Theorem qsdrng 33525

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 20516	. . . . . 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 21264	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅))
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → 𝑀 ∈ (LIdeal‘𝑅))
8		drngnzr 20748	. . . . . . 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 2737	. . . . . . . . . . 11 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
12	10, 11	qusring 21285	. . . . . . . . . 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 7439	. . . . . . . . . . . . . 14 ⊢ (𝑀 = (Base‘𝑅) → (𝑅 ~QG 𝑀) = (𝑅 ~QG (Base‘𝑅)))
16	15	oveq2d 7447	. . . . . . . . . . . . 13 ⊢ (𝑀 = (Base‘𝑅) → (𝑅 /s (𝑅 ~QG 𝑀)) = (𝑅 /s (𝑅 ~QG (Base‘𝑅))))
17	10, 16	eqtrid 2789	. . . . . . . . . . . 12 ⊢ (𝑀 = (Base‘𝑅) → 𝑄 = (𝑅 /s (𝑅 ~QG (Base‘𝑅))))
18	17	fveq2d 6910	. . . . . . . . . . 11 ⊢ (𝑀 = (Base‘𝑅) → (Base‘𝑄) = (Base‘(𝑅 /s (𝑅 ~QG (Base‘𝑅)))))
19	3	ringgrpd 20239	. . . . . . . . . . . 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 33392	. . . . . . . . . . . 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 2799	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 = (Base‘𝑅)) → (Base‘𝑄) = {(Base‘𝑅)})
25	24	fveq2d 6910	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = (♯‘{(Base‘𝑅)}))
26		fvex 6919	. . . . . . . . . 10 ⊢ (Base‘𝑅) ∈ V
27		hashsng 14408	. . . . . . . . . 10 ⊢ ((Base‘𝑅) ∈ V → (♯‘{(Base‘𝑅)}) = 1)
28	26, 27	ax-mp 5	. . . . . . . . 9 ⊢ (♯‘{(Base‘𝑅)}) = 1
29	25, 28	eqtrdi 2793	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = 1)
30		0ringnnzr 20525	. . . . . . . . 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 817	. . . . 5 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → ¬ 𝑀 = (Base‘𝑅))
35	34	neqned 2947	. . . 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 2947	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑗 ≠ 𝑀)
39	38	necomd 2996	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑀 ≠ 𝑗)
40		pssdifn0 4368	. . . . . . . . . . 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 4353	. . . . . . . . . 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 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 33524	. . . . . . . . 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 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 3146	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
57	20	ismxidl 33490	. . . . 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 20347	. . . . . 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 21265	. . . 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 2947	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑗 ≠ 𝑀)
68	67	necomd 2996	. . . . . . . . . . 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 2737	. . . . . . . . . 10 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
72		eqid 2737	. . . . . . . . . 10 ⊢ (𝑂 /s (𝑂 ~QG 𝑀)) = (𝑂 /s (𝑂 ~QG 𝑀))
73	44	opprnzr 20522	. . . . . . . . . . . 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 33514	. . . . . . . . . . . 12 ⊢ (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂))
77	5, 76	eleqtrd 2843	. . . . . . . . . . 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 20341	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑂)
80		eqid 2737	. . . . . . . . . . . . 13 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
81	80	opprdrng 20764	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ DivRing ↔ (oppr‘𝑄) ∈ DivRing)
82	20, 44, 10, 3, 5	opprqusdrng 33521	. . . . . . . . . . . . 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 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 33524	. . . . . . . . 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 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 3146	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → ∀𝑗 ∈ (LIdeal‘𝑂)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
95	79	ismxidl 33490	. . . . 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 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 33523	. 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 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  Vcvv 3480   ∖ cdif 3948   ⊆ wss 3951  ∅c0 4333  {csn 4626  ‘cfv 6561  (class class class)co 7431  1c1 11156  ♯chash 14369  Basecbs 17247   /_s cqus 17550  Grpcgrp 18951   ~_QG cqg 19140  Ringcrg 20230  opp_rcoppr 20333  NzRingcnzr 20512  DivRingcdr 20729  LIdealclidl 21216  2Idealc2idl 21259  MaxIdealcmxidl 33487

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-ec 8747  df-qs 8751  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-xnn0 12600  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-imas 17553  df-qus 17554  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-nsg 19142  df-eqg 19143  df-ghm 19231  df-cntz 19335  df-oppg 19364  df-lsm 19654  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-nzr 20513  df-subrg 20570  df-drng 20731  df-lmod 20860  df-lss 20930  df-lsp 20970  df-lmhm 21021  df-lbs 21074  df-sra 21172  df-rgmod 21173  df-lidl 21218  df-rsp 21219  df-2idl 21260  df-dsmm 21752  df-frlm 21767  df-uvc 21803  df-mxidl 33488

This theorem is referenced by:  qsfld  33526