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 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 14246	. . . . 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 14364	. . . . . . . . 9 |- ( R e. LRing <-> O e. LRing )
8	7	a1i 9	. . . . . . . 8 |- ( R e. Ring -> ( R e. LRing <-> O e. LRing ) )
9		aprlring 14460	. . . . . . . 8 |- ( R e. Ring -> ( R e. LRing <-> (#r ` R ) Ap ( Base ` R ) ) )
10		aprlring 14460	. . . . . . . . . 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 2234	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
13	2, 12	opprbasg 14240	. . . . . . . . . 10 |- ( R e. 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 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 2234	. . . . . . . . . . . . . . . . 17 |- ( +g ` R ) = ( +g ` R )
19	2, 18	oppraddg 14241	. . . . . . . . . . . . . . . 16 |- ( R e. Ring -> ( +g ` R ) = ( +g ` O ) )
20	19	ad2antrr 488	. . . . . . . . . . . . . . 15 |- ( ( ( R e. Ring /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( +g ` R ) = ( +g ` O ) )
21		eqidd 2235	. . . . . . . . . . . . . . 15 |- ( ( ( R e. Ring /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> x = x )
22		eqid 2234	. . . . . . . . . . . . . . . . . 18 |- ( invg ` R ) = ( invg ` R )
23	2, 22	opprnegg 14249	. . . . . . . . . . . . . . . . 17 |- ( R e. Ring -> ( invg ` R ) = ( invg ` O ) )
24	23	ad2antrr 488	. . . . . . . . . . . . . . . 16 |- ( ( ( R e. Ring /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( invg ` R ) = ( invg ` O ) )
25	24	fveq1d 5674	. . . . . . . . . . . . . . 15 |- ( ( ( R e. Ring /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( ( invg ` R ) ` y ) = ( ( invg ` O ) ` y ) )
26	20, 21, 25	oveq123d 6073	. . . . . . . . . . . . . 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 529	. . . . . . . . . . . . . . 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 2234	. . . . . . . . . . . . . . . 16 |- ( -g ` R ) = ( -g ` R )
30	12, 18, 22, 29	grpsubval 13780	. . . . . . . . . . . . . . 15 |- ( ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( -g ` R ) y ) = ( x ( +g ` R ) ( ( invg ` R ) ` y ) ) )
31	27, 28, 30	syl2anc 411	. . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . 16 |- ( ( ( R e. Ring /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( Base ` R ) = ( Base ` O ) )
33	27, 32	eleqtrd 2313	. . . . . . . . . . . . . . 15 |- ( ( ( R e. Ring /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> x e. ( Base ` O ) )
34	28, 32	eleqtrd 2313	. . . . . . . . . . . . . . 15 |- ( ( ( R e. Ring /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> y e. ( 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 e. ( Base ` O ) /\ y e. ( Base ` O ) ) -> ( x ( -g ` O ) y ) = ( x ( +g ` O ) ( ( invg ` O ) ` y ) ) )
40	33, 34, 39	syl2anc 411	. . . . . . . . . . . . . 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 2277	. . . . . . . . . . . . 13 |- ( ( ( R e. Ring /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( x ( -g ` R ) y ) = ( x ( -g ` O ) y ) )
42	41	eleq1d 2303	. . . . . . . . . . . 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 2235	. . . . . . . . . . . . 13 |- ( ( ( R e. Ring /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( Base ` R ) = ( Base ` R ) )
44		eqidd 2235	. . . . . . . . . . . . 13 |- ( ( ( R e. Ring /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> (#r ` R ) = (#r ` R ) )
45		eqidd 2235	. . . . . . . . . . . . 13 |- ( ( ( R e. Ring /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( -g ` R ) = ( -g ` R ) )
46		eqidd 2235	. . . . . . . . . . . . 13 |- ( ( ( R e. Ring /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> (Unit ` R ) = (Unit ` R ) )
47		simpll 527	. . . . . . . . . . . . 13 |- ( ( ( R e. Ring /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> R e. Ring )
48	43, 44, 45, 46, 47, 27, 28	aprval 14451	. . . . . . . . . . . 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 2235	. . . . . . . . . . . . 13 |- ( ( ( R e. Ring /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> (#r ` O ) = (#r ` O ) )
50		eqidd 2235	. . . . . . . . . . . . 13 |- ( ( ( R e. Ring /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( -g ` O ) = ( -g ` O ) )
51		eqidd 2235	. . . . . . . . . . . . . . 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 14277	. . . . . . . . . . . . . 14 |- ( R e. Ring -> (Unit ` R ) = (Unit ` O ) )
55	54	ad2antrr 488	. . . . . . . . . . . . 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 14451	. . . . . . . . . . . 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 673	. . . . . . . . . 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 2540	. . . . . . . 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 2540	. . . . . . 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 473	. . . . . 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 7568	. . . . . 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 7568	. . . . . 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 7572	. . . . . 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 473	. . 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 712	. 2 |- ( ( R e. Ring /\ (#r ` R ) TAp ( Base ` R ) ) <-> ( O e. Ring /\ (#r ` O ) TAp ( Base ` O ) ) )
72		eqid 2234	. . 3 |- (#r ` R ) = (#r ` R )
73	12, 72	isdrngtap 14466	. 2 |- ( R e. DivRing <-> ( R e. Ring /\ (#r ` R ) TAp ( Base ` R ) ) )
74		eqid 2234	. . 3 |- (#r ` O ) = (#r ` O )
75	35, 74	isdrngtap 14466	. 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 1398   e. wcel 2205   A.wral 2522   class class class wbr 4111   ` cfv 5354  (class class class)co 6052   Ap wap 7560   TAp wtap 7567   Basecbs 13233   +g cplusg 13311   invgcminusg 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)