Theorem qsdrngi 33503

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 2735	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		qsdrng.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ NzRing)
4		nzrring 20533	. . . . 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 33471	. . . . . . 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 20364	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring)
11	5, 10	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ Ring)
12		qsdrngi.2	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑂))
13		eqid 2735	. . . . . . . 8 ⊢ (Base‘𝑂) = (Base‘𝑂)
14	13	mxidlidl 33471	. . . . . . 7 ⊢ ((𝑂 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑂)) → 𝑀 ∈ (LIdeal‘𝑂))
15	11, 12, 14	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑂))
16	8, 15	elind 4210	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ((LIdeal‘𝑅) ∩ (LIdeal‘𝑂)))
17		eqid 2735	. . . . . 6 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
18		eqid 2735	. . . . . 6 ⊢ (LIdeal‘𝑂) = (LIdeal‘𝑂)
19		eqid 2735	. . . . . 6 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
20	17, 9, 18, 19	2idlval 21279	. . . . 5 ⊢ (2Ideal‘𝑅) = ((LIdeal‘𝑅) ∩ (LIdeal‘𝑂))
21	16, 20	eleqtrrdi 2850	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (2Ideal‘𝑅))
22	2	mxidlnr 33472	. . . . 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 33463	. . 3 ⊢ (𝜑 → 𝑄 ∈ NzRing)
25		eqid 2735	. . . 4 ⊢ (1r‘𝑄) = (1r‘𝑄)
26		eqid 2735	. . . 4 ⊢ (0g‘𝑄) = (0g‘𝑄)
27	25, 26	nzrnz 20532	. . 3 ⊢ (𝑄 ∈ NzRing → (1r‘𝑄) ≠ (0g‘𝑄))
28	24, 27	syl 17	. 2 ⊢ (𝜑 → (1r‘𝑄) ≠ (0g‘𝑄))
29		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑄) = (Base‘𝑄)
30		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑄) = (.r‘𝑄)
31		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (Unit‘𝑄) = (Unit‘𝑄)
32	1, 19	qusring 21303	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (2Ideal‘𝑅)) → 𝑄 ∈ Ring)
33	5, 21, 32	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑄 ∈ Ring)
34	33	ad10antr 744	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 4141	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)}) → 𝑢 ∈ (Base‘𝑄))
37	36	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → 𝑢 ∈ (Base‘𝑄))
38	37	ad10antr 744	. . . . . . . . . . . . . 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 7464	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ~QG 𝑀) ∈ V
40	39	ecelqsi 8812	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ (Base‘𝑅) → [𝑟](𝑅 ~QG 𝑀) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
41	40	ad4antlr 733	. . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅))
44		ovexd 7466	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑅 ~QG 𝑀) ∈ V)
45	42, 43, 44, 3	qusbas 17592	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((Base‘𝑅) / (𝑅 ~QG 𝑀)) = (Base‘𝑄))
46	45	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → ((Base‘𝑅) / (𝑅 ~QG 𝑀)) = (Base‘𝑄))
47	46	ad10antr 744	. . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . 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 8812	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ (Base‘𝑅) → [𝑠](𝑅 ~QG 𝑀) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
50	49	ad2antlr 727	. . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . 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 794	. . . . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . 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 7449	. . . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . 16 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
59		eqid 2735	. . . . . . . . . . . . . . . 16 ⊢ (.r‘(oppr‘𝑄)) = (.r‘(oppr‘𝑄))
60	29, 30, 58, 59	opprmul 20354	. . . . . . . . . . . . . . 15 ⊢ ([𝑠](𝑅 ~QG 𝑀)(.r‘(oppr‘𝑄))𝑢) = (𝑢(.r‘𝑄)[𝑠](𝑅 ~QG 𝑀))
61		simp-5r 786	. . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑅 ∈ Ring)
63	62	ad8antr 740	. . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑀 ∈ (2Ideal‘𝑅))
65	64	ad8antr 740	. . . . . . . . . . . . . . . . . 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 33499	. . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 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 21240	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑀 ∈ (LIdeal‘𝑅) → 𝑀 ⊆ (Base‘𝑅))
69	8, 68	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑀 ⊆ (Base‘𝑅))
70	9, 2	oppreqg 33491	. . . . . . . . . . . . . . . . . . . . . . 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 744	. . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 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 8787	. . . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . . . 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 7449	. . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . 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 20367	. . . . . . . . . . . . . . . . . . 19 ⊢ (1r‘𝑄) = (1r‘(oppr‘𝑄))
79	2, 9, 1, 5, 21	opprqus1r 33500	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (1r‘(oppr‘𝑄)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀))))
80	78, 79	eqtrid 2787	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1r‘𝑄) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀))))
81	80	ad10antr 744	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 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 2785	. . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . 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 33278	. . . . . . . . . . . . 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 2786	. . . . . . . . . . . 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 7447	. . . . . . . . . . 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 7447	. . . . . . . . . . 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 2779	. . . . . . . . . 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 784	. . . . . . . . . . . 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 8804	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((Base‘𝑅) / (𝑅 ~QG 𝑀)) = ((Base‘𝑅) / (𝑂 ~QG 𝑀)))
