Theorem qsdrngi 33900

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 2760	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		qsdrng.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ NzRing)
4		nzrring 20679	. . . . 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 33869	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
8	5, 6, 7	syl2anc 596	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑅))
9		qsdrng.0	. . . . . . . . 9 ⊢ 𝑂 = (oppr‘𝑅)
10	9	opprring 20491	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring)
11	5, 10	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ Ring)
12		qsdrngi.2	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑂))
13		eqid 2760	. . . . . . . 8 ⊢ (Base‘𝑂) = (Base‘𝑂)
14	13	mxidlidl 33869	. . . . . . 7 ⊢ ((𝑂 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑂)) → 𝑀 ∈ (LIdeal‘𝑂))
15	11, 12, 14	syl2anc 596	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (LIdeal‘𝑂))
16	8, 15	elind 4146	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ((LIdeal‘𝑅) ∩ (LIdeal‘𝑂)))
17		eqid 2760	. . . . . 6 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
18		eqid 2760	. . . . . 6 ⊢ (LIdeal‘𝑂) = (LIdeal‘𝑂)
19		eqid 2760	. . . . . 6 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
20	17, 9, 18, 19	2idlval 21456	. . . . 5 ⊢ (2Ideal‘𝑅) = ((LIdeal‘𝑅) ∩ (LIdeal‘𝑂))
21	16, 20	eleqtrrdi 2871	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (2Ideal‘𝑅))
22	2	mxidlnr 33870	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ (Base‘𝑅))
23	5, 6, 22	syl2anc 596	. . . 4 ⊢ (𝜑 → 𝑀 ≠ (Base‘𝑅))
24	1, 2, 5, 3, 21, 23	qsnzr 21549	. . 3 ⊢ (𝜑 → 𝑄 ∈ NzRing)
25		eqid 2760	. . . 4 ⊢ (1r‘𝑄) = (1r‘𝑄)
26		eqid 2760	. . . 4 ⊢ (0g‘𝑄) = (0g‘𝑄)
27	25, 26	nzrnz 20678	. . 3 ⊢ (𝑄 ∈ NzRing → (1r‘𝑄) ≠ (0g‘𝑄))
28	24, 27	syl 18	. 2 ⊢ (𝜑 → (1r‘𝑄) ≠ (0g‘𝑄))
29		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑄) = (Base‘𝑄)
30		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑄) = (.r‘𝑄)
31		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (Unit‘𝑄) = (Unit‘𝑄)
32	1, 19	qusring 21480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (2Ideal‘𝑅)) → 𝑄 ∈ Ring)
33	5, 21, 32	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑄 ∈ Ring)
34	33	ad10antr 757	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 4078	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)}) → 𝑢 ∈ (Base‘𝑄))
37	36	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → 𝑢 ∈ (Base‘𝑄))
38	37	ad10antr 757	. . . . . . . . . . . . . 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 7447	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ~QG 𝑀) ∈ V
40	39	ecelqsi 8772	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ (Base‘𝑅) → [𝑟](𝑅 ~QG 𝑀) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
41	40	ad4antlr 746	. . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅))
44		ovexd 7449	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑅 ~QG 𝑀) ∈ V)
45	42, 43, 44, 3	qusbas 17634	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((Base‘𝑅) / (𝑅 ~QG 𝑀)) = (Base‘𝑄))
46	45	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → ((Base‘𝑅) / (𝑅 ~QG 𝑀)) = (Base‘𝑄))
47	46	ad10antr 757	. . . . . . . . . . . . . . 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 2862	. . . . . . . . . . . . . 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 8772	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ (Base‘𝑅) → [𝑠](𝑅 ~QG 𝑀) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
50	49	ad2antlr 740	. . . . . . . . . . . . . . 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 2862	. . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . . 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 806	. . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . . 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 802	. . . . . . . . . . . . . . 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 2797	. . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . 16 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
59		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (.r‘(oppr‘𝑄)) = (.r‘(oppr‘𝑄))
60	29, 30, 58, 59	opprmul 20484	. . . . . . . . . . . . . . 15 ⊢ ([𝑠](𝑅 ~QG 𝑀)(.r‘(oppr‘𝑄))𝑢) = (𝑢(.r‘𝑄)[𝑠](𝑅 ~QG 𝑀))
61		simp-5r 798	. . . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑅 ∈ Ring)
63	62	ad8antr 753	. . . . . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑀 ∈ (2Ideal‘𝑅))
