Theorem qsdrngi 33722

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 2769	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		qsdrng.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ NzRing)
4		nzrring 20599	. . . . 5 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
5	3, 4	syl 18	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
6		qsdrngi.1	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))
7	2	mxidlidl 33691	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
8	5, 6, 7	syl2anc 595	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅))
9		qsdrng.0	. . . . . . . . 9 ⊢ 𝑂 = (oppr‘𝑅)
10	9	opprring 20429	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring)
11	5, 10	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ Ring)
12		qsdrngi.2	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑂))
13		eqid 2769	. . . . . . . 8 ⊢ (Base‘𝑂) = (Base‘𝑂)
14	13	mxidlidl 33691	. . . . . . 7 ⊢ ((𝑂 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑂)) → 𝑀 ∈ (LIdeal‘𝑂))
15	11, 12, 14	syl2anc 595	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑂))
16	8, 15	elind 4161	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ((LIdeal‘𝑅) ∩ (LIdeal‘𝑂)))
17		eqid 2769	. . . . . 6 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
18		eqid 2769	. . . . . 6 ⊢ (LIdeal‘𝑂) = (LIdeal‘𝑂)
19		eqid 2769	. . . . . 6 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
20	17, 9, 18, 19	2idlval 21361	. . . . 5 ⊢ (2Ideal‘𝑅) = ((LIdeal‘𝑅) ∩ (LIdeal‘𝑂))
21	16, 20	eleqtrrdi 2880	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (2Ideal‘𝑅))
22	2	mxidlnr 33692	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ (Base‘𝑅))
23	5, 6, 22	syl2anc 595	. . . 4 ⊢ (𝜑 → 𝑀 ≠ (Base‘𝑅))
24	1, 2, 5, 3, 21, 23	qsnzr 21452	. . 3 ⊢ (𝜑 → 𝑄 ∈ NzRing)
25		eqid 2769	. . . 4 ⊢ (1r‘𝑄) = (1r‘𝑄)
26		eqid 2769	. . . 4 ⊢ (0g‘𝑄) = (0g‘𝑄)
27	25, 26	nzrnz 20598	. . 3 ⊢ (𝑄 ∈ NzRing → (1r‘𝑄) ≠ (0g‘𝑄))
28	24, 27	syl 18	. 2 ⊢ (𝜑 → (1r‘𝑄) ≠ (0g‘𝑄))
29		eqid 2769	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑄) = (Base‘𝑄)
30		eqid 2769	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑄) = (.r‘𝑄)
31		eqid 2769	. . . . . . . . . . . . . 14 ⊢ (Unit‘𝑄) = (Unit‘𝑄)
32	1, 19	qusring 21385	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (2Ideal‘𝑅)) → 𝑄 ∈ Ring)
33	5, 21, 32	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑄 ∈ Ring)
34	33	ad10antr 756	. . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . 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 4093	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)}) → 𝑢 ∈ (Base‘𝑄))
37	36	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → 𝑢 ∈ (Base‘𝑄))
38	37	ad10antr 756	. . . . . . . . . . . . . 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 7444	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ~QG 𝑀) ∈ V
40	39	ecelqsi 8767	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ (Base‘𝑅) → [𝑟](𝑅 ~QG 𝑀) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
41	40	ad4antlr 745	. . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅))
44		ovexd 7446	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑅 ~QG 𝑀) ∈ V)
45	42, 43, 44, 3	qusbas 17599	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((Base‘𝑅) / (𝑅 ~QG 𝑀)) = (Base‘𝑄))
46	45	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → ((Base‘𝑅) / (𝑅 ~QG 𝑀)) = (Base‘𝑄))
47	46	ad10antr 756	. . . . . . . . . . . . . . 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 2871	. . . . . . . . . . . . . 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 8767	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ (Base‘𝑅) → [𝑠](𝑅 ~QG 𝑀) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
50	49	ad2antlr 739	. . . . . . . . . . . . . . 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 2871	. . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . . 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 805	. . . . . . . . . . . . . . . . 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 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)) → [𝑥](𝑅 ~QG 𝑀) = 𝑢)
55	52, 54	oveq12d 7429	. . . . . . . . . . . . . . 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 801	. . . . . . . . . . . . . . 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 2806	. . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . 16 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
59		eqid 2769	. . . . . . . . . . . . . . . 16 ⊢ (.r‘(oppr‘𝑄)) = (.r‘(oppr‘𝑄))
60	29, 30, 58, 59	opprmul 20422	. . . . . . . . . . . . . . 15 ⊢ ([𝑠](𝑅 ~QG 𝑀)(.r‘(oppr‘𝑄))𝑢) = (𝑢(.r‘𝑄)[𝑠](𝑅 ~QG 𝑀))
61		simp-5r 797	. . . . . . . . . . . . . . . 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 742	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑅 ∈ Ring)
63	62	ad8antr 752	. . . . . . . . . . . . . . . . . 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 742	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑀 ∈ (2Ideal‘𝑅))
