = 𝑀 → 𝑗 = (Base‘𝑅)))
54	53	orrd 877	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅)))
55	54	ex 418	. . . . 5 ⊢ (((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) → (𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
56	55	ralrimiva 3156	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
57	20	ismxidl 33873	. . . . 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 483	. . . 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 1399	. . 3 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → 𝑀 ∈ (MaxIdeal‘𝑅))
60	44	opprring 20494	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring)
61	3, 60	syl 18	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ Ring)
62	61	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → 𝑂 ∈ Ring)
63	5	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → 𝑀 ∈ (2Ideal‘𝑅))
64	63, 44	2idlridld 21463	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → 𝑀 ∈ (LIdeal‘𝑂))
65		simplr 781	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑀 ⊆ 𝑗)
66		simpr 490	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → ¬ 𝑗 = 𝑀)
67	66	neqned 2964	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑗 ≠ 𝑀)
68	67	necomd 3012	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑀 ≠ 𝑗)
69	65, 68, 40	syl2anc 596	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → (𝑗 ∖ 𝑀) ≠ ∅)
70	69, 42	sylib 221	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → ∃𝑥 𝑥 ∈ (𝑗 ∖ 𝑀))
71		eqid 2762	. . . . . . . . . 10 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
72		eqid 2762	. . . . . . . . . 10 ⊢ (𝑂 /s (𝑂 ~QG 𝑀)) = (𝑂 /s (𝑂 ~QG 𝑀))
73	44	opprnzr 20689	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ NzRing → 𝑂 ∈ NzRing)
74	1, 73	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑂 ∈ NzRing)
75	74	ad5antr 747	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑂 ∈ NzRing)
76	44, 3	oppr2idl 33896	. . . . . . . . . . . 12 ⊢ (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂))
77	5, 76	eleqtrd 2864	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ (2Ideal‘𝑂))
78	77	ad5antr 747	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑀 ∈ (2Ideal‘𝑂))
79	44, 20	opprbas 20490	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑂)
80		eqid 2762	. . . . . . . . . . . . 13 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
81	80	opprdrng 20936	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ DivRing ↔ (oppr‘𝑄) ∈ DivRing)
82	20, 44, 10, 3, 5	opprqusdrng 33903	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((oppr‘𝑄) ∈ DivRing ↔ (𝑂 /s (𝑂 ~QG 𝑀)) ∈ DivRing))
83	82	biimpa 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (oppr‘𝑄) ∈ DivRing) → (𝑂 /s (𝑂 ~QG 𝑀)) ∈ DivRing)
84	81, 83	sylan2b 606	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → (𝑂 /s (𝑂 ~QG 𝑀)) ∈ DivRing)
85	84	ad4antr 745	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → (𝑂 /s (𝑂 ~QG 𝑀)) ∈ DivRing)
86		simp-4r 796	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑗 ∈ (LIdeal‘𝑂))
87	65	adantr 486	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑀 ⊆ 𝑗)
88		simpr 490	. . . . . . . . . 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 33906	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑗 = (Base‘𝑅))
90	70, 89	exlimddv 1968	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑗 = (Base‘𝑅))
91	90	ex 418	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) → (¬ 𝑗 = 𝑀 → 𝑗 = (Base‘𝑅)))
92	91	orrd 877	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅)))
93	92	ex 418	. . . . 5 ⊢ (((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) → (𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
94	93	ralrimiva 3156	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → ∀𝑗 ∈ (LIdeal‘𝑂)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
95	79	ismxidl 33873	. . . . 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 483	. . . 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 1399	. . 3 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → 𝑀 ∈ (MaxIdeal‘𝑂))
98	59, 97	jca 521	. 2 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂)))
99	1	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))) → 𝑅 ∈ NzRing)
100		simprl 783	. . 3 ⊢ ((𝜑 ∧ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))) → 𝑀 ∈ (MaxIdeal‘𝑅))
101		simprr 785	. . 3 ⊢ ((𝜑 ∧ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))) → 𝑀 ∈ (MaxIdeal‘𝑂))
102	44, 10, 99, 100, 101	qsdrngi 33905	. 2 ⊢ ((𝜑 ∧ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))) → 𝑄 ∈ DivRing)
103	98, 102	impbida 813	1 ⊢ (𝜑 → (𝑄 ∈ DivRing ↔ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  Vcvv 3453   ∖ cdif 3899   ⊆ wss 3902  ∅c0 4282  {csn 4587  ‘cfv 6537  (class class class)co 7417  1c1 11129  ♯chash 14398  Basecbs 17307   /_s cqus 17597  Grpcgrp 19063   ~_QG cqg 19251  Ringcrg 20378  opp_rcoppr 20483  NzRingcnzr 20678  DivRingcdr 20896  LIdealclidl 21399  2Idealc2idl 21457  MaxIdealcmxidl 33870

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  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 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-of 7682  df-om 7867  df-1st 7990  df-2nd 7991  df-supp 8163  df-tpos 8228  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-oadd 8463  df-er 8700  df-ec 8702  df-qs 8706  df-map 8832  df-ixp 8909  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-fsupp 9336  df-sup 9416  df-inf 9417  df-oi 9486  df-dju 9910  df-card 9948  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-2 12331  df-3 12332  df-4 12333  df-5 12334  df-6 12335  df-7 12336  df-8 12337  df-9 12338  df-n0 12533  df-xnn0 12606  df-z 12620  df-dec 12741  df-uz 12892  df-fz 13566  df-fzo 13714  df-seq 14070  df-hash 14399  df-struct 17245  df-sets 17262  df-slot 17280  df-ndx 17292  df-base 17308  df-ress 17329  df-plusg 17361  df-mulr 17362  df-sca 17364  df-vsca 17365  df-ip 17366  df-tset 17367  df-ple 17368  df-ds 17370  df-hom 17372  df-cco 17373  df-0g 17532  df-gsum 17533  df-prds 17538  df-pws 17540  df-imas 17600  df-qus 17601  df-mre 17676  df-mrc 17677  df-acs 17679  df-mgm 18736  df-sgrp 18827  df-mnd 18843  df-mhm 18897  df-submnd 18898  df-grp 19066  df-minusg 19067  df-sbg 19068  df-mulg 19197  df-subg 19252  df-nsg 19253  df-eqg 19254  df-ghm 19347  df-cntz 19450  df-oppg 19479  df-lsm 19769  df-cmn 19915  df-abl 19916  df-mgp 20280  df-rng 20294  df-ur 20327  df-ring 20380  df-oppr 20484  df-dvdsr 20504  df-unit 20505  df-invr 20535  df-nzr 20679  df-subrg 20738  df-drng 20898  df-lmod 21052  df-lss 21122  df-lsp 21162  df-lmhm 21212  df-lbs 21265  df-sra 21363  df-rgmod 21364  df-lidl 21401  df-rsp 21402  df-2idl 21458  df-dsmm 21951  df-frlm 21966  df-uvc 22002  df-mxidl 33871

This theorem is used by:  qsfld  33908