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

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   class class class wbr 4128   ` cfv 5375  (class class class)co 6079   Ap wap 7601   TAp wtap 7608   Basecbs 13335   +g cplusg 13414   invgcminusg 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)