Theorem opprqus0g 32450

Description: The group identity element of the quotient of the opposite ring is the same as the group identity element 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))

Assertion

Ref	Expression
opprqus0g	⊢ (𝜑 → (0g‘(oppr‘𝑄)) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼))))

Proof of Theorem opprqus0g

Dummy variables 𝑥 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opprqus.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅)
2		opprqus.o	. . . . . . . 8 ⊢ 𝑂 = (oppr‘𝑅)
3		opprqus.q	. . . . . . . 8 ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))
4		opprqus.i	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
5	4	elfvexd 6917	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ V)
6		nsgsubg 19010	. . . . . . . . 9 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅))
7	1	subgss 18979	. . . . . . . . 9 ⊢ (𝐼 ∈ (SubGrp‘𝑅) → 𝐼 ⊆ 𝐵)
8	4, 6, 7	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ⊆ 𝐵)
9	1, 2, 3, 5, 8	opprqusbas 32448	. . . . . . 7 ⊢ (𝜑 → (Base‘(oppr‘𝑄)) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
10	9	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) → (Base‘(oppr‘𝑄)) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
11	4	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → 𝐼 ∈ (NrmSGrp‘𝑅))
12		eqid 2731	. . . . . . . . 9 ⊢ (Base‘𝑄) = (Base‘𝑄)
13		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) → 𝑒 ∈ (Base‘(oppr‘𝑄)))
14		eqid 2731	. . . . . . . . . . . . 13 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
15	14, 12	opprbas 20109	. . . . . . . . . . . 12 ⊢ (Base‘𝑄) = (Base‘(oppr‘𝑄))
16	15	eqcomi 2740	. . . . . . . . . . 11 ⊢ (Base‘(oppr‘𝑄)) = (Base‘𝑄)
17	13, 16	eleqtrdi 2842	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) → 𝑒 ∈ (Base‘𝑄))
18	17	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → 𝑒 ∈ (Base‘𝑄))
19		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → 𝑥 ∈ (Base‘(oppr‘𝑄)))
20	19, 16	eleqtrdi 2842	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → 𝑥 ∈ (Base‘𝑄))
21	20	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → 𝑥 ∈ (Base‘𝑄))
22	1, 2, 3, 11, 12, 18, 21	opprqusplusg 32449	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → (𝑒(+g‘(oppr‘𝑄))𝑥) = (𝑒(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥))
23	22	eqeq1d 2733	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → ((𝑒(+g‘(oppr‘𝑄))𝑥) = 𝑥 ↔ (𝑒(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = 𝑥))
24	1, 2, 3, 11, 12, 21, 18	opprqusplusg 32449	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → (𝑥(+g‘(oppr‘𝑄))𝑒) = (𝑥(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑒))
25	24	eqeq1d 2733	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → ((𝑥(+g‘(oppr‘𝑄))𝑒) = 𝑥 ↔ (𝑥(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑒) = 𝑥))
26	23, 25	anbi12d 631	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → (((𝑒(+g‘(oppr‘𝑄))𝑥) = 𝑥 ∧ (𝑥(+g‘(oppr‘𝑄))𝑒) = 𝑥) ↔ ((𝑒(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = 𝑥 ∧ (𝑥(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑒) = 𝑥)))
27	10, 26	raleqbidva 3326	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) → (∀𝑥 ∈ (Base‘(oppr‘𝑄))((𝑒(+g‘(oppr‘𝑄))𝑥) = 𝑥 ∧ (𝑥(+g‘(oppr‘𝑄))𝑒) = 𝑥) ↔ ∀𝑥 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))((𝑒(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = 𝑥 ∧ (𝑥(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑒) = 𝑥)))
28	27	pm5.32da 579	. . . 4 ⊢ (𝜑 → ((𝑒 ∈ (Base‘(oppr‘𝑄)) ∧ ∀𝑥 ∈ (Base‘(oppr‘𝑄))((𝑒(+g‘(oppr‘𝑄))𝑥) = 𝑥 ∧ (𝑥(+g‘(oppr‘𝑄))𝑒) = 𝑥)) ↔ (𝑒 ∈ (Base‘(oppr‘𝑄)) ∧ ∀𝑥 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))((𝑒(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = 𝑥 ∧ (𝑥(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑒) = 𝑥))))
