Theorem opprqusdrng 32453

Description: The quotient of the opposite ring is a division ring iff the opposite of the quotient ring is. (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))
opprqus1r.r	⊢ (𝜑 → 𝑅 ∈ Ring)
opprqus1r.i	⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))

Assertion

Ref	Expression
opprqusdrng	⊢ (𝜑 → ((oppr‘𝑄) ∈ DivRing ↔ (𝑂 /s (𝑂 ~QG 𝐼)) ∈ DivRing))

Proof of Theorem opprqusdrng

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

Step	Hyp	Ref	Expression
1		eqid 2731	. . . . . 6 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
2		eqid 2731	. . . . . 6 ⊢ (1r‘𝑄) = (1r‘𝑄)
3	1, 2	oppr1 20116	. . . . 5 ⊢ (1r‘𝑄) = (1r‘(oppr‘𝑄))
4		opprqus.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
5		opprqus.o	. . . . . 6 ⊢ 𝑂 = (oppr‘𝑅)
6		opprqus.q	. . . . . 6 ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))
7		opprqus1r.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring)
8		opprqus1r.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))
9	4, 5, 6, 7, 8	opprqus1r 32452	. . . . 5 ⊢ (𝜑 → (1r‘(oppr‘𝑄)) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))
10	3, 9	eqtrid 2783	. . . 4 ⊢ (𝜑 → (1r‘𝑄) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))
11		eqid 2731	. . . . . 6 ⊢ (0g‘𝑄) = (0g‘𝑄)
12	1, 11	oppr0 20115	. . . . 5 ⊢ (0g‘𝑄) = (0g‘(oppr‘𝑄))
13	8	2idllidld 20805	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))
14		lidlnsg 32415	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
15	7, 13, 14	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
16	4, 5, 6, 15	opprqus0g 32450	. . . . 5 ⊢ (𝜑 → (0g‘(oppr‘𝑄)) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼))))
17	12, 16	eqtrid 2783	. . . 4 ⊢ (𝜑 → (0g‘𝑄) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼))))
18	10, 17	neeq12d 3001	. . 3 ⊢ (𝜑 → ((1r‘𝑄) ≠ (0g‘𝑄) ↔ (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ≠ (0g‘(𝑂 /s (𝑂 ~QG 𝐼)))))
19		eqid 2731	. . . . . . 7 ⊢ (Base‘𝑄) = (Base‘𝑄)
20	1, 19	opprbas 20109	. . . . . 6 ⊢ (Base‘𝑄) = (Base‘(oppr‘𝑄))
21		eqid 2731	. . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
22	4, 21	lidlss 20781	. . . . . . . 8 ⊢ (𝐼 ∈ (LIdeal‘𝑅) → 𝐼 ⊆ 𝐵)
23	13, 22	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐼 ⊆ 𝐵)
24	4, 5, 6, 7, 23	opprqusbas 32448	. . . . . 6 ⊢ (𝜑 → (Base‘(oppr‘𝑄)) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
25	20, 24	eqtrid 2783	. . . . 5 ⊢ (𝜑 → (Base‘𝑄) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
26	17	sneqd 4634	. . . . 5 ⊢ (𝜑 → {(0g‘𝑄)} = {(0g‘(𝑂 /s (𝑂 ~QG 𝐼)))})
27	25, 26	difeq12d 4119	. . . 4 ⊢ (𝜑 → ((Base‘𝑄) ∖ {(0g‘𝑄)}) = ((Base‘(𝑂 /s (𝑂 ~QG 𝐼))) ∖ {(0g‘(𝑂 /s (𝑂 ~QG 𝐼)))}))
28	25	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → (Base‘𝑄) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
29	7	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑅 ∈ Ring)
30	8	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝐼 ∈ (2Ideal‘𝑅))
31		simplr 767	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)}))
