Theorem opprqusplusg 33460

Description: The group operation of the quotient of the opposite ring is the same as the group operation of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 13-Mar-2025.)

Hypotheses

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

Assertion

Ref	Expression
opprqusplusg	⊢ (𝜑 → (𝑋(+g‘(oppr‘𝑄))𝑌) = (𝑋(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))

Proof of Theorem opprqusplusg

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

Step	Hyp	Ref	Expression
1		eqid 2729	. . . 4 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
2		eqid 2729	. . . 4 ⊢ (+g‘𝑄) = (+g‘𝑄)
3	1, 2	oppradd 20253	. . 3 ⊢ (+g‘𝑄) = (+g‘(oppr‘𝑄))
4	3	oveqi 7400	. 2 ⊢ (𝑋(+g‘𝑄)𝑌) = (𝑋(+g‘(oppr‘𝑄))𝑌)
5		opprqus.i	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
6	5	ad4antr 732	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝐼 ∈ (NrmSGrp‘𝑅))
7		simp-4r 783	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑝 ∈ 𝐵)
8		simplr 768	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑞 ∈ 𝐵)
9		opprqus.q	. . . . . . 7 ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))
10		opprqus.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
11		eqid 2729	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
12	9, 10, 11, 2	qusadd 19120	. . . . . 6 ⊢ ((𝐼 ∈ (NrmSGrp‘𝑅) ∧ 𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → ([𝑝](𝑅 ~QG 𝐼)(+g‘𝑄)[𝑞](𝑅 ~QG 𝐼)) = [(𝑝(+g‘𝑅)𝑞)](𝑅 ~QG 𝐼))
13	6, 7, 8, 12	syl3anc 1373	. . . . 5 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → ([𝑝](𝑅 ~QG 𝐼)(+g‘𝑄)[𝑞](𝑅 ~QG 𝐼)) = [(𝑝(+g‘𝑅)𝑞)](𝑅 ~QG 𝐼))
14		simpllr 775	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑋 = [𝑝](𝑅 ~QG 𝐼))
15		simpr 484	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑌 = [𝑞](𝑅 ~QG 𝐼))
16	14, 15	oveq12d 7405	. . . . 5 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → (𝑋(+g‘𝑄)𝑌) = ([𝑝](𝑅 ~QG 𝐼)(+g‘𝑄)[𝑞](𝑅 ~QG 𝐼)))
17	5	elfvexd 6897	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ V)
18		nsgsubg 19090	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅))
19	10	subgss 19059	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (SubGrp‘𝑅) → 𝐼 ⊆ 𝐵)
20	5, 18, 19	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ⊆ 𝐵)
21		opprqus.o	. . . . . . . . . . . 12 ⊢ 𝑂 = (oppr‘𝑅)
22	21, 10	oppreqg 33454	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ V ∧ 𝐼 ⊆ 𝐵) → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼))
23	17, 20, 22	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼))
24	23	eceq2d 8714	. . . . . . . . 9 ⊢ (𝜑 → [𝑝](𝑅 ~QG 𝐼) = [𝑝](𝑂 ~QG 𝐼))
25	23	eceq2d 8714	. . . . . . . . 9 ⊢ (𝜑 → [𝑞](𝑅 ~QG 𝐼) = [𝑞](𝑂 ~QG 𝐼))
26	24, 25	oveq12d 7405	. . . . . . . 8 ⊢ (𝜑 → ([𝑝](𝑅 ~QG 𝐼)(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))[𝑞](𝑅 ~QG 𝐼)) = ([𝑝](𝑂 ~QG 𝐼)(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))[𝑞](𝑂 ~QG 𝐼)))
27	26	ad4antr 732	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → ([𝑝](𝑅 ~QG 𝐼)(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))[𝑞](𝑅 ~QG 𝐼)) = ([𝑝](𝑂 ~QG 𝐼)(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))[𝑞](𝑂 ~QG 𝐼)))
28	21	opprnsg 33455	. . . . . . . . . 10 ⊢ (NrmSGrp‘𝑅) = (NrmSGrp‘𝑂)
29	5, 28	eleqtrdi 2838	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑂))
30	29	ad4antr 732	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝐼 ∈ (NrmSGrp‘𝑂))
31	7, 10	eleqtrdi 2838	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑝 ∈ (Base‘𝑅))
32	8, 10	eleqtrdi 2838	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑞 ∈ (Base‘𝑅))
33		eqid 2729	. . . . . . . . 9 ⊢ (𝑂 /s (𝑂 ~QG 𝐼)) = (𝑂 /s (𝑂 ~QG 𝐼))
34	21, 10	opprbas 20252	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑂)
35	10, 34	eqtr3i 2754	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑂)
