Theorem opprdrng 14603

Description: The opposite of a division ring is also a division ring. (Contributed by NM, 18-Oct-2014.)

Hypothesis

Ref	Expression
opprdrng.1	⊢ 𝑂 = (oppr‘𝑅)

Assertion

Ref	Expression
opprdrng	⊢ (𝑅 ∈ DivRing ↔ 𝑂 ∈ DivRing)

Proof of Theorem opprdrng

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

Step	Hyp	Ref	Expression
1		simpl 109	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (#r‘𝑅) TAp (Base‘𝑅)) → 𝑅 ∈ Ring)
2		opprdrng.1	. . . . . 6 ⊢ 𝑂 = (oppr‘𝑅)
3	2	opprringb 14369	. . . . 5 ⊢ (𝑅 ∈ Ring ↔ 𝑂 ∈ Ring)
4	3	biimpri 133	. . . 4 ⊢ (𝑂 ∈ Ring → 𝑅 ∈ Ring)
5	4	adantr 276	. . 3 ⊢ ((𝑂 ∈ Ring ∧ (#r‘𝑂) TAp (Base‘𝑂)) → 𝑅 ∈ Ring)
6	3	a1i 9	. . . 4 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ Ring ↔ 𝑂 ∈ Ring))
7	2	opprlring 14487	. . . . . . . . 9 ⊢ (𝑅 ∈ LRing ↔ 𝑂 ∈ LRing)
8	7	a1i 9	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ LRing ↔ 𝑂 ∈ LRing))
9		aprlring 14583	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ LRing ↔ (#r‘𝑅) Ap (Base‘𝑅)))
10		aprlring 14583	. . . . . . . . . 10 ⊢ (𝑂 ∈ Ring → (𝑂 ∈ LRing ↔ (#r‘𝑂) Ap (Base‘𝑂)))
11	3, 10	sylbi 121	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (𝑂 ∈ LRing ↔ (#r‘𝑂) Ap (Base‘𝑂)))
12		eqid 2238	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
13	2, 12	opprbasg 14363	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑂))
14		papeq2 7604	. . . . . . . . . 10 ⊢ ((Base‘𝑅) = (Base‘𝑂) → ((#r‘𝑂) Ap (Base‘𝑅) ↔ (#r‘𝑂) Ap (Base‘𝑂)))
15	13, 14	syl 14	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → ((#r‘𝑂) Ap (Base‘𝑅) ↔ (#r‘𝑂) Ap (Base‘𝑂)))
16	11, 15	bitr4d 191	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (𝑂 ∈ LRing ↔ (#r‘𝑂) Ap (Base‘𝑅)))
17	8, 9, 16	3bitr3d 218	. . . . . . 7 ⊢ (𝑅 ∈ Ring → ((#r‘𝑅) Ap (Base‘𝑅) ↔ (#r‘𝑂) Ap (Base‘𝑅)))
18		eqid 2238	. . . . . . . . . . . . . . . . 17 ⊢ (+g‘𝑅) = (+g‘𝑅)
19	2, 18	oppraddg 14364	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → (+g‘𝑅) = (+g‘𝑂))
20	19	ad2antrr 492	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (+g‘𝑅) = (+g‘𝑂))
21		eqidd 2239	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 = 𝑥)
22		eqid 2238	. . . . . . . . . . . . . . . . . 18 ⊢ (invg‘𝑅) = (invg‘𝑅)
23	2, 22	opprnegg 14372	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Ring → (invg‘𝑅) = (invg‘𝑂))
24	23	ad2antrr 492	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (invg‘𝑅) = (invg‘𝑂))
25	24	fveq1d 5695	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((invg‘𝑅)‘𝑦) = ((invg‘𝑂)‘𝑦))
26	20, 21, 25	oveq123d 6100	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)((invg‘𝑅)‘𝑦)) = (𝑥(+g‘𝑂)((invg‘𝑂)‘𝑦)))
27		simplr 533	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
28		simpr 110	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑅))
29		eqid 2238	. . . . . . . . . . . . . . . 16 ⊢ (-g‘𝑅) = (-g‘𝑅)
