Theorem opprdrng 14730

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 14496	. . . . 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 14614	. . . . . . . . 9 ⊢ (𝑅 ∈ LRing ↔ 𝑂 ∈ LRing)
8	7	a1i 9	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ LRing ↔ 𝑂 ∈ LRing))
9		aprlring 14710	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ LRing ↔ ( #r ‘𝑅) Ap (Base‘𝑅)))
10		aprlring 14710	. . . . . . . . . 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 14490	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑂))
14		papeq2 7617	. . . . . . . . . 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 14491	. . . . . . . . . . . . . . . 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 14499	. . . . . . . . . . . . . . . . 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 5700	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((invg‘𝑅)‘𝑦) = ((invg‘𝑂)‘𝑦))
26	20, 21, 25	oveq123d 6109	. . . . . . . . . . . . . 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 13911	. . . . . . . . . . . . . . 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 13911	. . . . . . . . . . . . . . 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 14701	. . . . . . . . . . . 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 14527	. . . . . . . . . . . . . 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 14701	. . . . . . . . . . . 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 7622	. . . . . 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 7622	. . . . . 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 7626	. . . . . 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 14716	. 2 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ ( #r ‘𝑅) TAp (Base‘𝑅)))
74		eqid 2238	. . 3 ⊢ ( #r ‘𝑂) = ( #r ‘𝑂)
75	35, 74	isdrngtap 14716	. 2 ⊢ (𝑂 ∈ DivRing ↔ (𝑂 ∈ Ring ∧ ( #r ‘𝑂) TAp (Base‘𝑂)))
76	71, 73, 75	3bitr4i 212	1 ⊢ (𝑅 ∈ DivRing ↔ 𝑂 ∈ DivRing)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528   class class class wbr 4131  ‘cfv 5380  (class class class)co 6088   Ap wap 7614   TAp wtap 7621  Basecbs 13411  +_gcplusg 13491  inv_gcminusg 13866  -_gcsg 13867  Ringcrg 14408  opp_rcoppr 14482  Unitcui 14503  LRingclring 14607   #_r capr 14699  DivRingcdr 14712

This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-addcom 8286  ax-addass 8288  ax-i2m1 8291  ax-0lt1 8292  ax-0id 8294  ax-rnegex 8295  ax-pre-ltirr 8298  ax-pre-lttrn 8300  ax-pre-ltadd 8302

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-tpos 6520  df-pap 7615  df-tap 7622  df-pnf 8369  df-mnf 8370  df-ltxr 8372  df-inn 9315  df-2 9373  df-3 9374  df-ndx 13414  df-slot 13415  df-base 13417  df-sets 13418  df-iress 13419  df-plusg 13504  df-mulr 13505  df-0g 13672  df-mgm 13736  df-sgrp 13777  df-mnd 13790  df-grp 13868  df-minusg 13869  df-sbg 13870  df-cmn 14180  df-abl 14181  df-mgp 14326  df-ur 14371  df-srg 14376  df-ring 14410  df-oppr 14483  df-dvdsr 14505  df-unit 14506  df-invr 14538  df-dvr 14549  df-nzr 14597  df-lring 14608  df-apr 14700  df-drngap 14714

This theorem is used by: (None)