36	21, 11	oppradd 20253	. . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑂)
37		eqid 2729	. . . . . . . . 9 ⊢ (+g‘(𝑂 /s (𝑂 ~QG 𝐼))) = (+g‘(𝑂 /s (𝑂 ~QG 𝐼)))
38	33, 35, 36, 37	qusadd 19120	. . . . . . . 8 ⊢ ((𝐼 ∈ (NrmSGrp‘𝑂) ∧ 𝑝 ∈ (Base‘𝑅) ∧ 𝑞 ∈ (Base‘𝑅)) → ([𝑝](𝑂 ~QG 𝐼)(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))[𝑞](𝑂 ~QG 𝐼)) = [(𝑝(+g‘𝑅)𝑞)](𝑂 ~QG 𝐼))
39	30, 31, 32, 38	syl3anc 1373	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → ([𝑝](𝑂 ~QG 𝐼)(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))[𝑞](𝑂 ~QG 𝐼)) = [(𝑝(+g‘𝑅)𝑞)](𝑂 ~QG 𝐼))
40	27, 39	eqtrd 2764	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → ([𝑝](𝑅 ~QG 𝐼)(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))[𝑞](𝑅 ~QG 𝐼)) = [(𝑝(+g‘𝑅)𝑞)](𝑂 ~QG 𝐼))
41	14, 15	oveq12d 7405	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → (𝑋(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌) = ([𝑝](𝑅 ~QG 𝐼)(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))[𝑞](𝑅 ~QG 𝐼)))
42	23	ad4antr 732	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼))
43	42	eceq2d 8714	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → [(𝑝(+g‘𝑅)𝑞)](𝑅 ~QG 𝐼) = [(𝑝(+g‘𝑅)𝑞)](𝑂 ~QG 𝐼))
44	40, 41, 43	3eqtr4d 2774	. . . . 5 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → (𝑋(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌) = [(𝑝(+g‘𝑅)𝑞)](𝑅 ~QG 𝐼))
45	13, 16, 44	3eqtr4d 2774	. . . 4 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → (𝑋(+g‘𝑄)𝑌) = (𝑋(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))
46		opprqusplusg.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐸)
47		opprqusplusg.e	. . . . . . . 8 ⊢ 𝐸 = (Base‘𝑄)
48	9	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)))
49	10	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (Base‘𝑅))
50		ovexd 7422	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 ~QG 𝐼) ∈ V)
51	48, 49, 50, 17	qusbas 17508	. . . . . . . 8 ⊢ (𝜑 → (𝐵 / (𝑅 ~QG 𝐼)) = (Base‘𝑄))
52	47, 51	eqtr4id 2783	. . . . . . 7 ⊢ (𝜑 → 𝐸 = (𝐵 / (𝑅 ~QG 𝐼)))
53	46, 52	eleqtrd 2830	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝐵 / (𝑅 ~QG 𝐼)))
54	53	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) → 𝑌 ∈ (𝐵 / (𝑅 ~QG 𝐼)))
55		elqsi 8739	. . . . 5 ⊢ (𝑌 ∈ (𝐵 / (𝑅 ~QG 𝐼)) → ∃𝑞 ∈ 𝐵 𝑌 = [𝑞](𝑅 ~QG 𝐼))
56	54, 55	syl 17	. . . 4 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) → ∃𝑞 ∈ 𝐵 𝑌 = [𝑞](𝑅 ~QG 𝐼))
57	45, 56	r19.29a 3141	. . 3 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) → (𝑋(+g‘𝑄)𝑌) = (𝑋(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))
58		opprqusplusg.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐸)
59	58, 52	eleqtrd 2830	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐵 / (𝑅 ~QG 𝐼)))
60		elqsi 8739	. . . 4 ⊢ (𝑋 ∈ (𝐵 / (𝑅 ~QG 𝐼)) → ∃𝑝 ∈ 𝐵 𝑋 = [𝑝](𝑅 ~QG 𝐼))
61	59, 60	syl 17	. . 3 ⊢ (𝜑 → ∃𝑝 ∈ 𝐵 𝑋 = [𝑝](𝑅 ~QG 𝐼))
62	57, 61	r19.29a 3141	. 2 ⊢ (𝜑 → (𝑋(+g‘𝑄)𝑌) = (𝑋(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))
63	4, 62	eqtr3id 2778	1 ⊢ (𝜑 → (𝑋(+g‘(oppr‘𝑄))𝑌) = (𝑋(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  Vcvv 3447   ⊆ wss 3914  ‘cfv 6511  (class class class)co 7387  [cec 8669   / cqs 8670  Basecbs 17179  +_gcplusg 17220   /_s cqus 17468  SubGrpcsubg 19052  NrmSGrpcnsg 19053   ~_QG cqg 19054  opp_rcoppr 20245

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-tpos 8205  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-ec 8673  df-qs 8677  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-sup 9393  df-inf 9394  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-0g 17404  df-imas 17471  df-qus 17472  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-grp 18868  df-minusg 18869  df-subg 19055  df-nsg 19056  df-eqg 19057  df-oppr 20246

This theorem is referenced by:  opprqus0g  33461