Theorem opprqusmulr 33521

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 2734	. . 3 ⊢ (.r‘𝑄) = (.r‘𝑄)
3		eqid 2734	. . 3 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
4		eqid 2734	. . 3 ⊢ (.r‘(oppr‘𝑄)) = (.r‘(oppr‘𝑄))
5	1, 2, 3, 4	opprmul 20274	. 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 2734	. . . . . 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 21232	. . . . 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 7374	. . . . 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 2734	. . . . . . 7 ⊢ (𝑂 /s (𝑂 ~QG 𝐼)) = (𝑂 /s (𝑂 ~QG 𝐼))
20		opprqus.o	. . . . . . . 8 ⊢ 𝑂 = (oppr‘𝑅)
21	20, 7	opprbas 20277	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑂)
22		eqid 2734	. . . . . . 7 ⊢ (.r‘𝑂) = (.r‘𝑂)
23		eqid 2734	. . . . . . 7 ⊢ (.r‘(𝑂 /s (𝑂 ~QG 𝐼))) = (.r‘(𝑂 /s (𝑂 ~QG 𝐼)))
24	20	opprring 20281	. . . . . . . . 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 33516	. . . . . . . . 9 ⊢ (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂))
28	11, 27	eleqtrd 2836	. . . . . . . 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 21232	. . . . . 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 21207	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))
32		eqid 2734	. . . . . . . . . . . . 13 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
33	7, 32	lidlss 21165	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (LIdeal‘𝑅) → 𝐼 ⊆ 𝐵)
34	31, 33	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ⊆ 𝐵)
35	20, 7	oppreqg 33513	. . . . . . . . . . 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 8676	. . . . . . . 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 2769	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑋 = [𝑝](𝑂 ~QG 𝐼))
40	37	eceq2d 8676	. . . . . . . 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 2769	. . . . . . 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 7374	. . . . . 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 20274	. . . . . . . . 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 8673	. . . . . . 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 8676	. . . . . . 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 2771	. . . . . 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 2779	. . . . 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 2779	. . . 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 20277	. . . . . . . 8 ⊢ 𝐸 = (Base‘(oppr‘𝑄))
52	50, 51	eleqtrdi 2844	. . . . . . 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 7391	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 ~QG 𝐼) ∈ V)
57	54, 55, 56, 9	qusbas 17464	. . . . . . . 8 ⊢ (𝜑 → (𝐵 / (𝑅 ~QG 𝐼)) = (Base‘𝑄))
58	1, 51	eqtr3i 2759	. . . . . . . 8 ⊢ (Base‘𝑄) = (Base‘(oppr‘𝑄))
59	57, 58	eqtr2di 2786	. . . . . . 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 2836	. . . . 5 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) → 𝑌 ∈ (𝐵 / (𝑅 ~QG 𝐼)))
62		elqsi 8701	. . . . 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 3142	. . 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 2844	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘(oppr‘𝑄)))
67	66, 59	eleqtrd 2836	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐵 / (𝑅 ~QG 𝐼)))
68		elqsi 8701	. . . 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 3142	. 2 ⊢ (𝜑 → (𝑌(.r‘𝑄)𝑋) = (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))
71	5, 70	eqtrid 2781	1 ⊢ (𝜑 → (𝑋(.r‘(oppr‘𝑄))𝑌) = (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3058  Vcvv 3438   ⊆ wss 3899  ‘cfv 6490  (class class class)co 7356  [cec 8631   / cqs 8632  Basecbs 17134  ._rcmulr 17176   /_s cqus 17424   ~_QG cqg 19050  Ringcrg 20166  opp_rcoppr 20270  LIdealclidl 21159  2Idealc2idl 21202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-tpos 8166  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-ec 8635  df-qs 8639  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-sup 9343  df-inf 9344  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-2 12206  df-3 12207  df-4 12208  df-5 12209  df-6 12210  df-7 12211  df-8 12212  df-9 12213  df-n0 12400  df-z 12487  df-dec 12606  df-uz 12750  df-fz 13422  df-struct 17072  df-sets 17089  df-slot 17107  df-ndx 17119  df-base 17135  df-ress 17156  df-plusg 17188  df-mulr 17189  df-sca 17191  df-vsca 17192  df-ip 17193  df-tset 17194  df-ple 17195  df-ds 17197  df-0g 17359  df-imas 17427  df-qus 17428  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-grp 18864  df-minusg 18865  df-sbg 18866  df-subg 19051  df-eqg 19053  df-cmn 19709  df-abl 19710  df-mgp 20074  df-rng 20086  df-ur 20115  df-ring 20168  df-oppr 20271  df-subrg 20501  df-lmod 20811  df-lss 20881  df-sra 21123  df-rgmod 21124  df-lidl 21161  df-2idl 21203

This theorem is referenced by:  opprqus1r  33522  opprqusdrng  33523  qsdrngi  33525