Theorem opprqus0g 33510

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 6920	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ V)
6		nsgsubg 19146	. . . . . . . . 9 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅))
7	1	subgss 19115	. . . . . . . . 9 ⊢ (𝐼 ∈ (SubGrp‘𝑅) → 𝐼 ⊆ 𝐵)
8	4, 6, 7	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ⊆ 𝐵)
9	1, 2, 3, 5, 8	opprqusbas 33508	. . . . . . 7 ⊢ (𝜑 → (Base‘(oppr‘𝑄)) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
10	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) → (Base‘(oppr‘𝑄)) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
11	4	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → 𝐼 ∈ (NrmSGrp‘𝑅))
12		eqid 2736	. . . . . . . . 9 ⊢ (Base‘𝑄) = (Base‘𝑄)
13		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) → 𝑒 ∈ (Base‘(oppr‘𝑄)))
14		eqid 2736	. . . . . . . . . . . . 13 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
15	14, 12	opprbas 20308	. . . . . . . . . . . 12 ⊢ (Base‘𝑄) = (Base‘(oppr‘𝑄))
16	15	eqcomi 2745	. . . . . . . . . . 11 ⊢ (Base‘(oppr‘𝑄)) = (Base‘𝑄)
17	13, 16	eleqtrdi 2845	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) → 𝑒 ∈ (Base‘𝑄))
18	17	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → 𝑒 ∈ (Base‘𝑄))
19		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → 𝑥 ∈ (Base‘(oppr‘𝑄)))
20	19, 16	eleqtrdi 2845	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → 𝑥 ∈ (Base‘𝑄))
21	20	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → 𝑥 ∈ (Base‘𝑄))
22	1, 2, 3, 11, 12, 18, 21	opprqusplusg 33509	. . . . . . . 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 2738	. . . . . . 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 33509	. . . . . . . 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 2738	. . . . . . 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 632	. . . . . 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 3315	. . . . 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 2821	. . . . 5 ⊢ (𝜑 → (𝑒 ∈ (Base‘(oppr‘𝑄)) ↔ 𝑒 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))))
30	29	anbi1d 631	. . . 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 279	. . 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 6520	. 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 2736	. . . . 5 ⊢ (+g‘𝑄) = (+g‘𝑄)
34	14, 33	oppradd 20309	. . . 4 ⊢ (+g‘𝑄) = (+g‘(oppr‘𝑄))
35	34	eqcomi 2745	. . 3 ⊢ (+g‘(oppr‘𝑄)) = (+g‘𝑄)
36		eqid 2736	. . . . 5 ⊢ (0g‘𝑄) = (0g‘𝑄)
37	14, 36	oppr0 20314	. . . 4 ⊢ (0g‘𝑄) = (0g‘(oppr‘𝑄))
38	37	eqcomi 2745	. . 3 ⊢ (0g‘(oppr‘𝑄)) = (0g‘𝑄)
39	16, 35, 38	grpidval 18644	. 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 2736	. . 3 ⊢ (Base‘(𝑂 /s (𝑂 ~QG 𝐼))) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))
41		eqid 2736	. . 3 ⊢ (+g‘(𝑂 /s (𝑂 ~QG 𝐼))) = (+g‘(𝑂 /s (𝑂 ~QG 𝐼)))
42		eqid 2736	. . 3 ⊢ (0g‘(𝑂 /s (𝑂 ~QG 𝐼))) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼)))
43	40, 41, 42	grpidval 18644	. 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 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052  Vcvv 3464   ⊆ wss 3931  ℩cio 6487  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  +_gcplusg 17276  0_gc0g 17458   /_s cqus 17524  SubGrpcsubg 19108  NrmSGrpcnsg 19109   ~_QG cqg 19110  opp_rcoppr 20301

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-tpos 8230  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-ec 8726  df-qs 8730  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9459  df-inf 9460  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-z 12594  df-dec 12714  df-uz 12858  df-fz 13530  df-struct 17171  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-ress 17257  df-plusg 17289  df-mulr 17290  df-sca 17292  df-vsca 17293  df-ip 17294  df-tset 17295  df-ple 17296  df-ds 17298  df-0g 17460  df-imas 17527  df-qus 17528  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-grp 18924  df-minusg 18925  df-subg 19111  df-nsg 19112  df-eqg 19113  df-oppr 20302

This theorem is referenced by:  opprqusdrng  33513