Theorem opprqusmulr 33435

Description: The multiplication operation of the quotient of the opposite ring is the same as the multiplication operation of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)

Hypotheses

Ref	Expression
opprqus.b	⊢ 𝐵 = (Base‘𝑅)
opprqus.o	⊢ 𝑂 = (oppr‘𝑅)
opprqus.q	⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))
opprqus1r.r	⊢ (𝜑 → 𝑅 ∈ Ring)
opprqus1r.i	⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))
opprqusmulr.e	⊢ 𝐸 = (Base‘𝑄)
opprqusmulr.x	⊢ (𝜑 → 𝑋 ∈ 𝐸)
opprqusmulr.y	⊢ (𝜑 → 𝑌 ∈ 𝐸)

Assertion

Ref	Expression
opprqusmulr	⊢ (𝜑 → (𝑋(.r‘(oppr‘𝑄))𝑌) = (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))

Proof of Theorem opprqusmulr

Dummy variables 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opprqusmulr.e	. . 3 ⊢ 𝐸 = (Base‘𝑄)
2		eqid 2729	. . 3 ⊢ (.r‘𝑄) = (.r‘𝑄)
3		eqid 2729	. . 3 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
4		eqid 2729	. . 3 ⊢ (.r‘(oppr‘𝑄)) = (.r‘(oppr‘𝑄))
5	1, 2, 3, 4	opprmul 20225	. 2 ⊢ (𝑋(.r‘(oppr‘𝑄))𝑌) = (𝑌(.r‘𝑄)𝑋)
6		opprqus.q	. . . . . 6 ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))
7		opprqus.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
8		eqid 2729	. . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅)
9		opprqus1r.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
10	9	ad4antr 732	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑅 ∈ Ring)
11		opprqus1r.i	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))
12	11	ad4antr 732	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝐼 ∈ (2Ideal‘𝑅))
13		simplr 768	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑞 ∈ 𝐵)
14		simp-4r 783	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑝 ∈ 𝐵)
15	6, 7, 8, 2, 10, 12, 13, 14	qusmul2idl 21165	. . . . 5 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → ([𝑞](𝑅 ~QG 𝐼)(.r‘𝑄)[𝑝](𝑅 ~QG 𝐼)) = [(𝑞(.r‘𝑅)𝑝)](𝑅 ~QG 𝐼))
16		simpr 484	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑌 = [𝑞](𝑅 ~QG 𝐼))
17		simpllr 775	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑋 = [𝑝](𝑅 ~QG 𝐼))
18	16, 17	oveq12d 7387	. . . . 5 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → (𝑌(.r‘𝑄)𝑋) = ([𝑞](𝑅 ~QG 𝐼)(.r‘𝑄)[𝑝](𝑅 ~QG 𝐼)))
19		eqid 2729	. . . . . . 7 ⊢ (𝑂 /s (𝑂 ~QG 𝐼)) = (𝑂 /s (𝑂 ~QG 𝐼))
20		opprqus.o	. . . . . . . 8 ⊢ 𝑂 = (oppr‘𝑅)
21	20, 7	opprbas 20228	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑂)
22		eqid 2729	. . . . . . 7 ⊢ (.r‘𝑂) = (.r‘𝑂)
23		eqid 2729	. . . . . . 7 ⊢ (.r‘(𝑂 /s (𝑂 ~QG 𝐼))) = (.r‘(𝑂 /s (𝑂 ~QG 𝐼)))
24	20	opprring 20232	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring)
25	9, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑂 ∈ Ring)
26	25	ad4antr 732	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑂 ∈ Ring)
27	20, 9	oppr2idl 33430	. . . . . . . . 9 ⊢ (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂))
28	11, 27	eleqtrd 2830	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑂))
29	28	ad4antr 732	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝐼 ∈ (2Ideal‘𝑂))
30	19, 21, 22, 23, 26, 29, 14, 13	qusmul2idl 21165	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → ([𝑝](𝑂 ~QG 𝐼)(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))[𝑞](𝑂 ~QG 𝐼)) = [(𝑝(.r‘𝑂)𝑞)](𝑂 ~QG 𝐼))
31	11	2idllidld 21140	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))
32		eqid 2729	. . . . . . . . . . . . 13 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
33	7, 32	lidlss 21098	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (LIdeal‘𝑅) → 𝐼 ⊆ 𝐵)
34	31, 33	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ⊆ 𝐵)
35	20, 7	oppreqg 33427	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ⊆ 𝐵) → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼))
36	9, 34, 35	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼))
37	36	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼))
38	37	eceq2d 8691	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → [𝑝](𝑅 ~QG 𝐼) = [𝑝](𝑂 ~QG 𝐼))
39	17, 38	eqtrd 2764	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑋 = [𝑝](𝑂 ~QG 𝐼))
40	37	eceq2d 8691	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → [𝑞](𝑅 ~QG 𝐼) = [𝑞](𝑂 ~QG 𝐼))
