Theorem opprqusdrng 34017

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 2761	. . . . . 6 ⊢ (oppr‘𝑄) = (oppr‘𝑄)
2		eqid 2761	. . . . . 6 ⊢ (1r‘𝑄) = (1r‘𝑄)
3	1, 2	oppr1 20580	. . . . 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 34016	. . . . 5 ⊢ (𝜑 → (1r‘(oppr‘𝑄)) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))
10	3, 9	eqtrid 2808	. . . 4 ⊢ (𝜑 → (1r‘𝑄) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))
11		eqid 2761	. . . . . 6 ⊢ (0g‘𝑄) = (0g‘𝑄)
12	1, 11	oppr0 20579	. . . . 5 ⊢ (0g‘𝑄) = (0g‘(oppr‘𝑄))
13	8	2idllidld 21547	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))
14		lidlnsg 21536	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
15	7, 13, 14	syl2anc 596	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
16	4, 5, 6, 15	opprqus0g 34014	. . . . 5 ⊢ (𝜑 → (0g‘(oppr‘𝑄)) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼))))
17	12, 16	eqtrid 2808	. . . 4 ⊢ (𝜑 → (0g‘𝑄) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼))))
18	10, 17	neeq12d 3017	. . 3 ⊢ (𝜑 → ((1r‘𝑄) ≠ (0g‘𝑄) ↔ (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ≠ (0g‘(𝑂 /s (𝑂 ~QG 𝐼)))))
19		eqid 2761	. . . . . . 7 ⊢ (Base‘𝑄) = (Base‘𝑄)
20	1, 19	opprbas 20573	. . . . . 6 ⊢ (Base‘𝑄) = (Base‘(oppr‘𝑄))
21		eqid 2761	. . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
22	4, 21	lidlss 21490	. . . . . . . 8 ⊢ (𝐼 ∈ (LIdeal‘𝑅) → 𝐼 ⊆ 𝐵)
23	13, 22	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐼 ⊆ 𝐵)
24	4, 5, 6, 7, 23	opprqusbas 34012	. . . . . 6 ⊢ (𝜑 → (Base‘(oppr‘𝑄)) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
25	20, 24	eqtrid 2808	. . . . 5 ⊢ (𝜑 → (Base‘𝑄) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
26	17	sneqd 4596	. . . . 5 ⊢ (𝜑 → {(0g‘𝑄)} = {(0g‘(𝑂 /s (𝑂 ~QG 𝐼)))})
27	25, 26	difeq12d 4075	. . . 4 ⊢ (𝜑 → ((Base‘𝑄) ∖ {(0g‘𝑄)}) = ((Base‘(𝑂 /s (𝑂 ~QG 𝐼))) ∖ {(0g‘(𝑂 /s (𝑂 ~QG 𝐼)))}))
28	25	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) → (Base‘𝑄) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
29	7	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑅 ∈ Ring)
30	8	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝐼 ∈ (2Ideal‘𝑅))
31		simplr 781	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)}))
32	31	eldifad 3911	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑥 ∈ (Base‘𝑄))
33		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑦 ∈ (Base‘𝑄))
34	4, 5, 6, 29, 30, 19, 32, 33	opprqusmulr 34015	. . . . . . 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 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g‘𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → (1r‘𝑄) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))
36	34, 35	eqeq12d 2777	. . . . . 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 34015	. . . . . . 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 2777	. . . . . 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 644	. . . . 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 644	. 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 2761	. . 3 ⊢ (.r‘(oppr‘𝑄)) = (.r‘(oppr‘𝑄))
44		eqid 2761	. . . 4 ⊢ (Unit‘𝑄) = (Unit‘𝑄)
45	44, 1	opprunit 20607	. . 3 ⊢ (Unit‘𝑄) = (Unit‘(oppr‘𝑄))
46		eqid 2761	. . . . . 6 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅)
47	6, 46	qusring 21569	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (2Ideal‘𝑅)) → 𝑄 ∈ Ring)
48	7, 8, 47	syl2anc 596	. . . 4 ⊢ (𝜑 → 𝑄 ∈ Ring)
49	1	opprring 20577	. . . 4 ⊢ (𝑄 ∈ Ring → (oppr‘𝑄) ∈ Ring)
50	48, 49	syl 18	. . 3 ⊢ (𝜑 → (oppr‘𝑄) ∈ Ring)
51	20, 12, 3, 43, 45, 50	isdrng4 20992	. 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 2761	. . 3 ⊢ (Base‘(𝑂 /s (𝑂 ~QG 𝐼))) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))
53		eqid 2761	. . 3 ⊢ (0g‘(𝑂 /s (𝑂 ~QG 𝐼))) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼)))
54		eqid 2761	. . 3 ⊢ (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))
55		eqid 2761	. . 3 ⊢ (.r‘(𝑂 /s (𝑂 ~QG 𝐼))) = (.r‘(𝑂 /s (𝑂 ~QG 𝐼)))
56		eqid 2761	. . 3 ⊢ (Unit‘(𝑂 /s (𝑂 ~QG 𝐼))) = (Unit‘(𝑂 /s (𝑂 ~QG 𝐼)))
57	5	opprring 20577	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring)
58	7, 57	syl 18	. . . 4 ⊢ (𝜑 → 𝑂 ∈ Ring)
59	5, 7	oppr2idl 34010	. . . . 5 ⊢ (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂))
60	8, 59	eleqtrd 2863	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑂))
61		eqid 2761	. . . . 5 ⊢ (𝑂 /s (𝑂 ~QG 𝐼)) = (𝑂 /s (𝑂 ~QG 𝐼))
62		eqid 2761	. . . . 5 ⊢ (2Ideal‘𝑂) = (2Ideal‘𝑂)
63	61, 62	qusring 21569	. . . 4 ⊢ ((𝑂 ∈ Ring ∧ 𝐼 ∈ (2Ideal‘𝑂)) → (𝑂 /s (𝑂 ~QG 𝐼)) ∈ Ring)
64	58, 60, 63	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑂 /s (𝑂 ~QG 𝐼)) ∈ Ring)
65	52, 53, 54, 55, 56, 64	isdrng4 20992	. 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087   ∖ cdif 3896   ⊆ wss 3899  {csn 4584  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  ._rcmulr 17429  0_gc0g 17610   /_s cqus 17677  NrmSGrpcnsg 19331   ~_QG cqg 19332  1_rcur 20407  Ringcrg 20459  opp_rcoppr 20566  Unitcui 20585  DivRingcdr 20980  LIdealclidl 21484  2Idealc2idl 21542

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-tpos 8243  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-ec 8719  df-qs 8723  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-sup 9434  df-inf 9435  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-sca 17444  df-vsca 17445  df-ip 17446  df-tset 17447  df-ple 17448  df-ds 17450  df-0g 17612  df-imas 17680  df-qus 17681  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-grp 19147  df-minusg 19148  df-sbg 19149  df-subg 19333  df-nsg 19334  df-eqg 19335  df-cmn 19996  df-abl 19997  df-mgp 20361  df-rng 20375  df-ur 20408  df-ring 20461  df-oppr 20567  df-dvdsr 20587  df-unit 20588  df-invr 20618  df-subrg 20822  df-drng 20982  df-lmod 21137  df-lss 21207  df-sra 21448  df-rgmod 21449  df-lidl 21486  df-2idl 21543

This theorem is used by:  qsdrng  34021