65	64	ad8antr 752	. . . . . . . . . . . . . . . . . 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 33718	. . . . . . . . . . . . . . . . 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 489	. . . . . . . . . . . . . . . . . 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 21314	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑀 ∈ (LIdeal‘𝑅) → 𝑀 ⊆ (Base‘𝑅))
69	8, 68	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑀 ⊆ (Base‘𝑅))
70	9, 2	oppreqg 33710	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ⊆ (Base‘𝑅)) → (𝑅 ~QG 𝑀) = (𝑂 ~QG 𝑀))
71	5, 69, 70	syl2anc 595	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑅 ~QG 𝑀) = (𝑂 ~QG 𝑀))
72	71	ad10antr 756	. . . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . 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 8738	. . . . . . . . . . . . . . . . . . 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 2805	. . . . . . . . . . . . . . . . . 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 7429	. . . . . . . . . . . . . . . . 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 2807	. . . . . . . . . . . . . . . 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 20432	. . . . . . . . . . . . . . . . . . 19 ⊢ (1r‘𝑄) = (1r‘(oppr‘𝑄))
79	2, 9, 1, 5, 21	opprqus1r 33719	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (1r‘(oppr‘𝑄)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀))))
80	78, 79	eqtrid 2816	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1r‘𝑄) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀))))
81	80	ad10antr 756	. . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . 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 2814	. . . . . . . . . . . . . . 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 2818	. . . . . . . . . . . . . 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 33558	. . . . . . . . . . . . 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 2815	. . . . . . . . . . . 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 7427	. . . . . . . . . . 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 7427	. . . . . . . . . . 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 2808	. . . . . . . . . 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 795	. . . . . . . . . . . 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 8758	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((Base‘𝑅) / (𝑅 ~QG 𝑀)) = ((Base‘𝑅) / (𝑂 ~QG 𝑀)))
92	91	ad9antr 754	. . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . 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 20425	. . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . 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 6896	. . . . . . . . . . . . . . 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 17599	. . . . . . . . . . . . 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 2805	. . . . . . . . . . . 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 2871	. . . . . . . . . . 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 8763	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)) → ∃𝑠 ∈ (Base‘𝑅)𝑤 = [𝑠](𝑅 ~QG 𝑀))
103	101, 102	syl 18	. . . . . . . . . 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 3179	. . . . . . . . 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 795	. . . . . . . . . . 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 748	. . . . . . . . . . 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 2872	. . . . . . . . . 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 8763	. . . . . . . . . 10 ⊢ (𝑣 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)) → ∃𝑟 ∈ (Base‘𝑅)𝑣 = [𝑟](𝑅 ~QG 𝑀))
109	107, 108	syl 18	. . . . . . . . 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 3179	. . . . . . . 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 2769	. . . . . . . . . 10 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
112		eqid 2769	. . . . . . . . . 10 ⊢ (𝑂 /s (𝑂 ~QG 𝑀)) = (𝑂 /s (𝑂 ~QG 𝑀))
113	3	ad3antrrr 742	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑅 ∈ NzRing)
114	9	opprnzr 20606	. . . . . . . . . . 11 ⊢ (𝑅 ∈ NzRing → 𝑂 ∈ NzRing)
115	113, 114	syl 18	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑂 ∈ NzRing)
116	12	ad3antrrr 742	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑀 ∈ (MaxIdeal‘𝑂))
117	6	ad3antrrr 742	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑀 ∈ (MaxIdeal‘𝑅))
118	9, 62, 117	opprmxidlabs 33714	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑀 ∈ (MaxIdeal‘(oppr‘𝑂)))
119		simplr 780	. . . . . . . . . . 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 2871	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑥 ∈ (Base‘𝑂))
122		simplr 780	. . . . . . . . . . . 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 20324	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ Grp)
124	123	ad4antr 744	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → 𝑅 ∈ Grp)
125		lidlnsg 21356	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) → 𝑀 ∈ (NrmSGrp‘𝑅))
126	5, 8, 125	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ (NrmSGrp‘𝑅))
127		nsgsubg 19224	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (NrmSGrp‘𝑅) → 𝑀 ∈ (SubGrp‘𝑅))
