Theorem qsdrngi 32519

Description: A quotient by a maximal left and maximal right ideal is a division ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)

Hypotheses

Ref	Expression
qsdrng.0	⊢ 𝑂 = (oppr‘𝑅)
qsdrng.q	⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))
qsdrng.r	⊢ (𝜑 → 𝑅 ∈ NzRing)
qsdrngi.1	⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))
qsdrngi.2	⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑂))

Assertion

Ref	Expression
qsdrngi	⊢ (𝜑 → 𝑄 ∈ DivRing)

Proof of Theorem qsdrngi

Dummy variables 𝑟 𝑢 𝑣 𝑥 𝑠 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qsdrng.q	. . . 4 ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))
2		eqid 2732	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		qsdrng.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ NzRing)
4		nzrring 20247	. . . . 5 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
6		qsdrngi.1	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))
7	2	mxidlidl 32494	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
8	5, 6, 7	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅))
9		qsdrng.0	. . . . . . . . 9 ⊢ 𝑂 = (oppr‘𝑅)
10	9	opprring 20115	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring)
11	5, 10	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ Ring)
12		qsdrngi.2	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑂))
13		eqid 2732	. . . . . . . 8 ⊢ (Base‘𝑂) = (Base‘𝑂)
14	13	mxidlidl 32494	. . . . . . 7 ⊢ ((𝑂 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑂)) → 𝑀 ∈ (LIdeal‘𝑂))
15	11, 12, 14	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑂))
16	8, 15	elind 4191	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ((LIdeal‘𝑅) ∩ (LIdeal‘𝑂)))
17		eqid 2732	. . . . . 6 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
18		eqid 2732	. . . . . 6 ⊢ (LIdeal‘𝑂) = (LIdeal‘𝑂)
19		eqid 2732	. . . . . 6 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
20	17, 9, 18, 19	2idlval 20806	. . . . 5 ⊢ (2Ideal‘𝑅) = ((LIdeal‘𝑅) ∩ (LIdeal‘𝑂))
21	16, 20	eleqtrrdi 2844	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (2Ideal‘𝑅))
22	2	mxidlnr 32495	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ (Base‘𝑅))
23	5, 6, 22	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝑀 ≠ (Base‘𝑅))
24	1, 2, 5, 3, 21, 23	qsnzr 32489	. . 3 ⊢ (𝜑 → 𝑄 ∈ NzRing)
25		eqid 2732	. . . 4 ⊢ (1r‘𝑄) = (1r‘𝑄)
26		eqid 2732	. . . 4 ⊢ (0g‘𝑄) = (0g‘𝑄)
27	25, 26	nzrnz 20246	. . 3 ⊢ (𝑄 ∈ NzRing → (1r‘𝑄) ≠ (0g‘𝑄))
28	24, 27	syl 17	. 2 ⊢ (𝜑 → (1r‘𝑄) ≠ (0g‘𝑄))
29		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑄) = (Base‘𝑄)
30		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑄) = (.r‘𝑄)
31		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (Unit‘𝑄) = (Unit‘𝑄)
32	1, 19	qusring 20811	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (2Ideal‘𝑅)) → 𝑄 ∈ Ring)
33	5, 21, 32	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑄 ∈ Ring)
34	33	ad10antr 742	. . . . . . . . . . . . . . 15 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) → 𝑄 ∈ Ring)
35	34	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → 𝑄 ∈ Ring)
36		eldifi 4123	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)}) → 𝑢 ∈ (Base‘𝑄))
37	36	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → 𝑢 ∈ (Base‘𝑄))
38	37	ad10antr 742	. . . . . . . . . . . . . 14 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → 𝑢 ∈ (Base‘𝑄))
