Theorem opprqusmulr 33448

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 2731	. . 3 ⊢ (.r‘𝑄) = (.r‘𝑄)
3		eqid 2731	. . 3 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
4		eqid 2731	. . 3 ⊢ (.r‘(oppr‘𝑄)) = (.r‘(oppr‘𝑄))
5	1, 2, 3, 4	opprmul 20253	. 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 2731	. . . . . 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 21211	. . . . 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 7359	. . . . 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 2731	. . . . . . 7 ⊢ (𝑂 /s (𝑂 ~QG 𝐼)) = (𝑂 /s (𝑂 ~QG 𝐼))
20		opprqus.o	. . . . . . . 8 ⊢ 𝑂 = (oppr‘𝑅)
21	20, 7	opprbas 20256	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑂)
22		eqid 2731	. . . . . . 7 ⊢ (.r‘𝑂) = (.r‘𝑂)
23		eqid 2731	. . . . . . 7 ⊢ (.r‘(𝑂 /s (𝑂 ~QG 𝐼))) = (.r‘(𝑂 /s (𝑂 ~QG 𝐼)))
24	20	opprring 20260	. . . . . . . . 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 33443	. . . . . . . . 9 ⊢ (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂))
28	11, 27	eleqtrd 2833	. . . . . . . 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 21211	. . . . . 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 21186	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))
32		eqid 2731	. . . . . . . . . . . . 13 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
33	7, 32	lidlss 21144	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (LIdeal‘𝑅) → 𝐼 ⊆ 𝐵)
34	31, 33	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ⊆ 𝐵)
35	20, 7	oppreqg 33440	. . . . . . . . . . 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 8660	. . . . . . . 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 2766	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑋 = [𝑝](𝑂 ~QG 𝐼))
40	37	eceq2d 8660	. . . . . . . 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 2766	. . . . . . 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 7359	. . . . . 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 20253	. . . . . . . . 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 8657	. . . . . . 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 8660	. . . . . . 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 2768	. . . . . 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 2776	. . . . 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 2776	. . . 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 20256	. . . . . . . 8 ⊢ 𝐸 = (Base‘(oppr‘𝑄))
52	50, 51	eleqtrdi 2841	. . . . . . 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 7376	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 ~QG 𝐼) ∈ V)
57	54, 55, 56, 9	qusbas 17444	. . . . . . . 8 ⊢ (𝜑 → (𝐵 / (𝑅 ~QG 𝐼)) = (Base‘𝑄))
58	1, 51	eqtr3i 2756	. . . . . . . 8 ⊢ (Base‘𝑄) = (Base‘(oppr‘𝑄))
59	57, 58	eqtr2di 2783	. . . . . . 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 2833	. . . . 5 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) → 𝑌 ∈ (𝐵 / (𝑅 ~QG 𝐼)))
62		elqsi 8685	. . . . 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 3140	. . 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 2841	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘(oppr‘𝑄)))
67	66, 59	eleqtrd 2833	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐵 / (𝑅 ~QG 𝐼)))
68		elqsi 8685	. . . 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 3140	. 2 ⊢ (𝜑 → (𝑌(.r‘𝑄)𝑋) = (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))
71	5, 70	eqtrid 2778	1 ⊢ (𝜑 → (𝑋(.r‘(oppr‘𝑄))𝑌) = (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  Vcvv 3436   ⊆ wss 3897  ‘cfv 6476  (class class class)co 7341  [cec 8615   / cqs 8616  Basecbs 17115  ._rcmulr 17157   /_s cqus 17404   ~_QG cqg 19030  Ringcrg 20146  opp_rcoppr 20249  LIdealclidl 21138  2Idealc2idl 21181

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-tp 4576  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-tpos 8151  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-er 8617  df-ec 8619  df-qs 8623  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-sup 9321  df-inf 9322  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-nn 12121  df-2 12183  df-3 12184  df-4 12185  df-5 12186  df-6 12187  df-7 12188  df-8 12189  df-9 12190  df-n0 12377  df-z 12464  df-dec 12584  df-uz 12728  df-fz 13403  df-struct 17053  df-sets 17070  df-slot 17088  df-ndx 17100  df-base 17116  df-ress 17137  df-plusg 17169  df-mulr 17170  df-sca 17172  df-vsca 17173  df-ip 17174  df-tset 17175  df-ple 17176  df-ds 17178  df-0g 17340  df-imas 17407  df-qus 17408  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 19689  df-abl 19690  df-mgp 20054  df-rng 20066  df-ur 20095  df-ring 20148  df-oppr 20250  df-subrg 20480  df-lmod 20790  df-lss 20860  df-sra 21102  df-rgmod 21103  df-lidl 21140  df-2idl 21182

This theorem is referenced by:  opprqus1r  33449  opprqusdrng  33450  qsdrngi  33452