Theorem qsdrng 33572

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 20484	. . . . . 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 21244	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅))
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → 𝑀 ∈ (LIdeal‘𝑅))
8		drngnzr 20716	. . . . . . 7 ⊢ (𝑄 ∈ DivRing → 𝑄 ∈ NzRing)
9	8	ad2antlr 728	. . . . . 6 ⊢ (((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑀 = (Base‘𝑅)) → 𝑄 ∈ NzRing)
10		qsdrng.q	. . . . . . . . . . 11 ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))
11		eqid 2737	. . . . . . . . . . 11 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
12	10, 11	qusring 21265	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (2Ideal‘𝑅)) → 𝑄 ∈ Ring)
13	3, 5, 12	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ Ring)
14	13	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 = (Base‘𝑅)) → 𝑄 ∈ Ring)
15		oveq2 7368	. . . . . . . . . . . . . 14 ⊢ (𝑀 = (Base‘𝑅) → (𝑅 ~QG 𝑀) = (𝑅 ~QG (Base‘𝑅)))
16	15	oveq2d 7376	. . . . . . . . . . . . 13 ⊢ (𝑀 = (Base‘𝑅) → (𝑅 /s (𝑅 ~QG 𝑀)) = (𝑅 /s (𝑅 ~QG (Base‘𝑅))))
17	10, 16	eqtrid 2784	. . . . . . . . . . . 12 ⊢ (𝑀 = (Base‘𝑅) → 𝑄 = (𝑅 /s (𝑅 ~QG (Base‘𝑅))))
18	17	fveq2d 6838	. . . . . . . . . . 11 ⊢ (𝑀 = (Base‘𝑅) → (Base‘𝑄) = (Base‘(𝑅 /s (𝑅 ~QG (Base‘𝑅)))))
19	3	ringgrpd 20214	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ Grp)
20		eqid 2737	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
21		eqid 2737	. . . . . . . . . . . . 13 ⊢ (𝑅 /s (𝑅 ~QG (Base‘𝑅))) = (𝑅 /s (𝑅 ~QG (Base‘𝑅)))
22	20, 21	qustriv 33439	. . . . . . . . . . . 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 2794	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 = (Base‘𝑅)) → (Base‘𝑄) = {(Base‘𝑅)})
25	24	fveq2d 6838	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = (♯‘{(Base‘𝑅)}))
26		fvex 6847	. . . . . . . . . 10 ⊢ (Base‘𝑅) ∈ V
27		hashsng 14322	. . . . . . . . . 10 ⊢ ((Base‘𝑅) ∈ V → (♯‘{(Base‘𝑅)}) = 1)
28	26, 27	ax-mp 5	. . . . . . . . 9 ⊢ (♯‘{(Base‘𝑅)}) = 1
29	25, 28	eqtrdi 2788	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = 1)
30		0ringnnzr 20493	. . . . . . . . 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 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 = (Base‘𝑅)) → ¬ 𝑄 ∈ NzRing)
33	32	adantlr 716	. . . . . 6 ⊢ (((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑀 = (Base‘𝑅)) → ¬ 𝑄 ∈ NzRing)
34	9, 33	pm2.65da 817	. . . . 5 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → ¬ 𝑀 = (Base‘𝑅))
35	34	neqned 2940	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → 𝑀 ≠ (Base‘𝑅))
36		simplr 769	. . . . . . . . . . 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 2940	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑗 ≠ 𝑀)
39	38	necomd 2988	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑀 ≠ 𝑗)
40		pssdifn0 4309	. . . . . . . . . . 11 ⊢ ((𝑀 ⊆ 𝑗 ∧ 𝑀 ≠ 𝑗) → (𝑗 ∖ 𝑀) ≠ ∅)
41	36, 39, 40	syl2anc 585	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → (𝑗 ∖ 𝑀) ≠ ∅)
42		n0 4294	. . . . . . . . . 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 735	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑅 ∈ NzRing)
46	5	ad5antr 735	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑀 ∈ (2Ideal‘𝑅))
47		simp-5r 786	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑄 ∈ DivRing)
48		simp-4r 784	. . . . . . . . . 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 33571	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑗 = (Base‘𝑅))
52	43, 51	exlimddv 1937	. . . . . . . 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 864	. . . . . 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 3130	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
57	20	ismxidl 33537	. . . . 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 1375	. . 3 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → 𝑀 ∈ (MaxIdeal‘𝑅))
60	44	opprring 20318	. . . . . 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 21245	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → 𝑀 ∈ (LIdeal‘𝑂))
65		simplr 769	. . . . . . . . . . 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 2940	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑗 ≠ 𝑀)