30	12, 18, 22, 29	grpsubval 13834	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(-g‘𝑅)𝑦) = (𝑥(+g‘𝑅)((invg‘𝑅)‘𝑦)))
31	27, 28, 30	syl2anc 415	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(-g‘𝑅)𝑦) = (𝑥(+g‘𝑅)((invg‘𝑅)‘𝑦)))
32	13	ad2antrr 492	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (Base‘𝑅) = (Base‘𝑂))
33	27, 32	eleqtrd 2317	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑂))
34	28, 32	eleqtrd 2317	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑂))
35		eqid 2238	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑂) = (Base‘𝑂)
36		eqid 2238	. . . . . . . . . . . . . . . 16 ⊢ (+g‘𝑂) = (+g‘𝑂)
37		eqid 2238	. . . . . . . . . . . . . . . 16 ⊢ (invg‘𝑂) = (invg‘𝑂)
38		eqid 2238	. . . . . . . . . . . . . . . 16 ⊢ (-g‘𝑂) = (-g‘𝑂)
39	35, 36, 37, 38	grpsubval 13834	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (Base‘𝑂) ∧ 𝑦 ∈ (Base‘𝑂)) → (𝑥(-g‘𝑂)𝑦) = (𝑥(+g‘𝑂)((invg‘𝑂)‘𝑦)))
40	33, 34, 39	syl2anc 415	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(-g‘𝑂)𝑦) = (𝑥(+g‘𝑂)((invg‘𝑂)‘𝑦)))
41	26, 31, 40	3eqtr4d 2281	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(-g‘𝑅)𝑦) = (𝑥(-g‘𝑂)𝑦))
42	41	eleq1d 2307	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅) ↔ (𝑥(-g‘𝑂)𝑦) ∈ (Unit‘𝑅)))
43		eqidd 2239	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (Base‘𝑅) = (Base‘𝑅))
44		eqidd 2239	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (#r‘𝑅) = (#r‘𝑅))
45		eqidd 2239	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (-g‘𝑅) = (-g‘𝑅))
46		eqidd 2239	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (Unit‘𝑅) = (Unit‘𝑅))
47		simpll 531	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ Ring)
48	43, 44, 45, 46, 47, 27, 28	aprval 14574	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(#r‘𝑅)𝑦 ↔ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅)))
49		eqidd 2239	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (#r‘𝑂) = (#r‘𝑂))
50		eqidd 2239	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (-g‘𝑂) = (-g‘𝑂))
51		eqidd 2239	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ Ring → (Unit‘𝑅) = (Unit‘𝑅))
52	2	a1i 9	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ Ring → 𝑂 = (oppr‘𝑅))
53		id 19	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Ring)
54	51, 52, 53	opprunitd 14400	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → (Unit‘𝑅) = (Unit‘𝑂))
55	54	ad2antrr 492	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (Unit‘𝑅) = (Unit‘𝑂))
56	47, 3	sylib 122	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑂 ∈ Ring)
57	32, 49, 50, 55, 56, 27, 28	aprval 14574	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(#r‘𝑂)𝑦 ↔ (𝑥(-g‘𝑂)𝑦) ∈ (Unit‘𝑅)))
58	42, 48, 57	3bitr4d 220	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(#r‘𝑅)𝑦 ↔ 𝑥(#r‘𝑂)𝑦))
59	58	notbid 677	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (¬ 𝑥(#r‘𝑅)𝑦 ↔ ¬ 𝑥(#r‘𝑂)𝑦))
60	59	imbi1d 231	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((¬ 𝑥(#r‘𝑅)𝑦 → 𝑥 = 𝑦) ↔ (¬ 𝑥(#r‘𝑂)𝑦 → 𝑥 = 𝑦)))
