Theorem opprqusplusg 33433

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 20229	. . 3 ⊢ (+g‘𝑄) = (+g‘(oppr‘𝑄))
4	3	oveqi 7382	. 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 19096	. . . . . 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 7387	. . . . 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 6879	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ V)
18		nsgsubg 19066	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅))
19	10	subgss 19035	. . . . . . . . . . . 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 33427	. . . . . . . . . . 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 8691	. . . . . . . . 9 ⊢ (𝜑 → [𝑝](𝑅 ~QG 𝐼) = [𝑝](𝑂 ~QG 𝐼))
25	23	eceq2d 8691	. . . . . . . . 9 ⊢ (𝜑 → [𝑞](𝑅 ~QG 𝐼) = [𝑞](𝑂 ~QG 𝐼))
26	24, 25	oveq12d 7387	. . . . . . . 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 33428	. . . . . . . . . 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 20228	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑂)
35	10, 34	eqtr3i 2754	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑂)
36	21, 11	oppradd 20229	. . . . . . . . 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 19096	. . . . . . . 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 7387	. . . . . 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 8691	. . . . . 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 7404	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 ~QG 𝐼) ∈ V)
51	48, 49, 50, 17	qusbas 17484	. . . . . . . 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 8716	. . . . 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 8716	. . . 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 3444   ⊆ wss 3911  ‘cfv 6499  (class class class)co 7369  [cec 8646   / cqs 8647  Basecbs 17155  +_gcplusg 17196   /_s cqus 17444  SubGrpcsubg 19028  NrmSGrpcnsg 19029   ~_QG cqg 19030  opp_rcoppr 20221

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-subg 19031  df-nsg 19032  df-eqg 19033  df-oppr 20222

This theorem is referenced by:  opprqus0g  33434