32	31	eldifad 3956	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑥 ∈ (Base‘𝑄))
33		simpr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑦 ∈ (Base‘𝑄))
34	4, 5, 6, 29, 30, 19, 32, 33	opprqusmulr 32451	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → (𝑥(.r‘(oppr‘𝑄))𝑦) = (𝑥(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑦))
35	10	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → (1r‘𝑄) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))
36	34, 35	eqeq12d 2747	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → ((𝑥(.r‘(oppr‘𝑄))𝑦) = (1r‘𝑄) ↔ (𝑥(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑦) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))))
37	4, 5, 6, 29, 30, 19, 33, 32	opprqusmulr 32451	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → (𝑦(.r‘(oppr‘𝑄))𝑥) = (𝑦(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥))
38	37, 35	eqeq12d 2747	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → ((𝑦(.r‘(oppr‘𝑄))𝑥) = (1r‘𝑄) ↔ (𝑦(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))))
39	36, 38	anbi12d 631	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → (((𝑥(.r‘(oppr‘𝑄))𝑦) = (1r‘𝑄) ∧ (𝑦(.r‘(oppr‘𝑄))𝑥) = (1r‘𝑄)) ↔ ((𝑥(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑦) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ (𝑦(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))))
40	28, 39	rexeqbidva 3327	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → (∃𝑦 ∈ (Base‘𝑄)((𝑥(.r‘(oppr‘𝑄))𝑦) = (1r‘𝑄) ∧ (𝑦(.r‘(oppr‘𝑄))𝑥) = (1r‘𝑄)) ↔ ∃𝑦 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))((𝑥(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑦) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ (𝑦(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))))
41	27, 40	raleqbidva 3326	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})∃𝑦 ∈ (Base‘𝑄)((𝑥(.r‘(oppr‘𝑄))𝑦) = (1r‘𝑄) ∧ (𝑦(.r‘(oppr‘𝑄))𝑥) = (1r‘𝑄)) ↔ ∀𝑥 ∈ ((Base‘(𝑂 /s (𝑂 ~QG 𝐼))) ∖ {(0g‘(𝑂 /s (𝑂 ~QG 𝐼)))})∃𝑦 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))((𝑥(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑦) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ (𝑦(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))))
42	18, 41	anbi12d 631	. 2 ⊢ (𝜑 → (((1r‘𝑄) ≠ (0g‘𝑄) ∧ ∀𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})∃𝑦 ∈ (Base‘𝑄)((𝑥(.r‘(oppr‘𝑄))𝑦) = (1r‘𝑄) ∧ (𝑦(.r‘(oppr‘𝑄))𝑥) = (1r‘𝑄))) ↔ ((1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ≠ (0g‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ ∀𝑥 ∈ ((Base‘(𝑂 /s (𝑂 ~QG 𝐼))) ∖ {(0g‘(𝑂 /s (𝑂 ~QG 𝐼)))})∃𝑦 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))((𝑥(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑦) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ (𝑦(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))))))
43		eqid 2731	. . 3 ⊢ (.r‘(oppr‘𝑄)) = (.r‘(oppr‘𝑄))
44		eqid 2731	. . . 4 ⊢ (Unit‘𝑄) = (Unit‘𝑄)
45	44, 1	opprunit 20143	. . 3 ⊢ (Unit‘𝑄) = (Unit‘(oppr‘𝑄))