128	126, 127	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ (SubGrp‘𝑅))
129	128	ad4antr 744	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → 𝑀 ∈ (SubGrp‘𝑅))
130		simpr 489	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → 𝑥 ∈ 𝑀)
131		eqid 2769	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ~QG 𝑀) = (𝑅 ~QG 𝑀)
132	131	eqg0el 19254	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Grp ∧ 𝑀 ∈ (SubGrp‘𝑅)) → ([𝑥](𝑅 ~QG 𝑀) = 𝑀 ↔ 𝑥 ∈ 𝑀))
133	132	biimpar 482	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Grp ∧ 𝑀 ∈ (SubGrp‘𝑅)) ∧ 𝑥 ∈ 𝑀) → [𝑥](𝑅 ~QG 𝑀) = 𝑀)
134	124, 129, 130, 133	syl21anc 850	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → [𝑥](𝑅 ~QG 𝑀) = 𝑀)
135		eqid 2769	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝑅) = (0g‘𝑅)
136	2, 131, 135	eqgid 19248	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (SubGrp‘𝑅) → [(0g‘𝑅)](𝑅 ~QG 𝑀) = 𝑀)
137	129, 136	syl 18	. . . . . . . . . . . . 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 2807	. . . . . . . . . . . 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 19260	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (NrmSGrp‘𝑅) → [(0g‘𝑅)](𝑅 ~QG 𝑀) = (0g‘𝑄))
140	126, 139	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → [(0g‘𝑅)](𝑅 ~QG 𝑀) = (0g‘𝑄))
141	140	ad4antr 744	. . . . . . . . . . . 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 2808	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → 𝑢 = (0g‘𝑄))
143		eldifsnneq 4763	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)}) → ¬ 𝑢 = (0g‘𝑄))
144	143	ad4antlr 745	. . . . . . . . . . 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 828	. . . . . . . . . 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 33721	. . . . . . . . 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 738	. . . . . . . 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 3179	. . . . . . 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 787	. . . . . . . . 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 7427	. . . . . . . 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 489	. . . . . . . 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 2804	. . . . . . 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 520	. . . . . 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 471	. . . . 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 33721	. . . . 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 3187	. . . 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 2872	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → 𝑢 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
158		elqsi 8763	. . . . 5 ⊢ (𝑢 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)) → ∃𝑥 ∈ (Base‘𝑅)𝑢 = [𝑥](𝑅 ~QG 𝑀))
159	157, 158	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → ∃𝑥 ∈ (Base‘𝑅)𝑢 = [𝑥](𝑅 ~QG 𝑀))
160	156, 159	r19.29a 3179	. . 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 3163	. 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 33559	. 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 725	1 ⊢ (𝜑 → 𝑄 ∈ DivRing)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  Vcvv 3463   ∖ cdif 3910   ∩ cin 3912   ⊆ wss 3913  {csn 4594  ‘cfv 6537  (class class class)co 7411  [cec 8692   / cqs 8693  Basecbs 17269  ._rcmulr 17311  0_gc0g 17492   /_s cqus 17559  Grpcgrp 19000  SubGrpcsubg 19186  NrmSGrpcnsg 19187   ~_QG cqg 19188  1_rcur 20263  Ringcrg 20315  opp_rcoppr 20418  Unitcui 20437  NzRingcnzr 20595  DivRingcdr 20813  LIdealclidl 21308  2Idealc2idl 21359  MaxIdealcmxidl 33687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7675  df-om 7863  df-1st 7986  df-2nd 7987  df-supp 8157  df-tpos 8222  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-er 8694  df-ec 8696  df-qs 8700  df-map 8826  df-ixp 8896  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-fsupp 9322  df-sup 9402  df-inf 9403  df-oi 9472  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12505  df-z 12592  df-dec 12712  df-uz 12863  df-fz 13536  df-fzo 13683  df-seq 14038  df-hash 14367  df-struct 17207  df-sets 17224  df-slot 17242  df-ndx 17254  df-base 17270  df-ress 17291  df-plusg 17323  df-mulr 17324  df-sca 17326  df-vsca 17327  df-ip 17328  df-tset 17329  df-ple 17330  df-ds 17332  df-hom 17334  df-cco 17335  df-0g 17494  df-gsum 17495  df-prds 17500  df-pws 17502  df-imas 17562  df-qus 17563  df-mre 17638  df-mrc 17639  df-acs 17641  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-mhm 18841  df-submnd 18842  df-grp 19003  df-minusg 19004  df-sbg 19005  df-mulg 19134  df-subg 19189  df-nsg 19190  df-eqg 19191  df-ghm 19284  df-cntz 19387  df-cmn 19852  df-abl 19853  df-mgp 20217  df-rng 20231  df-ur 20264  df-ring 20317  df-oppr 20419  df-dvdsr 20439  df-unit 20440  df-invr 20470  df-nzr 20596  df-subrg 20655  df-drng 20815  df-lmod 20961  df-lss 21031  df-lsp 21071  df-lmhm 21121  df-lbs 21174  df-sra 21272  df-rgmod 21273  df-lidl 21310  df-rsp 21311  df-2idl 21360  df-dsmm 21851  df-frlm 21866  df-uvc 21902  df-mxidl 33688

This theorem is referenced by:  qsdrng  33724  algextdeglem4  34055