Theorem opprqus0g 33840

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 6921	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ V)
6		nsgsubg 19272	. . . . . . . . 9 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅))
7	1	subgss 19241	. . . . . . . . 9 ⊢ (𝐼 ∈ (SubGrp‘𝑅) → 𝐼 ⊆ 𝐵)
8	4, 6, 7	3syl 19	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ⊆ 𝐵)
9	1, 2, 3, 5, 8	opprqusbas 33838	. . . . . . 7 ⊢ (𝜑 → (Base‘(oppr‘𝑄)) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
10	9	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) → (Base‘(oppr‘𝑄)) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
11	4	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → 𝐼 ∈ (NrmSGrp‘𝑅))
12		eqid 2765	. . . . . . . . 9 ⊢ (Base‘𝑄) = (Base‘𝑄)
13		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) → 𝑒 ∈ (Base‘(oppr‘𝑄)))
14		eqid 2765	. . . . . . . . . . . . 13 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
15	14, 12	opprbas 20475	. . . . . . . . . . . 12 ⊢ (Base‘𝑄) = (Base‘(oppr‘𝑄))
16	15	eqcomi 2774	. . . . . . . . . . 11 ⊢ (Base‘(oppr‘𝑄)) = (Base‘𝑄)
17	13, 16	eleqtrdi 2875	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) → 𝑒 ∈ (Base‘𝑄))
18	17	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → 𝑒 ∈ (Base‘𝑄))
19		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → 𝑥 ∈ (Base‘(oppr‘𝑄)))
20	19, 16	eleqtrdi 2875	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → 𝑥 ∈ (Base‘𝑄))
21	20	adantlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ (Base‘(oppr‘𝑄))) ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → 𝑥 ∈ (Base‘𝑄))
22	1, 2, 3, 11, 12, 18, 21	opprqusplusg 33839	. . . . . . . 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 2767	. . . . . . 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 33839	. . . . . . . 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 2767	. . . . . . 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 644	. . . . . 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 3331	. . . . 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 590	. . . 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 2851	. . . . 5 ⊢ (𝜑 → (𝑒 ∈ (Base‘(oppr‘𝑄)) ↔ 𝑒 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))))
30	29	anbi1d 643	. . . 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 282	. . 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 6524	. 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 2765	. . . . 5 ⊢ (+g‘𝑄) = (+g‘𝑄)
34	14, 33	oppradd 20476	. . . 4 ⊢ (+g‘𝑄) = (+g‘(oppr‘𝑄))
35	34	eqcomi 2774	. . 3 ⊢ (+g‘(oppr‘𝑄)) = (+g‘𝑄)
36		eqid 2765	. . . . 5 ⊢ (0g‘𝑄) = (0g‘𝑄)
37	14, 36	oppr0 20481	. . . 4 ⊢ (0g‘𝑄) = (0g‘(oppr‘𝑄))
38	37	eqcomi 2774	. . 3 ⊢ (0g‘(oppr‘𝑄)) = (0g‘𝑄)
39	16, 35, 38	grpidval 18748	. 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 2765	. . 3 ⊢ (Base‘(𝑂 /s (𝑂 ~QG 𝐼))) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))
41		eqid 2765	. . 3 ⊢ (+g‘(𝑂 /s (𝑂 ~QG 𝐼))) = (+g‘(𝑂 /s (𝑂 ~QG 𝐼)))
42		eqid 2765	. . 3 ⊢ (0g‘(𝑂 /s (𝑂 ~QG 𝐼))) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼)))
43	40, 41, 42	grpidval 18748	. 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 2825	1 ⊢ (𝜑 → (0g‘(oppr‘𝑄)) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  Vcvv 3457   ⊆ wss 3906  ℩cio 6494  ‘cfv 6540  (class class class)co 7419  Basecbs 17295  +_gcplusg 17336  0_gc0g 17518   /_s cqus 17585  SubGrpcsubg 19234  NrmSGrpcnsg 19235   ~_QG cqg 19236  opp_rcoppr 20468

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11175  ax-resscn 11176  ax-1cn 11177  ax-icn 11178  ax-addcl 11179  ax-addrcl 11180  ax-mulcl 11181  ax-mulrcl 11182  ax-mulcom 11183  ax-addass 11184  ax-mulass 11185  ax-distr 11186  ax-i2m1 11187  ax-1ne0 11188  ax-1rid 11189  ax-rnegex 11190  ax-rrecex 11191  ax-cnre 11192  ax-pre-lttri 11193  ax-pre-lttrn 11194  ax-pre-ltadd 11195  ax-pre-mulgt0 11196

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-tpos 8228  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-ec 8702  df-qs 8706  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-sup 9409  df-inf 9410  df-pnf 11264  df-mnf 11265  df-xr 11266  df-ltxr 11267  df-le 11268  df-sub 11462  df-neg 11463  df-nn 12253  df-2 12322  df-3 12323  df-4 12324  df-5 12325  df-6 12326  df-7 12327  df-8 12328  df-9 12329  df-n0 12524  df-z 12611  df-dec 12732  df-uz 12883  df-fz 13556  df-struct 17233  df-sets 17250  df-slot 17268  df-ndx 17280  df-base 17296  df-ress 17317  df-plusg 17349  df-mulr 17350  df-sca 17352  df-vsca 17353  df-ip 17354  df-tset 17355  df-ple 17356  df-ds 17358  df-0g 17520  df-imas 17588  df-qus 17589  df-mgm 18724  df-sgrp 18813  df-mnd 18829  df-grp 19051  df-minusg 19052  df-subg 19237  df-nsg 19238  df-eqg 19239  df-oppr 20469

This theorem is used by:  opprqusdrng  33843