Theorem opprqusdrng 33775

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 2763	. . . . . 6 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
2		eqid 2763	. . . . . 6 ⊢ (1r‘𝑄) = (1r‘𝑄)
3	1, 2	oppr1 20428	. . . . 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 33774	. . . . 5 ⊢ (𝜑 → (1r‘(oppr‘𝑄)) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))
10	3, 9	eqtrid 2810	. . . 4 ⊢ (𝜑 → (1r‘𝑄) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))
11		eqid 2763	. . . . . 6 ⊢ (0g‘𝑄) = (0g‘𝑄)
12	1, 11	oppr0 20427	. . . . 5 ⊢ (0g‘𝑄) = (0g‘(oppr‘𝑄))
13	8	2idllidld 21393	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))
14		lidlnsg 21382	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
15	7, 13, 14	syl2anc 595	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
16	4, 5, 6, 15	opprqus0g 33772	. . . . 5 ⊢ (𝜑 → (0g‘(oppr‘𝑄)) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼))))
17	12, 16	eqtrid 2810	. . . 4 ⊢ (𝜑 → (0g‘𝑄) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼))))
18	10, 17	neeq12d 3019	. . 3 ⊢ (𝜑 → ((1r‘𝑄) ≠ (0g‘𝑄) ↔ (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ≠ (0g‘(𝑂 /s (𝑂 ~QG 𝐼)))))
19		eqid 2763	. . . . . . 7 ⊢ (Base‘𝑄) = (Base‘𝑄)
20	1, 19	opprbas 20421	. . . . . 6 ⊢ (Base‘𝑄) = (Base‘(oppr‘𝑄))
21		eqid 2763	. . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
22	4, 21	lidlss 21336	. . . . . . . 8 ⊢ (𝐼 ∈ (LIdeal‘𝑅) → 𝐼 ⊆ 𝐵)
23	13, 22	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐼 ⊆ 𝐵)
24	4, 5, 6, 7, 23	opprqusbas 33770	. . . . . 6 ⊢ (𝜑 → (Base‘(oppr‘𝑄)) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
25	20, 24	eqtrid 2810	. . . . 5 ⊢ (𝜑 → (Base‘𝑄) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
26	17	sneqd 4601	. . . . 5 ⊢ (𝜑 → {(0g‘𝑄)} = {(0g‘(𝑂 /s (𝑂 ~QG 𝐼)))})
27	25, 26	difeq12d 4082	. . . 4 ⊢ (𝜑 → ((Base‘𝑄) ∖ {(0g‘𝑄)}) = ((Base‘(𝑂 /s (𝑂 ~QG 𝐼))) ∖ {(0g‘(𝑂 /s (𝑂 ~QG 𝐼)))}))
28	25	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → (Base‘𝑄) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
29	7	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑅 ∈ Ring)
30	8	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝐼 ∈ (2Ideal‘𝑅))
31		simplr 780	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)}))
32	31	eldifad 3917	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑥 ∈ (Base‘𝑄))
33		simpr 489	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑦 ∈ (Base‘𝑄))
34	4, 5, 6, 29, 30, 19, 32, 33	opprqusmulr 33773	. . . . . . 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 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → (1r‘𝑄) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))
36	34, 35	eqeq12d 2779	. . . . . 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 33773	. . . . . . 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 2779	. . . . . 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 643	. . . . 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 3330	. . . 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 3329	. . 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 643	. 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 2763	. . 3 ⊢ (.r‘(oppr‘𝑄)) = (.r‘(oppr‘𝑄))