39		ovex 7427	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ~QG 𝑀) ∈ V
40	39	ecelqsi 8752	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ (Base‘𝑅) → [𝑟](𝑅 ~QG 𝑀) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
41	40	ad4antlr 731	. . . . . . . . . . . . . . 15 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → [𝑟](𝑅 ~QG 𝑀) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
42	1	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀)))
43		eqidd 2733	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅))
44		ovexd 7429	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑅 ~QG 𝑀) ∈ V)
45	42, 43, 44, 3	qusbas 17475	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((Base‘𝑅) / (𝑅 ~QG 𝑀)) = (Base‘𝑄))
46	45	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → ((Base‘𝑅) / (𝑅 ~QG 𝑀)) = (Base‘𝑄))
47	46	ad10antr 742	. . . . . . . . . . . . . . 15 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → ((Base‘𝑅) / (𝑅 ~QG 𝑀)) = (Base‘𝑄))
48	41, 47	eleqtrd 2835	. . . . . . . . . . . . . 14 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → [𝑟](𝑅 ~QG 𝑀) ∈ (Base‘𝑄))
49	39	ecelqsi 8752	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ (Base‘𝑅) → [𝑠](𝑅 ~QG 𝑀) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
50	49	ad2antlr 725	. . . . . . . . . . . . . . 15 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → [𝑠](𝑅 ~QG 𝑀) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
51	50, 47	eleqtrd 2835	. . . . . . . . . . . . . 14 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → [𝑠](𝑅 ~QG 𝑀) ∈ (Base‘𝑄))
52		simpllr 774	. . . . . . . . . . . . . . . 16 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → 𝑣 = [𝑟](𝑅 ~QG 𝑀))
53		simp-9r 792	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → 𝑢 = [𝑥](𝑅 ~QG 𝑀))
54	53	eqcomd 2738	. . . . . . . . . . . . . . . 16 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → [𝑥](𝑅 ~QG 𝑀) = 𝑢)
55	52, 54	oveq12d 7412	. . . . . . . . . . . . . . 15 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = ([𝑟](𝑅 ~QG 𝑀)(.r‘𝑄)𝑢))
56		simp-7r 788	. . . . . . . . . . . . . . 15 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄))
57	55, 56	eqtr3d 2774	. . . . . . . . . . . . . 14 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → ([𝑟](𝑅 ~QG 𝑀)(.r‘𝑄)𝑢) = (1r‘𝑄))
58		eqid 2732	. . . . . . . . . . . . . . . 16 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
59		eqid 2732	. . . . . . . . . . . . . . . 16 ⊢ (.r‘(oppr‘𝑄)) = (.r‘(oppr‘𝑄))
60	29, 30, 58, 59	opprmul 20107	. . . . . . . . . . . . . . 15 ⊢ ([𝑠](𝑅 ~QG 𝑀)(.r‘(oppr‘𝑄))𝑢) = (𝑢(.r‘𝑄)[𝑠](𝑅 ~QG 𝑀))
61		simp-5r 784	. . . . . . . . . . . . . . . 16 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀))))
62	5	ad3antrrr 728	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑅 ∈ Ring)
63	62	ad8antr 738	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → 𝑅 ∈ Ring)
64	21	ad3antrrr 728	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑀 ∈ (2Ideal‘𝑅))
65	64	ad8antr 738	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → 𝑀 ∈ (2Ideal‘𝑅))
66	2, 9, 1, 63, 65, 29, 51, 38	opprqusmulr 32515	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → ([𝑠](𝑅 ~QG 𝑀)(.r‘(oppr‘𝑄))𝑢) = ([𝑠](𝑅 ~QG 𝑀)(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))𝑢))
67		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → 𝑤 = [𝑠](𝑅 ~QG 𝑀))
68	2, 17	lidlss 20783	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑀 ∈ (LIdeal‘𝑅) → 𝑀 ⊆ (Base‘𝑅))
