Theorem opprqus0g 33483

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 6959	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ V)
6		nsgsubg 19198	. . . . . . . . 9 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅))
7	1	subgss 19167	. . . . . . . . 9 ⊢ (𝐼 ∈ (SubGrp‘𝑅) → 𝐼 ⊆ 𝐵)
8	4, 6, 7	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ⊆ 𝐵)
9	1, 2, 3, 5, 8	opprqusbas 33481	. . . . . . 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 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → 𝐼 ∈ (NrmSGrp‘𝑅))
12		eqid 2740	. . . . . . . . 9 ⊢ (Base‘𝑄) = (Base‘𝑄)
13		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) → 𝑒 ∈ (Base‘(oppr‘𝑄)))
14		eqid 2740	. . . . . . . . . . . . 13 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
15	14, 12	opprbas 20367	. . . . . . . . . . . 12 ⊢ (Base‘𝑄) = (Base‘(oppr‘𝑄))
16	15	eqcomi 2749	. . . . . . . . . . 11 ⊢ (Base‘(oppr‘𝑄)) = (Base‘𝑄)
17	13, 16	eleqtrdi 2854	. . . . . . . . . 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 2854	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → 𝑥 ∈ (Base‘𝑄))
21	20	adantlr 714	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → 𝑥 ∈ (Base‘𝑄))
22	1, 2, 3, 11, 12, 18, 21	opprqusplusg 33482	. . . . . . . 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 2742	. . . . . . 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 33482	. . . . . . . 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 2742	. . . . . . 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 3340	. . . . 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 578	. . . 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 2830	. . . . 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 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 6557	. 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 2740	. . . . 5 ⊢ (+g‘𝑄) = (+g‘𝑄)
34	14, 33	oppradd 20369	. . . 4 ⊢ (+g‘𝑄) = (+g‘(oppr‘𝑄))
35	34	eqcomi 2749	. . 3 ⊢ (+g‘(oppr‘𝑄)) = (+g‘𝑄)
36		eqid 2740	. . . . 5 ⊢ (0g‘𝑄) = (0g‘𝑄)
37	14, 36	oppr0 20375	. . . 4 ⊢ (0g‘𝑄) = (0g‘(oppr‘𝑄))
38	37	eqcomi 2749	. . 3 ⊢ (0g‘(oppr‘𝑄)) = (0g‘𝑄)
39	16, 35, 38	grpidval 18699	. 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 2740	. . 3 ⊢ (Base‘(𝑂 /s (𝑂 ~QG 𝐼))) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))
41		eqid 2740	. . 3 ⊢ (+g‘(𝑂 /s (𝑂 ~QG 𝐼))) = (+g‘(𝑂 /s (𝑂 ~QG 𝐼)))
42		eqid 2740	. . 3 ⊢ (0g‘(𝑂 /s (𝑂 ~QG 𝐼))) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼)))
43	40, 41, 42	grpidval 18699	. 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 2805	1 ⊢ (𝜑 → (0g‘(oppr‘𝑄)) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ⊆ wss 3976  ℩cio 6523  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  +_gcplusg 17311  0_gc0g 17499   /_s cqus 17565  SubGrpcsubg 19160  NrmSGrpcnsg 19161   ~_QG cqg 19162  opp_rcoppr 20359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-tpos 8267  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-ec 8765  df-qs 8769  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-0g 17501  df-imas 17568  df-qus 17569  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-minusg 18977  df-subg 19163  df-nsg 19164  df-eqg 19165  df-oppr 20360

This theorem is referenced by:  opprqusdrng  33486