29	9	eleq2d 2818	. . . . 5 ⊢ (𝜑 → (𝑒 ∈ (Base‘(oppr‘𝑄)) ↔ 𝑒 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))))
30	29	anbi1d 630	. . . 4 ⊢ (𝜑 → ((𝑒 ∈ (Base‘(oppr‘𝑄)) ∧ ∀𝑥 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))((𝑒(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = 𝑥 ∧ (𝑥(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑒) = 𝑥)) ↔ (𝑒 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ ∀𝑥 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))((𝑒(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = 𝑥 ∧ (𝑥(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑒) = 𝑥))))
31	28, 30	bitrd 278	. . 3 ⊢ (𝜑 → ((𝑒 ∈ (Base‘(oppr‘𝑄)) ∧ ∀𝑥 ∈ (Base‘(oppr‘𝑄))((𝑒(+g‘(oppr‘𝑄))𝑥) = 𝑥 ∧ (𝑥(+g‘(oppr‘𝑄))𝑒) = 𝑥)) ↔ (𝑒 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ ∀𝑥 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))((𝑒(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = 𝑥 ∧ (𝑥(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑒) = 𝑥))))
32	31	iotabidv 6516	. 2 ⊢ (𝜑 → (℩𝑒(𝑒 ∈ (Base‘(oppr‘𝑄)) ∧ ∀𝑥 ∈ (Base‘(oppr‘𝑄))((𝑒(+g‘(oppr‘𝑄))𝑥) = 𝑥 ∧ (𝑥(+g‘(oppr‘𝑄))𝑒) = 𝑥))) = (℩𝑒(𝑒 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ ∀𝑥 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))((𝑒(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = 𝑥 ∧ (𝑥(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑒) = 𝑥))))
33		eqid 2731	. . . . 5 ⊢ (+g‘𝑄) = (+g‘𝑄)
34	14, 33	oppradd 20111	. . . 4 ⊢ (+g‘𝑄) = (+g‘(oppr‘𝑄))
35	34	eqcomi 2740	. . 3 ⊢ (+g‘(oppr‘𝑄)) = (+g‘𝑄)
36		eqid 2731	. . . . 5 ⊢ (0g‘𝑄) = (0g‘𝑄)
37	14, 36	oppr0 20115	. . . 4 ⊢ (0g‘𝑄) = (0g‘(oppr‘𝑄))
38	37	eqcomi 2740	. . 3 ⊢ (0g‘(oppr‘𝑄)) = (0g‘𝑄)
39	16, 35, 38	grpidval 18562	. 2 ⊢ (0g‘(oppr‘𝑄)) = (℩𝑒(𝑒 ∈ (Base‘(oppr‘𝑄)) ∧ ∀𝑥 ∈ (Base‘(oppr‘𝑄))((𝑒(+g‘(oppr‘𝑄))𝑥) = 𝑥 ∧ (𝑥(+g‘(oppr‘𝑄))𝑒) = 𝑥)))
40		eqid 2731	. . 3 ⊢ (Base‘(𝑂 /s (𝑂 ~QG 𝐼))) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))
41		eqid 2731	. . 3 ⊢ (+g‘(𝑂 /s (𝑂 ~QG 𝐼))) = (+g‘(𝑂 /s (𝑂 ~QG 𝐼)))
42		eqid 2731	. . 3 ⊢ (0g‘(𝑂 /s (𝑂 ~QG 𝐼))) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼)))
43	40, 41, 42	grpidval 18562	. 2 ⊢ (0g‘(𝑂 /s (𝑂 ~QG 𝐼))) = (℩𝑒(𝑒 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ ∀𝑥 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))((𝑒(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = 𝑥 ∧ (𝑥(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑒) = 𝑥)))
44	32, 39, 43	3eqtr4g 2796	1 ⊢ (𝜑 → (0g‘(oppr‘𝑄)) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060  Vcvv 3473   ⊆ wss 3944  ℩cio 6482  ‘cfv 6532  (class class class)co 7393  Basecbs 17126  +_gcplusg 17179  0_gc0g 17367   /_s cqus 17433  SubGrpcsubg 18972  NrmSGrpcnsg 18973   ~_QG cqg 18974  opp_rcoppr 20101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-tp 4627  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-tpos 8193  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-er 8686  df-ec 8688  df-qs 8692  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-sup 9419  df-inf 9420  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-nn 12195  df-2 12257  df-3 12258  df-4 12259  df-5 12260  df-6 12261  df-7 12262  df-8 12263  df-9 12264  df-n0 12455  df-z 12541  df-dec 12660  df-uz 12805  df-fz 13467  df-struct 17062  df-sets 17079  df-slot 17097  df-ndx 17109  df-base 17127  df-ress 17156  df-plusg 17192  df-mulr 17193  df-sca 17195  df-vsca 17196  df-ip 17197  df-tset 17198  df-ple 17199  df-ds 17201  df-0g 17369  df-imas 17436  df-qus 17437  df-mgm 18543  df-sgrp 18592  df-mnd 18603  df-grp 18797  df-minusg 18798  df-subg 18975  df-nsg 18976  df-eqg 18977  df-oppr 20102

This theorem is referenced by:  opprqusdrng  32453