Theorem opprqusdrng 33471

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 2730	. . . . . 6 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
2		eqid 2730	. . . . . 6 ⊢ (1r‘𝑄) = (1r‘𝑄)
3	1, 2	oppr1 20266	. . . . 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 33470	. . . . 5 ⊢ (𝜑 → (1r‘(oppr‘𝑄)) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))
10	3, 9	eqtrid 2777	. . . 4 ⊢ (𝜑 → (1r‘𝑄) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))
11		eqid 2730	. . . . . 6 ⊢ (0g‘𝑄) = (0g‘𝑄)
12	1, 11	oppr0 20265	. . . . 5 ⊢ (0g‘𝑄) = (0g‘(oppr‘𝑄))
13	8	2idllidld 21171	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))
14		lidlnsg 21165	. . . . . . 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 33468	. . . . 5 ⊢ (𝜑 → (0g‘(oppr‘𝑄)) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼))))
17	12, 16	eqtrid 2777	. . . 4 ⊢ (𝜑 → (0g‘𝑄) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼))))
18	10, 17	neeq12d 2987	. . 3 ⊢ (𝜑 → ((1r‘𝑄) ≠ (0g‘𝑄) ↔ (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ≠ (0g‘(𝑂 /s (𝑂 ~QG 𝐼)))))
19		eqid 2730	. . . . . . 7 ⊢ (Base‘𝑄) = (Base‘𝑄)
20	1, 19	opprbas 20259	. . . . . 6 ⊢ (Base‘𝑄) = (Base‘(oppr‘𝑄))
21		eqid 2730	. . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
22	4, 21	lidlss 21129	. . . . . . . 8 ⊢ (𝐼 ∈ (LIdeal‘𝑅) → 𝐼 ⊆ 𝐵)
23	13, 22	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐼 ⊆ 𝐵)
24	4, 5, 6, 7, 23	opprqusbas 33466	. . . . . 6 ⊢ (𝜑 → (Base‘(oppr‘𝑄)) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
25	20, 24	eqtrid 2777	. . . . 5 ⊢ (𝜑 → (Base‘𝑄) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
26	17	sneqd 4604	. . . . 5 ⊢ (𝜑 → {(0g‘𝑄)} = {(0g‘(𝑂 /s (𝑂 ~QG 𝐼)))})
27	25, 26	difeq12d 4093	. . . 4 ⊢ (𝜑 → ((Base‘𝑄) ∖ {(0g‘𝑄)}) = ((Base‘(𝑂 /s (𝑂 ~QG 𝐼))) ∖ {(0g‘(𝑂 /s (𝑂 ~QG 𝐼)))}))
28	25	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → (Base‘𝑄) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
29	7	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑅 ∈ Ring)
30	8	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝐼 ∈ (2Ideal‘𝑅))
31		simplr 768	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)}))
32	31	eldifad 3929	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑥 ∈ (Base‘𝑄))
33		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑦 ∈ (Base‘𝑄))
34	4, 5, 6, 29, 30, 19, 32, 33	opprqusmulr 33469	. . . . . . 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 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → (1r‘𝑄) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))
36	34, 35	eqeq12d 2746	. . . . . 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 33469	. . . . . . 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 2746	. . . . . 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 632	. . . . 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 3308	. . . 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 3307	. . 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 632	. 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 2730	. . 3 ⊢ (.r‘(oppr‘𝑄)) = (.r‘(oppr‘𝑄))
44		eqid 2730	. . . 4 ⊢ (Unit‘𝑄) = (Unit‘𝑄)
45	44, 1	opprunit 20293	. . 3 ⊢ (Unit‘𝑄) = (Unit‘(oppr‘𝑄))
46		eqid 2730	. . . . . 6 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
47	6, 46	qusring 21192	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (2Ideal‘𝑅)) → 𝑄 ∈ Ring)
48	7, 8, 47	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝑄 ∈ Ring)
49	1	opprring 20263	. . . 4 ⊢ (𝑄 ∈ Ring → (oppr‘𝑄) ∈ Ring)
50	48, 49	syl 17	. . 3 ⊢ (𝜑 → (oppr‘𝑄) ∈ Ring)
51	20, 12, 3, 43, 45, 50	isdrng4 33252	. 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 2730	. . 3 ⊢ (Base‘(𝑂 /s (𝑂 ~QG 𝐼))) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))
53		eqid 2730	. . 3 ⊢ (0g‘(𝑂 /s (𝑂 ~QG 𝐼))) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼)))
54		eqid 2730	. . 3 ⊢ (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))
55		eqid 2730	. . 3 ⊢ (.r‘(𝑂 /s (𝑂 ~QG 𝐼))) = (.r‘(𝑂 /s (𝑂 ~QG 𝐼)))
56		eqid 2730	. . 3 ⊢ (Unit‘(𝑂 /s (𝑂 ~QG 𝐼))) = (Unit‘(𝑂 /s (𝑂 ~QG 𝐼)))
57	5	opprring 20263	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring)
58	7, 57	syl 17	. . . 4 ⊢ (𝜑 → 𝑂 ∈ Ring)
59	5, 7	oppr2idl 33464	. . . . 5 ⊢ (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂))
60	8, 59	eleqtrd 2831	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑂))
61		eqid 2730	. . . . 5 ⊢ (𝑂 /s (𝑂 ~QG 𝐼)) = (𝑂 /s (𝑂 ~QG 𝐼))
62		eqid 2730	. . . . 5 ⊢ (2Ideal‘𝑂) = (2Ideal‘𝑂)
63	61, 62	qusring 21192	. . . 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 33252	. 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 311	1 ⊢ (𝜑 → ((oppr‘𝑄) ∈ DivRing ↔ (𝑂 /s (𝑂 ~QG 𝐼)) ∈ DivRing))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  ∃wrex 3054   ∖ cdif 3914   ⊆ wss 3917  {csn 4592  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  ._rcmulr 17228  0_gc0g 17409   /_s cqus 17475  NrmSGrpcnsg 19060   ~_QG cqg 19061  1_rcur 20097  Ringcrg 20149  opp_rcoppr 20252  Unitcui 20271  DivRingcdr 20645  LIdealclidl 21123  2Idealc2idl 21166

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-tpos 8208  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-ec 8676  df-qs 8680  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-inf 9401  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-0g 17411  df-imas 17478  df-qus 17479  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-grp 18875  df-minusg 18876  df-sbg 18877  df-subg 19062  df-nsg 19063  df-eqg 19064  df-cmn 19719  df-abl 19720  df-mgp 20057  df-rng 20069  df-ur 20098  df-ring 20151  df-oppr 20253  df-dvdsr 20273  df-unit 20274  df-invr 20304  df-subrg 20486  df-drng 20647  df-lmod 20775  df-lss 20845  df-sra 21087  df-rgmod 21088  df-lidl 21125  df-2idl 21167

This theorem is referenced by:  qsdrng  33475