44		eqid 2763	. . . 4 ⊢ (Unit‘𝑄) = (Unit‘𝑄)
45	44, 1	opprunit 20455	. . 3 ⊢ (Unit‘𝑄) = (Unit‘(oppr‘𝑄))
46		eqid 2763	. . . . . 6 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
47	6, 46	qusring 21414	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (2Ideal‘𝑅)) → 𝑄 ∈ Ring)
48	7, 8, 47	syl2anc 595	. . . 4 ⊢ (𝜑 → 𝑄 ∈ Ring)
49	1	opprring 20425	. . . 4 ⊢ (𝑄 ∈ Ring → (oppr‘𝑄) ∈ Ring)
50	48, 49	syl 18	. . 3 ⊢ (𝜑 → (oppr‘𝑄) ∈ Ring)
51	20, 12, 3, 43, 45, 50	isdrng4 20839	. 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 2763	. . 3 ⊢ (Base‘(𝑂 /s (𝑂 ~QG 𝐼))) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))
53		eqid 2763	. . 3 ⊢ (0g‘(𝑂 /s (𝑂 ~QG 𝐼))) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼)))
54		eqid 2763	. . 3 ⊢ (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))
55		eqid 2763	. . 3 ⊢ (.r‘(𝑂 /s (𝑂 ~QG 𝐼))) = (.r‘(𝑂 /s (𝑂 ~QG 𝐼)))
56		eqid 2763	. . 3 ⊢ (Unit‘(𝑂 /s (𝑂 ~QG 𝐼))) = (Unit‘(𝑂 /s (𝑂 ~QG 𝐼)))
57	5	opprring 20425	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring)
58	7, 57	syl 18	. . . 4 ⊢ (𝜑 → 𝑂 ∈ Ring)
59	5, 7	oppr2idl 33768	. . . . 5 ⊢ (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂))
60	8, 59	eleqtrd 2865	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑂))
61		eqid 2763	. . . . 5 ⊢ (𝑂 /s (𝑂 ~QG 𝐼)) = (𝑂 /s (𝑂 ~QG 𝐼))
62		eqid 2763	. . . . 5 ⊢ (2Ideal‘𝑂) = (2Ideal‘𝑂)
63	61, 62	qusring 21414	. . . 4 ⊢ ((𝑂 ∈ Ring ∧ 𝐼 ∈ (2Ideal‘𝑂)) → (𝑂 /s (𝑂 ~QG 𝐼)) ∈ Ring)
64	58, 60, 63	syl2anc 595	. . 3 ⊢ (𝜑 → (𝑂 /s (𝑂 ~QG 𝐼)) ∈ Ring)
65	52, 53, 54, 55, 56, 64	isdrng4 20839	. 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 314	1 ⊢ (𝜑 → ((oppr‘𝑄) ∈ DivRing ↔ (𝑂 /s (𝑂 ~QG 𝐼)) ∈ DivRing))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089   ∖ cdif 3902   ⊆ wss 3905  {csn 4589  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  ._rcmulr 17306  0_gc0g 17487   /_s cqus 17554  NrmSGrpcnsg 19182   ~_QG cqg 19183  1_rcur 20258  Ringcrg 20310  opp_rcoppr 20414  Unitcui 20433  DivRingcdr 20827  LIdealclidl 21330  2Idealc2idl 21388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-tpos 8218  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-ec 8692  df-qs 8696  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-inf 9399  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-z 12587  df-dec 12707  df-uz 12858  df-fz 13531  df-struct 17202  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-ress 17286  df-plusg 17318  df-mulr 17319  df-sca 17321  df-vsca 17322  df-ip 17323  df-tset 17324  df-ple 17325  df-ds 17327  df-0g 17489  df-imas 17557  df-qus 17558  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-grp 18998  df-minusg 18999  df-sbg 19000  df-subg 19184  df-nsg 19185  df-eqg 19186  df-cmn 19847  df-abl 19848  df-mgp 20212  df-rng 20226  df-ur 20259  df-ring 20312  df-oppr 20415  df-dvdsr 20435  df-unit 20436  df-invr 20466  df-subrg 20669  df-drng 20829  df-lmod 20983  df-lss 21053  df-sra 21294  df-rgmod 21295  df-lidl 21332  df-2idl 21389

This theorem is referenced by:  qsdrng  33779