69	8, 68	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑀 ⊆ (Base‘𝑅))
70	9, 2	oppreqg 32507	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ⊆ (Base‘𝑅)) → (𝑅 ~QG 𝑀) = (𝑂 ~QG 𝑀))
71	5, 69, 70	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑅 ~QG 𝑀) = (𝑂 ~QG 𝑀))
72	71	ad10antr 742	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) → (𝑅 ~QG 𝑀) = (𝑂 ~QG 𝑀))
73	72	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → (𝑅 ~QG 𝑀) = (𝑂 ~QG 𝑀))
74	73	eceq2d 8730	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → [𝑥](𝑅 ~QG 𝑀) = [𝑥](𝑂 ~QG 𝑀))
75	53, 74	eqtr2d 2773	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → [𝑥](𝑂 ~QG 𝑀) = 𝑢)
76	67, 75	oveq12d 7412	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = ([𝑠](𝑅 ~QG 𝑀)(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))𝑢))
77	66, 76	eqtr4d 2775	. . . . . . . . . . . . . . . 16 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → ([𝑠](𝑅 ~QG 𝑀)(.r‘(oppr‘𝑄))𝑢) = (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)))
78	58, 25	oppr1 20118	. . . . . . . . . . . . . . . . . . 19 ⊢ (1r‘𝑄) = (1r‘(oppr‘𝑄))
79	2, 9, 1, 5, 21	opprqus1r 32516	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (1r‘(oppr‘𝑄)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀))))
80	78, 79	eqtrid 2784	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1r‘𝑄) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀))))
81	80	ad10antr 742	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) → (1r‘𝑄) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀))))
82	81	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → (1r‘𝑄) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀))))
83	61, 77, 82	3eqtr4d 2782	. . . . . . . . . . . . . . 15 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → ([𝑠](𝑅 ~QG 𝑀)(.r‘(oppr‘𝑄))𝑢) = (1r‘𝑄))
84	60, 83	eqtr3id 2786	. . . . . . . . . . . . . 14 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → (𝑢(.r‘𝑄)[𝑠](𝑅 ~QG 𝑀)) = (1r‘𝑄))
85	29, 26, 25, 30, 31, 35, 38, 48, 51, 57, 84	ringinveu 32320	. . . . . . . . . . . . 13 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → [𝑠](𝑅 ~QG 𝑀) = [𝑟](𝑅 ~QG 𝑀))
86	85, 67, 52	3eqtr4rd 2783	. . . . . . . . . . . 12 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → 𝑣 = 𝑤)
87	86	oveq2d 7410	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → (𝑢(.r‘𝑄)𝑣) = (𝑢(.r‘𝑄)𝑤))
88	67	oveq2d 7410	. . . . . . . . . . 11 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → (𝑢(.r‘𝑄)𝑤) = (𝑢(.r‘𝑄)[𝑠](𝑅 ~QG 𝑀)))
89	87, 88, 84	3eqtrd 2776	. . . . . . . . . 10 ⊢ ((((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → (𝑢(.r‘𝑄)𝑣) = (1r‘𝑄))
90		simp-4r 782	. . . . . . . . . . . 12 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) → 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀))))
91	71	qseq2d 8745	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((Base‘𝑅) / (𝑅 ~QG 𝑀)) = ((Base‘𝑅) / (𝑂 ~QG 𝑀)))
92	91	ad9antr 740	. . . . . . . . . . . . 13 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) → ((Base‘𝑅) / (𝑅 ~QG 𝑀)) = ((Base‘𝑅) / (𝑂 ~QG 𝑀)))
93		eqidd 2733	. . . . . . . . . . . . . 14 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) → (𝑂 /s (𝑂 ~QG 𝑀)) = (𝑂 /s (𝑂 ~QG 𝑀)))
94	9, 2	opprbas 20111	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑅) = (Base‘𝑂)
95	94	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) → (Base‘𝑅) = (Base‘𝑂))
96		ovexd 7429	. . . . . . . . . . . . . 14 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) → (𝑂 ~QG 𝑀) ∈ V)