65	64	ad8antr 753	. . . . . . . . . . . . . . . . . 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 33896	. . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . 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 21402	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑀 ∈ (LIdeal‘𝑅) → 𝑀 ⊆ (Base‘𝑅))
69	8, 68	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑀 ⊆ (Base‘𝑅))
70	9, 2	oppreqg 33888	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ⊆ (Base‘𝑅)) → (𝑅 ~QG 𝑀) = (𝑂 ~QG 𝑀))
71	5, 69, 70	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑅 ~QG 𝑀) = (𝑂 ~QG 𝑀))
72	71	ad10antr 757	. . . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . 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 8743	. . . . . . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . 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 20494	. . . . . . . . . . . . . . . . . . 19 ⊢ (1r‘𝑄) = (1r‘(oppr‘𝑄))
79	2, 9, 1, 5, 21	opprqus1r 33897	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (1r‘(oppr‘𝑄)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀))))
80	78, 79	eqtrid 2807	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1r‘𝑄) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀))))
81	80	ad10antr 757	. . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . 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 2805	. . . . . . . . . . . . . . 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 2809	. . . . . . . . . . . . . 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 20904	. . . . . . . . . . . . 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 2806	. . . . . . . . . . . 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 7430	. . . . . . . . . . 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 7430	. . . . . . . . . . 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 2799	. . . . . . . . . 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 796	. . . . . . . . . . . 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 8763	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((Base‘𝑅) / (𝑅 ~QG 𝑀)) = ((Base‘𝑅) / (𝑂 ~QG 𝑀)))
92	91	ad9antr 755	. . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . 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 20487	. . . . . . . . . . . . . . 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 7449	. . . . . . . . . . . . . 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 17634	. . . . . . . . . . . . 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 2796	. . . . . . . . . . . 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 2862	. . . . . . . . . . 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 8768	. . . . . . . . . . 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 3170	. . . . . . . . 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 796	. . . . . . . . . . 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 749	. . . . . . . . . . 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 2863	. . . . . . . . . 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 8768	. . . . . . . . . 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 3170	. . . . . . . 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 2760	. . . . . . . . . 10 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
112		eqid 2760	. . . . . . . . . 10 ⊢ (𝑂 /s (𝑂 ~QG 𝑀)) = (𝑂 /s (𝑂 ~QG 𝑀))
113	3	ad3antrrr 743	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑅 ∈ NzRing)
114	9	opprnzr 20686	. . . . . . . . . . 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 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑀 ∈ (MaxIdeal‘𝑂))
117	6	ad3antrrr 743	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑀 ∈ (MaxIdeal‘𝑅))
118	9, 62, 117	opprmxidlabs 33892	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑀 ∈ (MaxIdeal‘(oppr‘𝑂)))
119		simplr 781	. . . . . . . . . . 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 2862	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑥 ∈ (Base‘𝑂))
122		simplr 781	. . . . . . . . . . . 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 20384	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ Grp)
124	123	ad4antr 745	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → 𝑅 ∈ Grp)
125		lidlnsg 21448	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) → 𝑀 ∈ (NrmSGrp‘𝑅))
126	5, 8, 125	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ (NrmSGrp‘𝑅))
127		nsgsubg 19284	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (NrmSGrp‘𝑅) → 𝑀 ∈ (SubGrp‘𝑅))
128	126, 127	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ (SubGrp‘𝑅))
129	128	ad4antr 745	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → 𝑀 ∈ (SubGrp‘𝑅))