41	16, 40	eqtrd 2764	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑌 = [𝑞](𝑂 ~QG 𝐼))
42	39, 41	oveq12d 7387	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌) = ([𝑝](𝑂 ~QG 𝐼)(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))[𝑞](𝑂 ~QG 𝐼)))
43	7, 8, 20, 22	opprmul 20225	. . . . . . . . 9 ⊢ (𝑝(.r‘𝑂)𝑞) = (𝑞(.r‘𝑅)𝑝)
44	43	a1i 11	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → (𝑝(.r‘𝑂)𝑞) = (𝑞(.r‘𝑅)𝑝))
45	44	eceq1d 8688	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → [(𝑝(.r‘𝑂)𝑞)](𝑅 ~QG 𝐼) = [(𝑞(.r‘𝑅)𝑝)](𝑅 ~QG 𝐼))
46	37	eceq2d 8691	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → [(𝑝(.r‘𝑂)𝑞)](𝑅 ~QG 𝐼) = [(𝑝(.r‘𝑂)𝑞)](𝑂 ~QG 𝐼))
47	45, 46	eqtr3d 2766	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → [(𝑞(.r‘𝑅)𝑝)](𝑅 ~QG 𝐼) = [(𝑝(.r‘𝑂)𝑞)](𝑂 ~QG 𝐼))
48	30, 42, 47	3eqtr4d 2774	. . . . 5 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌) = [(𝑞(.r‘𝑅)𝑝)](𝑅 ~QG 𝐼))
49	15, 18, 48	3eqtr4d 2774	. . . 4 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → (𝑌(.r‘𝑄)𝑋) = (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))
50		opprqusmulr.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐸)
51	3, 1	opprbas 20228	. . . . . . . 8 ⊢ 𝐸 = (Base‘(oppr‘𝑄))
52	50, 51	eleqtrdi 2838	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Base‘(oppr‘𝑄)))
53	52	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) → 𝑌 ∈ (Base‘(oppr‘𝑄)))
54	6	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)))
55	7	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (Base‘𝑅))
56		ovexd 7404	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 ~QG 𝐼) ∈ V)
57	54, 55, 56, 9	qusbas 17484	. . . . . . . 8 ⊢ (𝜑 → (𝐵 / (𝑅 ~QG 𝐼)) = (Base‘𝑄))
58	1, 51	eqtr3i 2754	. . . . . . . 8 ⊢ (Base‘𝑄) = (Base‘(oppr‘𝑄))
59	57, 58	eqtr2di 2781	. . . . . . 7 ⊢ (𝜑 → (Base‘(oppr‘𝑄)) = (𝐵 / (𝑅 ~QG 𝐼)))
60	59	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) → (Base‘(oppr‘𝑄)) = (𝐵 / (𝑅 ~QG 𝐼)))
61	53, 60	eleqtrd 2830	. . . . 5 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) → 𝑌 ∈ (𝐵 / (𝑅 ~QG 𝐼)))
62		elqsi 8716	. . . . 5 ⊢ (𝑌 ∈ (𝐵 / (𝑅 ~QG 𝐼)) → ∃𝑞 ∈ 𝐵 𝑌 = [𝑞](𝑅 ~QG 𝐼))
63	61, 62	syl 17	. . . 4 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) → ∃𝑞 ∈ 𝐵 𝑌 = [𝑞](𝑅 ~QG 𝐼))
64	49, 63	r19.29a 3141	. . 3 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) → (𝑌(.r‘𝑄)𝑋) = (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))
65		opprqusmulr.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐸)
66	65, 51	eleqtrdi 2838	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘(oppr‘𝑄)))
67	66, 59	eleqtrd 2830	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐵 / (𝑅 ~QG 𝐼)))
68		elqsi 8716	. . . 4 ⊢ (𝑋 ∈ (𝐵 / (𝑅 ~QG 𝐼)) → ∃𝑝 ∈ 𝐵 𝑋 = [𝑝](𝑅 ~QG 𝐼))
69	67, 68	syl 17	. . 3 ⊢ (𝜑 → ∃𝑝 ∈ 𝐵 𝑋 = [𝑝](𝑅 ~QG 𝐼))
70	64, 69	r19.29a 3141	. 2 ⊢ (𝜑 → (𝑌(.r‘𝑄)𝑋) = (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))
71	5, 70	eqtrid 2776	1 ⊢ (𝜑 → (𝑋(.r‘(oppr‘𝑄))𝑌) = (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  Vcvv 3444   ⊆ wss 3911  ‘cfv 6499  (class class class)co 7369  [cec 8646   / cqs 8647  Basecbs 17155  ._rcmulr 17197   /_s cqus 17444   ~_QG cqg 19030  Ringcrg 20118  opp_rcoppr 20221  LIdealclidl 21092  2Idealc2idl 21135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-tpos 8182  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-ec 8650  df-qs 8654  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9369  df-inf 9370  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-uz 12770  df-fz 13445  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-mulr 17210  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-0g 17380  df-imas 17447  df-qus 17448  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-grp 18844  df-minusg 18845  df-sbg 18846  df-subg 19031  df-eqg 19033  df-cmn 19688  df-abl 19689  df-mgp 20026  df-rng 20038  df-ur 20067  df-ring 20120  df-oppr 20222  df-subrg 20455  df-lmod 20744  df-lss 20814  df-sra 21056  df-rgmod 21057  df-lidl 21094  df-2idl 21136

This theorem is referenced by:  opprqus1r  33436  opprqusdrng  33437  qsdrngi  33439