97	9	fvexi 6893	. . . . . . . . . . . . . . 15 ⊢ 𝑂 ∈ V
98	97	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) → 𝑂 ∈ V)
99	93, 95, 96, 98	qusbas 17475	. . . . . . . . . . . . 13 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) → ((Base‘𝑅) / (𝑂 ~QG 𝑀)) = (Base‘(𝑂 /s (𝑂 ~QG 𝑀))))
100	92, 99	eqtr2d 2773	. . . . . . . . . . . 12 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) → (Base‘(𝑂 /s (𝑂 ~QG 𝑀))) = ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
101	90, 100	eleqtrd 2835	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) → 𝑤 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
102		elqsi 8749	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)) → ∃𝑠 ∈ (Base‘𝑅)𝑤 = [𝑠](𝑅 ~QG 𝑀))
103	101, 102	syl 17	. . . . . . . . . 10 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) → ∃𝑠 ∈ (Base‘𝑅)𝑤 = [𝑠](𝑅 ~QG 𝑀))
104	89, 103	r19.29a 3162	. . . . . . . . 9 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) → (𝑢(.r‘𝑄)𝑣) = (1r‘𝑄))
105		simp-4r 782	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) → 𝑣 ∈ (Base‘𝑄))
106	46	ad6antr 734	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) → ((Base‘𝑅) / (𝑅 ~QG 𝑀)) = (Base‘𝑄))
107	105, 106	eleqtrrd 2836	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) → 𝑣 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
108		elqsi 8749	. . . . . . . . . 10 ⊢ (𝑣 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)) → ∃𝑟 ∈ (Base‘𝑅)𝑣 = [𝑟](𝑅 ~QG 𝑀))
109	107, 108	syl 17	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) → ∃𝑟 ∈ (Base‘𝑅)𝑣 = [𝑟](𝑅 ~QG 𝑀))
110	104, 109	r19.29a 3162	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) → (𝑢(.r‘𝑄)𝑣) = (1r‘𝑄))
111		eqid 2732	. . . . . . . . . 10 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
112		eqid 2732	. . . . . . . . . 10 ⊢ (𝑂 /s (𝑂 ~QG 𝑀)) = (𝑂 /s (𝑂 ~QG 𝑀))
113	3	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑅 ∈ NzRing)
114	9	opprnzr 20251	. . . . . . . . . . 11 ⊢ (𝑅 ∈ NzRing → 𝑂 ∈ NzRing)
115	113, 114	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑂 ∈ NzRing)
116	12	ad3antrrr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑀 ∈ (MaxIdeal‘𝑂))
117	6	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑀 ∈ (MaxIdeal‘𝑅))
118	9, 62, 117	opprmxidlabs 32511	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑀 ∈ (MaxIdeal‘(oppr‘𝑂)))
119		simplr 767	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑥 ∈ (Base‘𝑅))
120	94	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → (Base‘𝑅) = (Base‘𝑂))
121	119, 120	eleqtrd 2835	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑥 ∈ (Base‘𝑂))
122		simplr 767	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → 𝑢 = [𝑥](𝑅 ~QG 𝑀))
123	5	ringgrpd 20025	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ Grp)
124	123	ad4antr 730	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → 𝑅 ∈ Grp)
125		lidlnsg 32479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) → 𝑀 ∈ (NrmSGrp‘𝑅))
126	5, 8, 125	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ (NrmSGrp‘𝑅))
127		nsgsubg 19012	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (NrmSGrp‘𝑅) → 𝑀 ∈ (SubGrp‘𝑅))
128	126, 127	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ (SubGrp‘𝑅))
129	128	ad4antr 730	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → 𝑀 ∈ (SubGrp‘𝑅))
130		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → 𝑥 ∈ 𝑀)