61	60	ralbidva 2546	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → (∀𝑦 ∈ (Base‘𝑅)(¬ 𝑥(#r‘𝑅)𝑦 → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ (Base‘𝑅)(¬ 𝑥(#r‘𝑂)𝑦 → 𝑥 = 𝑦)))
62	61	ralbidva 2546	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(¬ 𝑥(#r‘𝑅)𝑦 → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(¬ 𝑥(#r‘𝑂)𝑦 → 𝑥 = 𝑦)))
63	17, 62	anbi12d 477	. . . . . 6 ⊢ (𝑅 ∈ Ring → (((#r‘𝑅) Ap (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(¬ 𝑥(#r‘𝑅)𝑦 → 𝑥 = 𝑦)) ↔ ((#r‘𝑂) Ap (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(¬ 𝑥(#r‘𝑂)𝑦 → 𝑥 = 𝑦))))
64		df-tap 7609	. . . . . 6 ⊢ ((#r‘𝑅) TAp (Base‘𝑅) ↔ ((#r‘𝑅) Ap (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(¬ 𝑥(#r‘𝑅)𝑦 → 𝑥 = 𝑦)))
65		df-tap 7609	. . . . . 6 ⊢ ((#r‘𝑂) TAp (Base‘𝑅) ↔ ((#r‘𝑂) Ap (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(¬ 𝑥(#r‘𝑂)𝑦 → 𝑥 = 𝑦)))
66	63, 64, 65	3bitr4g 223	. . . . 5 ⊢ (𝑅 ∈ Ring → ((#r‘𝑅) TAp (Base‘𝑅) ↔ (#r‘𝑂) TAp (Base‘𝑅)))
67		tapeq2 7613	. . . . . 6 ⊢ ((Base‘𝑅) = (Base‘𝑂) → ((#r‘𝑂) TAp (Base‘𝑅) ↔ (#r‘𝑂) TAp (Base‘𝑂)))
68	13, 67	syl 14	. . . . 5 ⊢ (𝑅 ∈ Ring → ((#r‘𝑂) TAp (Base‘𝑅) ↔ (#r‘𝑂) TAp (Base‘𝑂)))
69	66, 68	bitrd 188	. . . 4 ⊢ (𝑅 ∈ Ring → ((#r‘𝑅) TAp (Base‘𝑅) ↔ (#r‘𝑂) TAp (Base‘𝑂)))
70	6, 69	anbi12d 477	. . 3 ⊢ (𝑅 ∈ Ring → ((𝑅 ∈ Ring ∧ (#r‘𝑅) TAp (Base‘𝑅)) ↔ (𝑂 ∈ Ring ∧ (#r‘𝑂) TAp (Base‘𝑂))))
71	1, 5, 70	pm5.21nii 716	. 2 ⊢ ((𝑅 ∈ Ring ∧ (#r‘𝑅) TAp (Base‘𝑅)) ↔ (𝑂 ∈ Ring ∧ (#r‘𝑂) TAp (Base‘𝑂)))
72		eqid 2238	. . 3 ⊢ (#r‘𝑅) = (#r‘𝑅)
73	12, 72	isdrngtap 14589	. 2 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (#r‘𝑅) TAp (Base‘𝑅)))
74		eqid 2238	. . 3 ⊢ (#r‘𝑂) = (#r‘𝑂)
75	35, 74	isdrngtap 14589	. 2 ⊢ (𝑂 ∈ DivRing ↔ (𝑂 ∈ Ring ∧ (#r‘𝑂) TAp (Base‘𝑂)))
76	71, 73, 75	3bitr4i 212	1 ⊢ (𝑅 ∈ DivRing ↔ 𝑂 ∈ DivRing)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528   class class class wbr 4128  ‘cfv 5375  (class class class)co 6079   Ap wap 7601   TAp wtap 7608  Basecbs 13335  +_gcplusg 13414  inv_gcminusg 13789  -_gcsg 13790  Ringcrg 14283  opp_rcoppr 14355  Unitcui 14376  LRingclring 14480  #_rcapr 14572  DivRingcdr 14585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-pre-ltirr 8285  ax-pre-lttrn 8287  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-tpos 6510  df-pap 7602  df-tap 7609  df-pnf 8356  df-mnf 8357  df-ltxr 8359  df-inn 9288  df-2 9346  df-3 9347  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-iress 13343  df-plusg 13427  df-mulr 13428  df-0g 13595  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-grp 13791  df-minusg 13792  df-sbg 13793  df-cmn 14072  df-abl 14073  df-mgp 14201  df-ur 14246  df-srg 14251  df-ring 14285  df-oppr 14356  df-dvdsr 14378  df-unit 14379  df-invr 14411  df-dvr 14422  df-nzr 14470  df-lring 14481  df-apr 14573  df-drngap 14587

This theorem is referenced by: (None)