Theorem opprqusplusg 33572

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 2739	. . . 4 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
2		eqid 2739	. . . 4 ⊢ (+g‘𝑄) = (+g‘𝑄)
3	1, 2	oppradd 20315	. . 3 ⊢ (+g‘𝑄) = (+g‘(oppr‘𝑄))
4	3	oveqi 7369	. 2 ⊢ (𝑋(+g‘𝑄)𝑌) = (𝑋(+g‘(oppr‘𝑄))𝑌)
5		opprqus.i	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
6	5	ad4antr 738	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝐼 ∈ (NrmSGrp‘𝑅))
7		simp-4r 789	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑝 ∈ 𝐵)
8		simplr 774	. . . . . 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 2739	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
12	9, 10, 11, 2	qusadd 19154	. . . . . 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 1379	. . . . 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 781	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑋 = [𝑝](𝑅 ~QG 𝐼))
15		simpr 485	. . . . . 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 7374	. . . . 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 6863	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ V)
18		nsgsubg 19124	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅))
19	10	subgss 19094	. . . . . . . . . . . 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 33566	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ V ∧ 𝐼 ⊆ 𝐵) → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼))
23	17, 20, 22	syl2anc 590	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼))
24	23	eceq2d 8677	. . . . . . . . 9 ⊢ (𝜑 → [𝑝](𝑅 ~QG 𝐼) = [𝑝](𝑂 ~QG 𝐼))
25	23	eceq2d 8677	. . . . . . . . 9 ⊢ (𝜑 → [𝑞](𝑅 ~QG 𝐼) = [𝑞](𝑂 ~QG 𝐼))
26	24, 25	oveq12d 7374	. . . . . . . 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 738	. . . . . . 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 33567	. . . . . . . . . 10 ⊢ (NrmSGrp‘𝑅) = (NrmSGrp‘𝑂)
29	5, 28	eleqtrdi 2849	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑂))
30	29	ad4antr 738	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝐼 ∈ (NrmSGrp‘𝑂))
31	7, 10	eleqtrdi 2849	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑝 ∈ (Base‘𝑅))
32	8, 10	eleqtrdi 2849	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑞 ∈ (Base‘𝑅))
33		eqid 2739	. . . . . . . . 9 ⊢ (𝑂 /s (𝑂 ~QG 𝐼)) = (𝑂 /s (𝑂 ~QG 𝐼))
34	21, 10	opprbas 20314	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑂)
35	10, 34	eqtr3i 2764	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑂)
36	21, 11	oppradd 20315	. . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑂)
37		eqid 2739	. . . . . . . . 9 ⊢ (+g‘(𝑂 /s (𝑂 ~QG 𝐼))) = (+g‘(𝑂 /s (𝑂 ~QG 𝐼)))
38	33, 35, 36, 37	qusadd 19154	. . . . . . . 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 1379	. . . . . . 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 2774	. . . . . 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 7374	. . . . . 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 738	. . . . . . 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 8677	. . . . . 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 2784	. . . . 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 2784	. . . 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 7391	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 ~QG 𝐼) ∈ V)
51	48, 49, 50, 17	qusbas 17500	. . . . . . . 8 ⊢ (𝜑 → (𝐵 / (𝑅 ~QG 𝐼)) = (Base‘𝑄))
52	47, 51	eqtr4id 2793	. . . . . . 7 ⊢ (𝜑 → 𝐸 = (𝐵 / (𝑅 ~QG 𝐼)))
53	46, 52	eleqtrd 2841	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝐵 / (𝑅 ~QG 𝐼)))
54	53	ad2antrr 732	. . . . 5 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) → 𝑌 ∈ (𝐵 / (𝑅 ~QG 𝐼)))
55		elqsi 8702	. . . . 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 3147	. . 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 2841	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐵 / (𝑅 ~QG 𝐼)))
60		elqsi 8702	. . . 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 3147	. 2 ⊢ (𝜑 → (𝑋(+g‘𝑄)𝑌) = (𝑋(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))
63	4, 62	eqtr3id 2788	1 ⊢ (𝜑 → (𝑋(+g‘(oppr‘𝑄))𝑌) = (𝑋(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃wrex 3063  Vcvv 3431   ⊆ wss 3883  ‘cfv 6485  (class class class)co 7356  [cec 8631   / cqs 8632  Basecbs 17170  +_gcplusg 17211   /_s cqus 17460  SubGrpcsubg 19087  NrmSGrpcnsg 19088   ~_QG cqg 19089  opp_rcoppr 20307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  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 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-inf 9346  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-sca 17227  df-vsca 17228  df-ip 17229  df-tset 17230  df-ple 17231  df-ds 17233  df-0g 17395  df-imas 17463  df-qus 17464  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18903  df-minusg 18904  df-subg 19090  df-nsg 19091  df-eqg 19092  df-oppr 20308

This theorem is referenced by:  opprqus0g  33573