46		eqid 2731	. . . . . 6 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
47	6, 46	qusring 20809	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (2Ideal‘𝑅)) → 𝑄 ∈ Ring)
48	7, 8, 47	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝑄 ∈ Ring)
49	1	opprring 20113	. . . 4 ⊢ (𝑄 ∈ Ring → (oppr‘𝑄) ∈ Ring)
50	48, 49	syl 17	. . 3 ⊢ (𝜑 → (oppr‘𝑄) ∈ Ring)
51	20, 12, 3, 43, 45, 50	isdrng4 32257	. 2 ⊢ (𝜑 → ((oppr‘𝑄) ∈ DivRing ↔ ((1r‘𝑄) ≠ (0g‘𝑄) ∧ ∀𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})∃𝑦 ∈ (Base‘𝑄)((𝑥(.r‘(oppr‘𝑄))𝑦) = (1r‘𝑄) ∧ (𝑦(.r‘(oppr‘𝑄))𝑥) = (1r‘𝑄)))))
52		eqid 2731	. . 3 ⊢ (Base‘(𝑂 /s (𝑂 ~QG 𝐼))) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))
53		eqid 2731	. . 3 ⊢ (0g‘(𝑂 /s (𝑂 ~QG 𝐼))) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼)))
54		eqid 2731	. . 3 ⊢ (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))
55		eqid 2731	. . 3 ⊢ (.r‘(𝑂 /s (𝑂 ~QG 𝐼))) = (.r‘(𝑂 /s (𝑂 ~QG 𝐼)))
56		eqid 2731	. . 3 ⊢ (Unit‘(𝑂 /s (𝑂 ~QG 𝐼))) = (Unit‘(𝑂 /s (𝑂 ~QG 𝐼)))
57	5	opprring 20113	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring)
58	7, 57	syl 17	. . . 4 ⊢ (𝜑 → 𝑂 ∈ Ring)
59	5, 7	oppr2idl 32446	. . . . 5 ⊢ (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂))
60	8, 59	eleqtrd 2834	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑂))
61		eqid 2731	. . . . 5 ⊢ (𝑂 /s (𝑂 ~QG 𝐼)) = (𝑂 /s (𝑂 ~QG 𝐼))
62		eqid 2731	. . . . 5 ⊢ (2Ideal‘𝑂) = (2Ideal‘𝑂)
63	61, 62	qusring 20809	. . . 4 ⊢ ((𝑂 ∈ Ring ∧ 𝐼 ∈ (2Ideal‘𝑂)) → (𝑂 /s (𝑂 ~QG 𝐼)) ∈ Ring)
64	58, 60, 63	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑂 /s (𝑂 ~QG 𝐼)) ∈ Ring)
65	52, 53, 54, 55, 56, 64	isdrng4 32257	. 2 ⊢ (𝜑 → ((𝑂 /s (𝑂 ~QG 𝐼)) ∈ DivRing ↔ ((1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ≠ (0g‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ ∀𝑥 ∈ ((Base‘(𝑂 /s (𝑂 ~QG 𝐼))) ∖ {(0g‘(𝑂 /s (𝑂 ~QG 𝐼)))})∃𝑦 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))((𝑥(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑦) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ (𝑦(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))))))
66	42, 51, 65	3bitr4d 310	1 ⊢ (𝜑 → ((oppr‘𝑄) ∈ DivRing ↔ (𝑂 /s (𝑂 ~QG 𝐼)) ∈ DivRing))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2939  ∀wral 3060  ∃wrex 3069   ∖ cdif 3941   ⊆ wss 3944  {csn 4622  ‘cfv 6532  (class class class)co 7393  Basecbs 17126  ._rcmulr 17180  0_gc0g 17367   /_s cqus 17433  NrmSGrpcnsg 18973   ~_QG cqg 18974  1_rcur 19963  Ringcrg 20014  opp_rcoppr 20101  Unitcui 20121  DivRingcdr 20265  LIdealclidl 20732  2Idealc2idl 20802

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-sbg 18799  df-subg 18975  df-nsg 18976  df-eqg 18977  df-cmn 19614  df-abl 19615  df-mgp 19947  df-ur 19964  df-ring 20016  df-oppr 20102  df-dvdsr 20123  df-unit 20124  df-invr 20154  df-drng 20267  df-subrg 20310  df-lmod 20422  df-lss 20492  df-sra 20734  df-rgmod 20735  df-lidl 20736  df-2idl 20803

This theorem is referenced by:  qsdrng  32457