92	91	ad9antr 742	. . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . 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 20358	. . . . . . . . . . . . . . 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 7466	. . . . . . . . . . . . . 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 6921	. . . . . . . . . . . . . . 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 17592	. . . . . . . . . . . . 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 2776	. . . . . . . . . . . 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 2841	. . . . . . . . . . 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 8809	. . . . . . . . . . 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 3160	. . . . . . . . 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 784	. . . . . . . . . . 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 736	. . . . . . . . . . 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 2842	. . . . . . . . . 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 8809	. . . . . . . . . 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 3160	. . . . . . . 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 2735	. . . . . . . . . 10 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
112		eqid 2735	. . . . . . . . . 10 ⊢ (𝑂 /s (𝑂 ~QG 𝑀)) = (𝑂 /s (𝑂 ~QG 𝑀))
113	3	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑅 ∈ NzRing)
114	9	opprnzr 20539	. . . . . . . . . . 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 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑀 ∈ (MaxIdeal‘𝑂))
117	6	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑀 ∈ (MaxIdeal‘𝑅))
118	9, 62, 117	opprmxidlabs 33495	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑀 ∈ (MaxIdeal‘(oppr‘𝑂)))
119		simplr 769	. . . . . . . . . . 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 2841	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑥 ∈ (Base‘𝑂))
122		simplr 769	. . . . . . . . . . . 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 20260	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ Grp)
124	123	ad4antr 732	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → 𝑅 ∈ Grp)
125		lidlnsg 21276	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) → 𝑀 ∈ (NrmSGrp‘𝑅))
126	5, 8, 125	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ (NrmSGrp‘𝑅))
127		nsgsubg 19189	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (NrmSGrp‘𝑅) → 𝑀 ∈ (SubGrp‘𝑅))
128	126, 127	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ (SubGrp‘𝑅))
129	128	ad4antr 732	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → 𝑀 ∈ (SubGrp‘𝑅))
130		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → 𝑥 ∈ 𝑀)
131		eqid 2735	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ~QG 𝑀) = (𝑅 ~QG 𝑀)
132	131	eqg0el 19214	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Grp ∧ 𝑀 ∈ (SubGrp‘𝑅)) → ([𝑥](𝑅 ~QG 𝑀) = 𝑀 ↔ 𝑥 ∈ 𝑀))
133	132	biimpar 477	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Grp ∧ 𝑀 ∈ (SubGrp‘𝑅)) ∧ 𝑥 ∈ 𝑀) → [𝑥](𝑅 ~QG 𝑀) = 𝑀)
134	124, 129, 130, 133	syl21anc 838	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → [𝑥](𝑅 ~QG 𝑀) = 𝑀)
135		eqid 2735	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝑅) = (0g‘𝑅)
136	2, 131, 135	eqgid 19211	. . . . . . . . . . . . . 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 2778	. . . . . . . . . . . 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 19220	. . . . . . . . . . . . . 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 732	. . . . . . . . . . . 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 2779	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → 𝑢 = (0g‘𝑄))
143		eldifsnneq 4796	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)}) → ¬ 𝑢 = (0g‘𝑄))
144	143	ad4antlr 733	. . . . . . . . . . 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 817	. . . . . . . . . 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 33502	. . . . . . . . 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 726	. . . . . . . 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 3160	. . . . . . 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 776	. . . . . . . . 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 7447	. . . . . . . 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 484	. . . . . . . 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 2775	. . . . . . 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 511	. . . . . 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 466	. . . . 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 33502	. . . . 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 3169	. . . 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 2842	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → 𝑢 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
158		elqsi 8809	. . . . 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 3160	. . 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 3144	. 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 33279	. 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 713	1 ⊢ (𝜑 → 𝑄 ∈ DivRing)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ∃wrex 3068  Vcvv 3478   ∖ cdif 3960   ∩ cin 3962   ⊆ wss 3963  {csn 4631  ‘cfv 6563  (class class class)co 7431  [cec 8742   / cqs 8743  Basecbs 17245  ._rcmulr 17299  0_gc0g 17486   /_s cqus 17552  Grpcgrp 18964  SubGrpcsubg 19151  NrmSGrpcnsg 19152   ~_QG cqg 19153  1_rcur 20199  Ringcrg 20251  opp_rcoppr 20350  Unitcui 20372  NzRingcnzr 20529  DivRingcdr 20746  LIdealclidl 21234  2Idealc2idl 21277  MaxIdealcmxidl 33467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-ec 8746  df-qs 8750  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-imas 17555  df-qus 17556  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-nsg 19155  df-eqg 19156  df-ghm 19244  df-cntz 19348  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-nzr 20530  df-subrg 20587  df-drng 20748  df-lmod 20877  df-lss 20948  df-lsp 20988  df-lmhm 21039  df-lbs 21092  df-sra 21190  df-rgmod 21191  df-lidl 21236  df-rsp 21237  df-2idl 21278  df-dsmm 21770  df-frlm 21785  df-uvc 21821  df-mxidl 33468

This theorem is referenced by:  qsdrng  33505  algextdeglem4  33726