Theorem qsdrng 34091

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 20796	. . . . . 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 21577	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅))
7	6	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → 𝑀 ∈ (LIdeal‘𝑅))
8		drngnzr 21032	. . . . . . 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 2763	. . . . . . . . . . 11 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
12	10, 11	qusring 21599	. . . . . . . . . 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 7451	. . . . . . . . . . . . . 14 ⊢ (𝑀 = (Base‘𝑅) → (𝑅 ~QG 𝑀) = (𝑅 ~QG (Base‘𝑅)))
16	15	oveq2d 7459	. . . . . . . . . . . . 13 ⊢ (𝑀 = (Base‘𝑅) → (𝑅 /s (𝑅 ~QG 𝑀)) = (𝑅 /s (𝑅 ~QG (Base‘𝑅))))
17	10, 16	eqtrid 2810	. . . . . . . . . . . 12 ⊢ (𝑀 = (Base‘𝑅) → 𝑄 = (𝑅 /s (𝑅 ~QG (Base‘𝑅))))
18	17	fveq2d 6912	. . . . . . . . . . 11 ⊢ (𝑀 = (Base‘𝑅) → (Base‘𝑄) = (Base‘(𝑅 /s (𝑅 ~QG (Base‘𝑅)))))
19	3	ringgrpd 20499	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ Grp)
20		eqid 2763	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
21		eqid 2763	. . . . . . . . . . . . 13 ⊢ (𝑅 /s (𝑅 ~QG (Base‘𝑅))) = (𝑅 /s (𝑅 ~QG (Base‘𝑅)))
22	20, 21	qustriv 19425	. . . . . . . . . . . 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 2820	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 = (Base‘𝑅)) → (Base‘𝑄) = {(Base‘𝑅)})
25	24	fveq2d 6912	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = (♯‘{(Base‘𝑅)}))
26		fvex 6921	. . . . . . . . . 10 ⊢ (Base‘𝑅) ∈ V
27		hashsng 14540	. . . . . . . . . 10 ⊢ ((Base‘𝑅) ∈ V → (♯‘{(Base‘𝑅)}) = 1)
28	26, 27	ax-mp 5	. . . . . . . . 9 ⊢ (♯‘{(Base‘𝑅)}) = 1
29	25, 28	eqtrdi 2814	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = 1)
30		0ringnnzr 20806	. . . . . . . . 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 2966	. . . 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 2966	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑗 ≠ 𝑀)
39	38	necomd 3014	. . . . . . . . . . 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 34090	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑗 = (Base‘𝑅))
52	43, 51	exlimddv 1970	. . . . . . . 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 3158	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
57	20	ismxidl 34057	. . . . 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 1400	. . 3 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → 𝑀 ∈ (MaxIdeal‘𝑅))
60	44	opprring 20607	. . . . . 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 21578	. . . 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 2966	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) → 𝑗 ≠ 𝑀)
68	67	necomd 3014	. . . . . . . . . . 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 2763	. . . . . . . . . 10 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
72		eqid 2763	. . . . . . . . . 10 ⊢ (𝑂 /s (𝑂 ~QG 𝑀)) = (𝑂 /s (𝑂 ~QG 𝑀))
73	44	opprnzr 20803	. . . . . . . . . . . 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 34080	. . . . . . . . . . . 12 ⊢ (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂))
77	5, 76	eleqtrd 2866	. . . . . . . . . . 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 20603	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑂)
80		eqid 2763	. . . . . . . . . . . . 13 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
81	80	opprdrng 21051	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ DivRing ↔ (oppr‘𝑄) ∈ DivRing)
82	20, 44, 10, 3, 5	opprqusdrng 34087	. . . . . . . . . . . . 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 34090	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑄 ∈ DivRing) ∧ 𝑗 ∈ (LIdeal‘𝑂)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = 𝑀) ∧ 𝑥 ∈ (𝑗 ∖ 𝑀)) → 𝑗 = (Base‘𝑅))
90	70, 89	exlimddv 1970	. . . . . . . 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 3158	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ DivRing) → ∀𝑗 ∈ (LIdeal‘𝑂)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
95	79	ismxidl 34057	. . . . 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 1400	. . 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 34089	. 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 2147   ≠ wne 2959  ∀wral 3080  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048  ∅c0 4281  {csn 4588  ‘cfv 6558  (class class class)co 7443  1c1 11228  ♯chash 14501  Basecbs 17416   /_s cqus 17706  Grpcgrp 19173   ~_QG cqg 19361  Ringcrg 20489  opp_rcoppr 20596  NzRingcnzr 20792  DivRingcdr 21010  LIdealclidl 21514  2Idealc2idl 21572  MaxIdealcmxidl 34054

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-of 7717  df-om 7902  df-1st 8025  df-2nd 8026  df-supp 8205  df-tpos 8271  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8745  df-ec 8747  df-qs 8751  df-map 8877  df-ixp 8954  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-fsupp 9382  df-sup 9461  df-inf 9462  df-oi 9531  df-dju 10009  df-card 10047  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-5 12435  df-6 12436  df-7 12437  df-8 12438  df-9 12439  df-n0 12634  df-xnn0 12707  df-z 12721  df-dec 12842  df-uz 12993  df-fz 13667  df-fzo 13816  df-seq 14172  df-hash 14502  df-struct 17354  df-sets 17371  df-slot 17389  df-ndx 17401  df-base 17417  df-ress 17438  df-plusg 17470  df-mulr 17471  df-sca 17473  df-vsca 17474  df-ip 17475  df-tset 17476  df-ple 17477  df-ds 17479  df-hom 17481  df-cco 17482  df-0g 17641  df-gsum 17642  df-prds 17647  df-pws 17649  df-imas 17709  df-qus 17710  df-mre 17785  df-mrc 17786  df-acs 17788  df-mgm 18845  df-sgrp 18937  df-mnd 18953  df-mhm 19007  df-submnd 19008  df-grp 19176  df-minusg 19177  df-sbg 19178  df-mulg 19307  df-subg 19362  df-nsg 19363  df-eqg 19364  df-ghm 19457  df-cntz 19561  df-oppg 19590  df-lsm 19880  df-cmn 20026  df-abl 20027  df-mgp 20391  df-rng 20405  df-ur 20438  df-ring 20491  df-oppr 20597  df-dvdsr 20617  df-unit 20618  df-invr 20648  df-nzr 20793  df-subrg 20852  df-drng 21012  df-lmod 21167  df-lss 21237  df-lsp 21277  df-lmhm 21327  df-lbs 21380  df-sra 21478  df-rgmod 21479  df-lidl 21516  df-rsp 21517  df-2idl 21573  df-dsmm 22068  df-frlm 22083  df-uvc 22119  df-mxidl 34055

This theorem is used by:  qsfld  34092