68	67	necomd 2988	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑀 ≠ 𝑗)
69	65, 68, 40	syl2anc 585	. . . . . . . . . 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 2737	. . . . . . . . . 10 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
72		eqid 2737	. . . . . . . . . 10 ⊢ (𝑂 /s (𝑂 ~QG 𝑀)) = (𝑂 /s (𝑂 ~QG 𝑀))
73	44	opprnzr 20490	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ NzRing → 𝑂 ∈ NzRing)
74	1, 73	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑂 ∈ NzRing)
75	74	ad5antr 735	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑂 ∈ NzRing)
76	44, 3	oppr2idl 33561	. . . . . . . . . . . 12 ⊢ (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂))
77	5, 76	eleqtrd 2839	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ (2Ideal‘𝑂))
78	77	ad5antr 735	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑀 ∈ (2Ideal‘𝑂))
79	44, 20	opprbas 20314	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑂)
80		eqid 2737	. . . . . . . . . . . . 13 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
81	80	opprdrng 20732	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ DivRing ↔ (oppr‘𝑄) ∈ DivRing)
82	20, 44, 10, 3, 5	opprqusdrng 33568	. . . . . . . . . . . . 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 595	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → (𝑂 /s (𝑂 ~QG 𝑀)) ∈ DivRing)
85	84	ad4antr 733	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → (𝑂 /s (𝑂 ~QG 𝑀)) ∈ DivRing)
86		simp-4r 784	. . . . . . . . . 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 33571	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑗 = (Base‘𝑅))
90	70, 89	exlimddv 1937	. . . . . . . 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 864	. . . . . 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 3130	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → ∀𝑗 ∈ (LIdeal‘𝑂)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
95	79	ismxidl 33537	. . . . 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 1375	. . 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 771	. . 3 ⊢ ((𝜑 ∧ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))) → 𝑀 ∈ (MaxIdeal‘𝑅))
101		simprr 773	. . 3 ⊢ ((𝜑 ∧ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))) → 𝑀 ∈ (MaxIdeal‘𝑂))
102	44, 10, 99, 100, 101	qsdrngi 33570	. 2 ⊢ ((𝜑 ∧ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))) → 𝑄 ∈ DivRing)
103	98, 102	impbida 801	1 ⊢ (𝜑 → (𝑄 ∈ DivRing ↔ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  Vcvv 3430   ∖ cdif 3887   ⊆ wss 3890  ∅c0 4274  {csn 4568  ‘cfv 6492  (class class class)co 7360  1c1 11030  ♯chash 14283  Basecbs 17170   /_s cqus 17460  Grpcgrp 18900   ~_QG cqg 19089  Ringcrg 20205  opp_rcoppr 20307  NzRingcnzr 20480  DivRingcdr 20697  LIdealclidl 21196  2Idealc2idl 21239  MaxIdealcmxidl 33534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8104  df-tpos 8169  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-oadd 8402  df-er 8636  df-ec 8638  df-qs 8642  df-map 8768  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9268  df-sup 9348  df-inf 9349  df-oi 9418  df-dju 9816  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-xnn0 12502  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-fzo 13600  df-seq 13955  df-hash 14284  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-sca 17227  df-vsca 17228  df-ip 17229  df-tset 17230  df-ple 17231  df-ds 17233  df-hom 17235  df-cco 17236  df-0g 17395  df-gsum 17396  df-prds 17401  df-pws 17403  df-imas 17463  df-qus 17464  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18742  df-submnd 18743  df-grp 18903  df-minusg 18904  df-sbg 18905  df-mulg 19035  df-subg 19090  df-nsg 19091  df-eqg 19092  df-ghm 19179  df-cntz 19283  df-oppg 19312  df-lsm 19602  df-cmn 19748  df-abl 19749  df-mgp 20113  df-rng 20125  df-ur 20154  df-ring 20207  df-oppr 20308  df-dvdsr 20328  df-unit 20329  df-invr 20359  df-nzr 20481  df-subrg 20538  df-drng 20699  df-lmod 20848  df-lss 20918  df-lsp 20958  df-lmhm 21009  df-lbs 21062  df-sra 21160  df-rgmod 21161  df-lidl 21198  df-rsp 21199  df-2idl 21240  df-dsmm 21722  df-frlm 21737  df-uvc 21773  df-mxidl 33535

This theorem is referenced by:  qsfld  33573