131		eqid 2732	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ~QG 𝑀) = (𝑅 ~QG 𝑀)
132	131	eqg0el 32399	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Grp ∧ 𝑀 ∈ (SubGrp‘𝑅)) → ([𝑥](𝑅 ~QG 𝑀) = 𝑀 ↔ 𝑥 ∈ 𝑀))
133	132	biimpar 478	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Grp ∧ 𝑀 ∈ (SubGrp‘𝑅)) ∧ 𝑥 ∈ 𝑀) → [𝑥](𝑅 ~QG 𝑀) = 𝑀)
134	124, 129, 130, 133	syl21anc 836	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → [𝑥](𝑅 ~QG 𝑀) = 𝑀)
135		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝑅) = (0g‘𝑅)
136	2, 131, 135	eqgid 19034	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (SubGrp‘𝑅) → [(0g‘𝑅)](𝑅 ~QG 𝑀) = 𝑀)
137	129, 136	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → [(0g‘𝑅)](𝑅 ~QG 𝑀) = 𝑀)
138	134, 137	eqtr4d 2775	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → [𝑥](𝑅 ~QG 𝑀) = [(0g‘𝑅)](𝑅 ~QG 𝑀))
139	1, 135	qus0 19040	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (NrmSGrp‘𝑅) → [(0g‘𝑅)](𝑅 ~QG 𝑀) = (0g‘𝑄))
140	126, 139	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → [(0g‘𝑅)](𝑅 ~QG 𝑀) = (0g‘𝑄))
141	140	ad4antr 730	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → [(0g‘𝑅)](𝑅 ~QG 𝑀) = (0g‘𝑄))
142	122, 138, 141	3eqtrd 2776	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → 𝑢 = (0g‘𝑄))
143		eldifsnneq 4788	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)}) → ¬ 𝑢 = (0g‘𝑄))
144	143	ad4antlr 731	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → ¬ 𝑢 = (0g‘𝑄))
145	142, 144	pm2.65da 815	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → ¬ 𝑥 ∈ 𝑀)
146	111, 112, 115, 116, 118, 121, 145	qsdrngilem 32518	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → ∃𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))(𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀))))
147	146	ad2antrr 724	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) → ∃𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))(𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀))))
148	110, 147	r19.29a 3162	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) → (𝑢(.r‘𝑄)𝑣) = (1r‘𝑄))
149		simpllr 774	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) → 𝑢 = [𝑥](𝑅 ~QG 𝑀))
150	149	oveq2d 7410	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) → (𝑣(.r‘𝑄)𝑢) = (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)))
151		simpr 485	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) → (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄))
152	150, 151	eqtrd 2772	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) → (𝑣(.r‘𝑄)𝑢) = (1r‘𝑄))
153	148, 152	jca 512	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄)) → ((𝑢(.r‘𝑄)𝑣) = (1r‘𝑄) ∧ (𝑣(.r‘𝑄)𝑢) = (1r‘𝑄)))
154	153	anasss 467	. . . . 5 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ (𝑣 ∈ (Base‘𝑄) ∧ (𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄))) → ((𝑢(.r‘𝑄)𝑣) = (1r‘𝑄) ∧ (𝑣(.r‘𝑄)𝑢) = (1r‘𝑄)))
155	9, 1, 113, 117, 116, 119, 145	qsdrngilem 32518	. . . . 5 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → ∃𝑣 ∈ (Base‘𝑄)(𝑣(.r‘𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r‘𝑄))
156	154, 155	reximddv 3171	. . . 4 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → ∃𝑣 ∈ (Base‘𝑄)((𝑢(.r‘𝑄)𝑣) = (1r‘𝑄) ∧ (𝑣(.r‘𝑄)𝑢) = (1r‘𝑄)))
157	37, 46	eleqtrrd 2836	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → 𝑢 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
158		elqsi 8749	. . . . 5 ⊢ (𝑢 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)) → ∃𝑥 ∈ (Base‘𝑅)𝑢 = [𝑥](𝑅 ~QG 𝑀))
