Theorem opprqusplusg 33072

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 2724	. . . 4 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
2		eqid 2724	. . . 4 ⊢ (+g‘𝑄) = (+g‘𝑄)
3	1, 2	oppradd 20235	. . 3 ⊢ (+g‘𝑄) = (+g‘(oppr‘𝑄))
4	3	oveqi 7414	. 2 ⊢ (𝑋(+g‘𝑄)𝑌) = (𝑋(+g‘(oppr‘𝑄))𝑌)
5		opprqus.i	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
6	5	ad4antr 729	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝐼 ∈ (NrmSGrp‘𝑅))
7		simp-4r 781	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑝 ∈ 𝐵)
8		simplr 766	. . . . . 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 2724	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
12	9, 10, 11, 2	qusadd 19104	. . . . . 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 1368	. . . . 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 773	. . . . . 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 7419	. . . . 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 6920	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ V)
18		nsgsubg 19075	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅))
19	10	subgss 19044	. . . . . . . . . . . 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 33066	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ V ∧ 𝐼 ⊆ 𝐵) → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼))
23	17, 20, 22	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼))
24	23	eceq2d 8741	. . . . . . . . 9 ⊢ (𝜑 → [𝑝](𝑅 ~QG 𝐼) = [𝑝](𝑂 ~QG 𝐼))
25	23	eceq2d 8741	. . . . . . . . 9 ⊢ (𝜑 → [𝑞](𝑅 ~QG 𝐼) = [𝑞](𝑂 ~QG 𝐼))
26	24, 25	oveq12d 7419	. . . . . . . 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 729	. . . . . . 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 33067	. . . . . . . . . 10 ⊢ (NrmSGrp‘𝑅) = (NrmSGrp‘𝑂)
29	5, 28	eleqtrdi 2835	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑂))
30	29	ad4antr 729	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝐼 ∈ (NrmSGrp‘𝑂))
31	7, 10	eleqtrdi 2835	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑝 ∈ (Base‘𝑅))
32	8, 10	eleqtrdi 2835	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑞 ∈ (Base‘𝑅))
33		eqid 2724	. . . . . . . . 9 ⊢ (𝑂 /s (𝑂 ~QG 𝐼)) = (𝑂 /s (𝑂 ~QG 𝐼))
34	21, 10	opprbas 20233	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑂)
35	10, 34	eqtr3i 2754	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑂)
36	21, 11	oppradd 20235	. . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑂)
37		eqid 2724	. . . . . . . . 9 ⊢ (+g‘(𝑂 /s (𝑂 ~QG 𝐼))) = (+g‘(𝑂 /s (𝑂 ~QG 𝐼)))
38	33, 35, 36, 37	qusadd 19104	. . . . . . . 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 1368	. . . . . . 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 7419	. . . . . 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 729	. . . . . . 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 8741	. . . . . 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 7436	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 ~QG 𝐼) ∈ V)
51	48, 49, 50, 17	qusbas 17490	. . . . . . . 8 ⊢ (𝜑 → (𝐵 / (𝑅 ~QG 𝐼)) = (Base‘𝑄))
52	47, 51	eqtr4id 2783	. . . . . . 7 ⊢ (𝜑 → 𝐸 = (𝐵 / (𝑅 ~QG 𝐼)))
53	46, 52	eleqtrd 2827	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝐵 / (𝑅 ~QG 𝐼)))
54	53	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) → 𝑌 ∈ (𝐵 / (𝑅 ~QG 𝐼)))
55		elqsi 8760	. . . . 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 3154	. . 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 2827	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐵 / (𝑅 ~QG 𝐼)))
60		elqsi 8760	. . . 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 3154	. 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 1533   ∈ wcel 2098  ∃wrex 3062  Vcvv 3466   ⊆ wss 3940  ‘cfv 6533  (class class class)co 7401  [cec 8697   / cqs 8698  Basecbs 17143  +_gcplusg 17196   /_s cqus 17450  SubGrpcsubg 19037  NrmSGrpcnsg 19038   ~_QG cqg 19039  opp_rcoppr 20225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-tpos 8206  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-ec 8701  df-qs 8705  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-inf 9434  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-z 12556  df-dec 12675  df-uz 12820  df-fz 13482  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  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 17386  df-imas 17453  df-qus 17454  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-grp 18856  df-minusg 18857  df-subg 19040  df-nsg 19041  df-eqg 19042  df-oppr 20226

This theorem is referenced by:  opprqus0g  33073