130		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → 𝑥 ∈ 𝑀)
131		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ~QG 𝑀) = (𝑅 ~QG 𝑀)
132	131	eqg0el 19314	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Grp ∧ 𝑀 ∈ (SubGrp‘𝑅)) → ([𝑥](𝑅 ~QG 𝑀) = 𝑀 ↔ 𝑥 ∈ 𝑀))
133	132	biimpar 483	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Grp ∧ 𝑀 ∈ (SubGrp‘𝑅)) ∧ 𝑥 ∈ 𝑀) → [𝑥](𝑅 ~QG 𝑀) = 𝑀)
134	124, 129, 130, 133	syl21anc 851	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → [𝑥](𝑅 ~QG 𝑀) = 𝑀)
135		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝑅) = (0g‘𝑅)
136	2, 131, 135	eqgid 19308	. . . . . . . . . . . . . 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 2798	. . . . . . . . . . . 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 19320	. . . . . . . . . . . . . 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 745	. . . . . . . . . . . 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 2799	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥 ∈ 𝑀) → 𝑢 = (0g‘𝑄))
143		eldifsnneq 4754	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)}) → ¬ 𝑢 = (0g‘𝑄))
144	143	ad4antlr 746	. . . . . . . . . . 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 829	. . . . . . . . . 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 33899	. . . . . . . . 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 739	. . . . . . . 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 3170	. . . . . . 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 788	. . . . . . . . 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 7430	. . . . . . . 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 490	. . . . . . . 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 2795	. . . . . . 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 521	. . . . . 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 472	. . . . 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 33899	. . . . 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 3178	. . . 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 2863	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → 𝑢 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
158		elqsi 8768	. . . . 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 3170	. . 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 3154	. 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 20905	. 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 726	1 ⊢ (𝜑 → 𝑄 ∈ DivRing)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  {csn 4584  ‘cfv 6533  (class class class)co 7414  [cec 8697   / cqs 8698  Basecbs 17304  ._rcmulr 17346  0_gc0g 17527   /_s cqus 17594  Grpcgrp 19060  SubGrpcsubg 19246  NrmSGrpcnsg 19247   ~_QG cqg 19248  1_rcur 20323  Ringcrg 20375  opp_rcoppr 20480  Unitcui 20499  NzRingcnzr 20675  DivRingcdr 20893  LIdealclidl 21396  2Idealc2idl 21454  MaxIdealcmxidl 33865

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  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 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7679  df-om 7864  df-1st 7987  df-2nd 7988  df-supp 8160  df-tpos 8225  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8458  df-2o 8459  df-er 8699  df-ec 8701  df-qs 8705  df-map 8831  df-ixp 8908  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-fsupp 9335  df-sup 9415  df-inf 9416  df-oi 9485  df-card 9947  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12532  df-z 12619  df-dec 12740  df-uz 12891  df-fz 13565  df-fzo 13713  df-seq 14069  df-hash 14398  df-struct 17242  df-sets 17259  df-slot 17277  df-ndx 17289  df-base 17305  df-ress 17326  df-plusg 17358  df-mulr 17359  df-sca 17361  df-vsca 17362  df-ip 17363  df-tset 17364  df-ple 17365  df-ds 17367  df-hom 17369  df-cco 17370  df-0g 17529  df-gsum 17530  df-prds 17535  df-pws 17537  df-imas 17597  df-qus 17598  df-mre 17673  df-mrc 17674  df-acs 17676  df-mgm 18733  df-sgrp 18824  df-mnd 18840  df-mhm 18894  df-submnd 18895  df-grp 19063  df-minusg 19064  df-sbg 19065  df-mulg 19194  df-subg 19249  df-nsg 19250  df-eqg 19251  df-ghm 19344  df-cntz 19447  df-cmn 19912  df-abl 19913  df-mgp 20277  df-rng 20291  df-ur 20324  df-ring 20377  df-oppr 20481  df-dvdsr 20501  df-unit 20502  df-invr 20532  df-nzr 20676  df-subrg 20735  df-drng 20895  df-lmod 21049  df-lss 21119  df-lsp 21159  df-lmhm 21209  df-lbs 21262  df-sra 21360  df-rgmod 21361  df-lidl 21398  df-rsp 21399  df-2idl 21455  df-dsmm 21948  df-frlm 21963  df-uvc 21999  df-mxidl 33866

This theorem is used by:  qsdrng  33902  algextdeglem4  34233