Theorem opprqus0g 33454

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 6879	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ V)
6		nsgsubg 19072	. . . . . . . . 9 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅))
7	1	subgss 19041	. . . . . . . . 9 ⊢ (𝐼 ∈ (SubGrp‘𝑅) → 𝐼 ⊆ 𝐵)
8	4, 6, 7	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ⊆ 𝐵)
9	1, 2, 3, 5, 8	opprqusbas 33452	. . . . . . 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 2729	. . . . . . . . 9 ⊢ (Base‘𝑄) = (Base‘𝑄)
13		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) → 𝑒 ∈ (Base‘(oppr‘𝑄)))
14		eqid 2729	. . . . . . . . . . . . 13 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
15	14, 12	opprbas 20263	. . . . . . . . . . . 12 ⊢ (Base‘𝑄) = (Base‘(oppr‘𝑄))
16	15	eqcomi 2738	. . . . . . . . . . 11 ⊢ (Base‘(oppr‘𝑄)) = (Base‘𝑄)
17	13, 16	eleqtrdi 2838	. . . . . . . . . 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 2838	. . . . . . . . . 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 33453	. . . . . . . 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 2731	. . . . . . 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 33453	. . . . . . . 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 2731	. . . . . . 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 3302	. . . . 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 2814	. . . . 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 6483	. 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 2729	. . . . 5 ⊢ (+g‘𝑄) = (+g‘𝑄)
34	14, 33	oppradd 20264	. . . 4 ⊢ (+g‘𝑄) = (+g‘(oppr‘𝑄))
35	34	eqcomi 2738	. . 3 ⊢ (+g‘(oppr‘𝑄)) = (+g‘𝑄)
36		eqid 2729	. . . . 5 ⊢ (0g‘𝑄) = (0g‘𝑄)
37	14, 36	oppr0 20269	. . . 4 ⊢ (0g‘𝑄) = (0g‘(oppr‘𝑄))
38	37	eqcomi 2738	. . 3 ⊢ (0g‘(oppr‘𝑄)) = (0g‘𝑄)
39	16, 35, 38	grpidval 18570	. 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 2729	. . 3 ⊢ (Base‘(𝑂 /s (𝑂 ~QG 𝐼))) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))
41		eqid 2729	. . 3 ⊢ (+g‘(𝑂 /s (𝑂 ~QG 𝐼))) = (+g‘(𝑂 /s (𝑂 ~QG 𝐼)))
42		eqid 2729	. . 3 ⊢ (0g‘(𝑂 /s (𝑂 ~QG 𝐼))) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼)))
43	40, 41, 42	grpidval 18570	. 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 2789	1 ⊢ (𝜑 → (0g‘(oppr‘𝑄)) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3444   ⊆ wss 3911  ℩cio 6450  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  +_gcplusg 17196  0_gc0g 17378   /_s cqus 17444  SubGrpcsubg 19034  NrmSGrpcnsg 19035   ~_QG cqg 19036  opp_rcoppr 20256

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-tpos 8182  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-ec 8650  df-qs 8654  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9369  df-inf 9370  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-uz 12770  df-fz 13445  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  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 17380  df-imas 17447  df-qus 17448  df-mgm 18549  df-sgrp 18628  df-mnd 18644  df-grp 18850  df-minusg 18851  df-subg 19037  df-nsg 19038  df-eqg 19039  df-oppr 20257

This theorem is referenced by:  opprqusdrng  33457