Theorem opprdrng 14480

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 14246	. . . . 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 14364	. . . . . . . . 9 ⊢ (𝑅 ∈ LRing ↔ 𝑂 ∈ LRing)
8	7	a1i 9	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ LRing ↔ 𝑂 ∈ LRing))
9		aprlring 14460	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ LRing ↔ (#r‘𝑅) Ap (Base‘𝑅)))
10		aprlring 14460	. . . . . . . . . 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 2234	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
13	2, 12	opprbasg 14240	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑂))
14		papeq2 7563	. . . . . . . . . 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 2234	. . . . . . . . . . . . . . . . 17 ⊢ (+g‘𝑅) = (+g‘𝑅)
19	2, 18	oppraddg 14241	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → (+g‘𝑅) = (+g‘𝑂))
20	19	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (+g‘𝑅) = (+g‘𝑂))
21		eqidd 2235	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 = 𝑥)
22		eqid 2234	. . . . . . . . . . . . . . . . . 18 ⊢ (invg‘𝑅) = (invg‘𝑅)
23	2, 22	opprnegg 14249	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Ring → (invg‘𝑅) = (invg‘𝑂))
24	23	ad2antrr 488	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (invg‘𝑅) = (invg‘𝑂))
25	24	fveq1d 5674	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((invg‘𝑅)‘𝑦) = ((invg‘𝑂)‘𝑦))
26	20, 21, 25	oveq123d 6073	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)((invg‘𝑅)‘𝑦)) = (𝑥(+g‘𝑂)((invg‘𝑂)‘𝑦)))
27		simplr 529	. . . . . . . . . . . . . . 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 2234	. . . . . . . . . . . . . . . 16 ⊢ (-g‘𝑅) = (-g‘𝑅)
30	12, 18, 22, 29	grpsubval 13780	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(-g‘𝑅)𝑦) = (𝑥(+g‘𝑅)((invg‘𝑅)‘𝑦)))
31	27, 28, 30	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(-g‘𝑅)𝑦) = (𝑥(+g‘𝑅)((invg‘𝑅)‘𝑦)))
32	13	ad2antrr 488	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (Base‘𝑅) = (Base‘𝑂))
33	27, 32	eleqtrd 2313	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑂))
34	28, 32	eleqtrd 2313	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑂))
35		eqid 2234	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑂) = (Base‘𝑂)
36		eqid 2234	. . . . . . . . . . . . . . . 16 ⊢ (+g‘𝑂) = (+g‘𝑂)
37		eqid 2234	. . . . . . . . . . . . . . . 16 ⊢ (invg‘𝑂) = (invg‘𝑂)
38		eqid 2234	. . . . . . . . . . . . . . . 16 ⊢ (-g‘𝑂) = (-g‘𝑂)
39	35, 36, 37, 38	grpsubval 13780	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (Base‘𝑂) ∧ 𝑦 ∈ (Base‘𝑂)) → (𝑥(-g‘𝑂)𝑦) = (𝑥(+g‘𝑂)((invg‘𝑂)‘𝑦)))
40	33, 34, 39	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(-g‘𝑂)𝑦) = (𝑥(+g‘𝑂)((invg‘𝑂)‘𝑦)))
41	26, 31, 40	3eqtr4d 2277	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(-g‘𝑅)𝑦) = (𝑥(-g‘𝑂)𝑦))
42	41	eleq1d 2303	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅) ↔ (𝑥(-g‘𝑂)𝑦) ∈ (Unit‘𝑅)))
43		eqidd 2235	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (Base‘𝑅) = (Base‘𝑅))
44		eqidd 2235	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (#r‘𝑅) = (#r‘𝑅))
45		eqidd 2235	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (-g‘𝑅) = (-g‘𝑅))
46		eqidd 2235	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (Unit‘𝑅) = (Unit‘𝑅))
47		simpll 527	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ Ring)
48	43, 44, 45, 46, 47, 27, 28	aprval 14451	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(#r‘𝑅)𝑦 ↔ (𝑥(-g‘𝑅)𝑦) ∈ (Unit‘𝑅)))
49		eqidd 2235	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (#r‘𝑂) = (#r‘𝑂))
50		eqidd 2235	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (-g‘𝑂) = (-g‘𝑂))
51		eqidd 2235	. . . . . . . . . . . . . . 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 14277	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → (Unit‘𝑅) = (Unit‘𝑂))
55	54	ad2antrr 488	. . . . . . . . . . . . 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 14451	. . . . . . . . . . . 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 673	. . . . . . . . . 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 2540	. . . . . . . 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 2540	. . . . . . 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 473	. . . . . 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 7568	. . . . . 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 7568	. . . . . 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 7572	. . . . . 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 473	. . 3 ⊢ (𝑅 ∈ Ring → ((𝑅 ∈ Ring ∧ (#r‘𝑅) TAp (Base‘𝑅)) ↔ (𝑂 ∈ Ring ∧ (#r‘𝑂) TAp (Base‘𝑂))))
71	1, 5, 70	pm5.21nii 712	. 2 ⊢ ((𝑅 ∈ Ring ∧ (#r‘𝑅) TAp (Base‘𝑅)) ↔ (𝑂 ∈ Ring ∧ (#r‘𝑂) TAp (Base‘𝑂)))
72		eqid 2234	. . 3 ⊢ (#r‘𝑅) = (#r‘𝑅)
73	12, 72	isdrngtap 14466	. 2 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (#r‘𝑅) TAp (Base‘𝑅)))
74		eqid 2234	. . 3 ⊢ (#r‘𝑂) = (#r‘𝑂)
75	35, 74	isdrngtap 14466	. 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 1398   ∈ wcel 2205  ∀wral 2522   class class class wbr 4111  ‘cfv 5354  (class class class)co 6052   Ap wap 7560   TAp wtap 7567  Basecbs 13233  +_gcplusg 13311  inv_gcminusg 13735  -_gcsg 13736  Ringcrg 14161  opp_rcoppr 14232  Unitcui 14253  LRingclring 14357  #_rcapr 14449  DivRingcdr 14462

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-addcom 8232  ax-addass 8234  ax-i2m1 8237  ax-0lt1 8238  ax-0id 8240  ax-rnegex 8241  ax-pre-ltirr 8244  ax-pre-lttrn 8246  ax-pre-ltadd 8248

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-tpos 6478  df-pap 7561  df-tap 7568  df-pnf 8315  df-mnf 8316  df-ltxr 8318  df-inn 9243  df-2 9301  df-3 9302  df-ndx 13236  df-slot 13237  df-base 13239  df-sets 13240  df-iress 13241  df-plusg 13324  df-mulr 13325  df-0g 13492  df-mgm 13590  df-sgrp 13636  df-mnd 13651  df-grp 13737  df-minusg 13738  df-sbg 13739  df-cmn 14024  df-abl 14025  df-mgp 14086  df-ur 14125  df-srg 14129  df-ring 14163  df-oppr 14233  df-dvdsr 14255  df-unit 14256  df-invr 14288  df-dvr 14299  df-nzr 14347  df-lring 14358  df-apr 14450  df-drngap 14464

This theorem is referenced by: (None)