Theorem opprqusplusg 33497

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 2735	. . . 4 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
2		eqid 2735	. . . 4 ⊢ (+g‘𝑄) = (+g‘𝑄)
3	1, 2	oppradd 20360	. . 3 ⊢ (+g‘𝑄) = (+g‘(oppr‘𝑄))
4	3	oveqi 7444	. 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 784	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑝 ∈ 𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞 ∈ 𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑝 ∈ 𝐵)
8		simplr 769	. . . . . 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 2735	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
12	9, 10, 11, 2	qusadd 19219	. . . . . 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 1370	. . . . 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 776	. . . . . 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 7449	. . . . 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 6946	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ V)
18		nsgsubg 19189	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅))
19	10	subgss 19158	. . . . . . . . . . . 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 33491	. . . . . . . . . . 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 8787	. . . . . . . . 9 ⊢ (𝜑 → [𝑝](𝑅 ~QG 𝐼) = [𝑝](𝑂 ~QG 𝐼))
25	23	eceq2d 8787	. . . . . . . . 9 ⊢ (𝜑 → [𝑞](𝑅 ~QG 𝐼) = [𝑞](𝑂 ~QG 𝐼))
26	24, 25	oveq12d 7449	. . . . . . . 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 33492	. . . . . . . . . 10 ⊢ (NrmSGrp‘𝑅) = (NrmSGrp‘𝑂)
29	5, 28	eleqtrdi 2849	. . . . . . . . 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 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 2735	. . . . . . . . 9 ⊢ (𝑂 /s (𝑂 ~QG 𝐼)) = (𝑂 /s (𝑂 ~QG 𝐼))
34	21, 10	opprbas 20358	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑂)
35	10, 34	eqtr3i 2765	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑂)
36	21, 11	oppradd 20360	. . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑂)
37		eqid 2735	. . . . . . . . 9 ⊢ (+g‘(𝑂 /s (𝑂 ~QG 𝐼))) = (+g‘(𝑂 /s (𝑂 ~QG 𝐼)))
38	33, 35, 36, 37	qusadd 19219	. . . . . . . 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 1370	. . . . . . 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 2775	. . . . . 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 7449	. . . . . 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 8787	. . . . . 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 2785	. . . . 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 2785	. . . 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 7466	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 ~QG 𝐼) ∈ V)
51	48, 49, 50, 17	qusbas 17592	. . . . . . . 8 ⊢ (𝜑 → (𝐵 / (𝑅 ~QG 𝐼)) = (Base‘𝑄))
52	47, 51	eqtr4id 2794	. . . . . . 7 ⊢ (𝜑 → 𝐸 = (𝐵 / (𝑅 ~QG 𝐼)))
53	46, 52	eleqtrd 2841	. . . . . 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 8809	. . . . 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 3160	. . 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 8809	. . . 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 3160	. 2 ⊢ (𝜑 → (𝑋(+g‘𝑄)𝑌) = (𝑋(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))
63	4, 62	eqtr3id 2789	1 ⊢ (𝜑 → (𝑋(+g‘(oppr‘𝑄))𝑌) = (𝑋(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∃wrex 3068  Vcvv 3478   ⊆ wss 3963  ‘cfv 6563  (class class class)co 7431  [cec 8742   / cqs 8743  Basecbs 17245  +_gcplusg 17298   /_s cqus 17552  SubGrpcsubg 19151  NrmSGrpcnsg 19152   ~_QG cqg 19153  opp_rcoppr 20350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-ec 8746  df-qs 8750  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-0g 17488  df-imas 17555  df-qus 17556  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-subg 19154  df-nsg 19155  df-eqg 19156  df-oppr 20351

This theorem is referenced by:  opprqus0g  33498