159	157, 158	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → ∃𝑥 ∈ (Base‘𝑅)𝑢 = [𝑥](𝑅 ~QG 𝑀))
160	156, 159	r19.29a 3162	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → ∃𝑣 ∈ (Base‘𝑄)((𝑢(.r‘𝑄)𝑣) = (1r‘𝑄) ∧ (𝑣(.r‘𝑄)𝑢) = (1r‘𝑄)))
161	160	ralrimiva 3146	. 2 ⊢ (𝜑 → ∀𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})∃𝑣 ∈ (Base‘𝑄)((𝑢(.r‘𝑄)𝑣) = (1r‘𝑄) ∧ (𝑣(.r‘𝑄)𝑢) = (1r‘𝑄)))
162	29, 26, 25, 30, 31, 33	isdrng4 32321	. 2 ⊢ (𝜑 → (𝑄 ∈ DivRing ↔ ((1r‘𝑄) ≠ (0g‘𝑄) ∧ ∀𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})∃𝑣 ∈ (Base‘𝑄)((𝑢(.r‘𝑄)𝑣) = (1r‘𝑄) ∧ (𝑣(.r‘𝑄)𝑢) = (1r‘𝑄)))))
163	28, 161, 162	mpbir2and 711	1 ⊢ (𝜑 → 𝑄 ∈ DivRing)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  Vcvv 3474   ∖ cdif 3942   ∩ cin 3944   ⊆ wss 3945  {csn 4623  ‘cfv 6533  (class class class)co 7394  [cec 8686   / cqs 8687  Basecbs 17128  ._rcmulr 17182  0_gc0g 17369   /_s cqus 17435  Grpcgrp 18796  SubGrpcsubg 18974  NrmSGrpcnsg 18975   ~_QG cqg 18976  1_rcur 19965  Ringcrg 20016  opp_rcoppr 20103  Unitcui 20123  NzRingcnzr 20243  DivRingcdr 20267  LIdealclidl 20734  2Idealc2idl 20804  MaxIdealcmxidl 32490

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709  ax-cnex 11150  ax-resscn 11151  ax-1cn 11152  ax-icn 11153  ax-addcl 11154  ax-addrcl 11155  ax-mulcl 11156  ax-mulrcl 11157  ax-mulcom 11158  ax-addass 11159  ax-mulass 11160  ax-distr 11161  ax-i2m1 11162  ax-1ne0 11163  ax-1rid 11164  ax-rnegex 11165  ax-rrecex 11166  ax-cnre 11167  ax-pre-lttri 11168  ax-pre-lttrn 11169  ax-pre-ltadd 11170  ax-pre-mulgt0 11171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-se 5626  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7350  df-ov 7397  df-oprab 7398  df-mpo 7399  df-of 7654  df-om 7840  df-1st 7959  df-2nd 7960  df-supp 8131  df-tpos 8195  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-rdg 8394  df-1o 8450  df-2o 8451  df-er 8688  df-ec 8690  df-qs 8694  df-map 8807  df-ixp 8877  df-en 8925  df-dom 8926  df-sdom 8927  df-fin 8928  df-fsupp 9347  df-sup 9421  df-inf 9422  df-oi 9489  df-card 9918  df-pnf 11234  df-mnf 11235  df-xr 11236  df-ltxr 11237  df-le 11238  df-sub 11430  df-neg 11431  df-nn 12197  df-2 12259  df-3 12260  df-4 12261  df-5 12262  df-6 12263  df-7 12264  df-8 12265  df-9 12266  df-n0 12457  df-z 12543  df-dec 12662  df-uz 12807  df-fz 13469  df-fzo 13612  df-seq 13951  df-hash 14275  df-struct 17064  df-sets 17081  df-slot 17099  df-ndx 17111  df-base 17129  df-ress 17158  df-plusg 17194  df-mulr 17195  df-sca 17197  df-vsca 17198  df-ip 17199  df-tset 17200  df-ple 17201  df-ds 17203  df-hom 17205  df-cco 17206  df-0g 17371  df-gsum 17372  df-prds 17377  df-pws 17379  df-imas 17438  df-qus 17439  df-mre 17514  df-mrc 17515  df-acs 17517  df-mgm 18545  df-sgrp 18594  df-mnd 18605  df-mhm 18649  df-submnd 18650  df-grp 18799  df-minusg 18800  df-sbg 18801  df-mulg 18925  df-subg 18977  df-nsg 18978  df-eqg 18979  df-ghm 19058  df-cntz 19149  df-cmn 19616  df-abl 19617  df-mgp 19949  df-ur 19966  df-ring 20018  df-oppr 20104  df-dvdsr 20125  df-unit 20126  df-invr 20156  df-nzr 20244  df-drng 20269  df-subrg 20312  df-lmod 20424  df-lss 20494  df-lsp 20534  df-lmhm 20584  df-lbs 20637  df-sra 20736  df-rgmod 20737  df-lidl 20738  df-rsp 20739  df-2idl 20805  df-dsmm 21222  df-frlm 21237  df-uvc 21273  df-mxidl 32491

This theorem is referenced by:  qsdrng  32521