Theorem opprdomnbg 14223

Description: A class is a domain if and only if its opposite is a domain, biconditional form of opprdomn 14224. (Contributed by SN, 15-Jun-2015.)

Hypothesis

Ref	Expression
opprdomn.1	|- O = (oppr ` R )

Assertion

Ref	Expression
opprdomnbg	|- ( R e. V -> ( R e. Domn <-> O e. Domn ) )

Proof of Theorem opprdomnbg

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

Step	Hyp	Ref	Expression
1		opprdomn.1	. . . 4 |- O = (oppr ` R )
2	1	opprnzrbg 14134	. . 3 |- ( R e. V -> ( R e. NzRing <-> O e. NzRing ) )
3		eqid 2229	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
4	1, 3	opprbasg 14024	. . . . 5 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
5		vex 2802	. . . . . . . . . 10 |- y e. _V
6		vex 2802	. . . . . . . . . 10 |- x e. _V
7		eqid 2229	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
8		eqid 2229	. . . . . . . . . . 11 |- ( .r ` O ) = ( .r ` O )
9	3, 7, 1, 8	opprmulg 14020	. . . . . . . . . 10 |- ( ( R e. V /\ y e. _V /\ x e. _V ) -> ( y ( .r ` O ) x ) = ( x ( .r ` R ) y ) )
10	5, 6, 9	mp3an23 1363	. . . . . . . . 9 |- ( R e. V -> ( y ( .r ` O ) x ) = ( x ( .r ` R ) y ) )
11	10	eqcomd 2235	. . . . . . . 8 |- ( R e. V -> ( x ( .r ` R ) y ) = ( y ( .r ` O ) x ) )
12		eqid 2229	. . . . . . . . 9 |- ( 0g ` R ) = ( 0g ` R )
13	1, 12	oppr0g 14030	. . . . . . . 8 |- ( R e. V -> ( 0g ` R ) = ( 0g ` O ) )
14	11, 13	eqeq12d 2244	. . . . . . 7 |- ( R e. V -> ( ( x ( .r ` R ) y ) = ( 0g ` R ) <-> ( y ( .r ` O ) x ) = ( 0g ` O ) ) )
15	13	eqeq2d 2241	. . . . . . . . 9 |- ( R e. V -> ( x = ( 0g ` R ) <-> x = ( 0g ` O ) ) )
16	13	eqeq2d 2241	. . . . . . . . 9 |- ( R e. V -> ( y = ( 0g ` R ) <-> y = ( 0g ` O ) ) )
17	15, 16	orbi12d 798	. . . . . . . 8 |- ( R e. V -> ( ( x = ( 0g ` R ) \/ y = ( 0g ` R ) ) <-> ( x = ( 0g ` O ) \/ y = ( 0g ` O ) ) ) )
18		orcom 733	. . . . . . . 8 |- ( ( x = ( 0g ` O ) \/ y = ( 0g ` O ) ) <-> ( y = ( 0g ` O ) \/ x = ( 0g ` O ) ) )
19	17, 18	bitrdi 196	. . . . . . 7 |- ( R e. V -> ( ( x = ( 0g ` R ) \/ y = ( 0g ` R ) ) <-> ( y = ( 0g ` O ) \/ x = ( 0g ` O ) ) ) )
20	14, 19	imbi12d 234	. . . . . 6 |- ( R e. V -> ( ( ( x ( .r ` R ) y ) = ( 0g ` R ) -> ( x = ( 0g ` R ) \/ y = ( 0g ` R ) ) ) <-> ( ( y ( .r ` O ) x ) = ( 0g ` O ) -> ( y = ( 0g ` O ) \/ x = ( 0g ` O ) ) ) ) )
21	4, 20	raleqbidv 2744	. . . . 5 |- ( R e. V -> ( A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) = ( 0g ` R ) -> ( x = ( 0g ` R ) \/ y = ( 0g ` R ) ) ) <-> A. y e. ( Base ` O ) ( ( y ( .r ` O ) x ) = ( 0g ` O ) -> ( y = ( 0g ` O ) \/ x = ( 0g ` O ) ) ) ) )
22	4, 21	raleqbidv 2744	. . . 4 |- ( R e. V -> ( A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) = ( 0g ` R ) -> ( x = ( 0g ` R ) \/ y = ( 0g ` R ) ) ) <-> A. x e. ( Base ` O ) A. y e. ( Base ` O ) ( ( y ( .r ` O ) x ) = ( 0g ` O ) -> ( y = ( 0g ` O ) \/ x = ( 0g ` O ) ) ) ) )
23		ralcom 2694	. . . 4 |- ( A. x e. ( Base ` O ) A. y e. ( Base ` O ) ( ( y ( .r ` O ) x ) = ( 0g ` O ) -> ( y = ( 0g ` O ) \/ x = ( 0g ` O ) ) ) <-> A. y e. ( Base ` O ) A. x e. ( Base ` O ) ( ( y ( .r ` O ) x ) = ( 0g ` O ) -> ( y = ( 0g ` O ) \/ x = ( 0g ` O ) ) ) )
24	22, 23	bitrdi 196	. . 3 |- ( R e. V -> ( A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) = ( 0g ` R ) -> ( x = ( 0g ` R ) \/ y = ( 0g ` R ) ) ) <-> A. y e. ( Base ` O ) A. x e. ( Base ` O ) ( ( y ( .r ` O ) x ) = ( 0g ` O ) -> ( y = ( 0g ` O ) \/ x = ( 0g ` O ) ) ) ) )
25	2, 24	anbi12d 473	. 2 |- ( R e. V -> ( ( R e. NzRing /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) = ( 0g ` R ) -> ( x = ( 0g ` R ) \/ y = ( 0g ` R ) ) ) ) <-> ( O e. NzRing /\ A. y e. ( Base ` O ) A. x e. ( Base ` O ) ( ( y ( .r ` O ) x ) = ( 0g ` O ) -> ( y = ( 0g ` O ) \/ x = ( 0g ` O ) ) ) ) ) )
26	3, 7, 12	isdomn 14218	. 2 |- ( R e. Domn <-> ( R e. NzRing /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) = ( 0g ` R ) -> ( x = ( 0g ` R ) \/ y = ( 0g ` R ) ) ) ) )
27		eqid 2229	. . 3 |- ( Base ` O ) = ( Base ` O )
28		eqid 2229	. . 3 |- ( 0g ` O ) = ( 0g ` O )
29	27, 8, 28	isdomn 14218	. 2 |- ( O e. Domn <-> ( O e. NzRing /\ A. y e. ( Base ` O ) A. x e. ( Base ` O ) ( ( y ( .r ` O ) x ) = ( 0g ` O ) -> ( y = ( 0g ` O ) \/ x = ( 0g ` O ) ) ) ) )
30	25, 26, 29	3bitr4g 223	1 |- ( R e. V -> ( R e. Domn <-> O e. Domn ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   = wceq 1395   e. wcel 2200   A.wral 2508   _Vcvv 2799   ` cfv 5314  (class class class)co 5994   Basecbs 13018   .rcmulr 13097   0gc0g 13275  opp_rcoppr 14016  NzRingcnzr 14128  Domncdomn 14205

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-addcom 8087  ax-addass 8089  ax-i2m1 8092  ax-0lt1 8093  ax-0id 8095  ax-rnegex 8096  ax-pre-ltirr 8099  ax-pre-lttrn 8101  ax-pre-ltadd 8103

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-tpos 6381  df-pnf 8171  df-mnf 8172  df-ltxr 8174  df-inn 9099  df-2 9157  df-3 9158  df-ndx 13021  df-slot 13022  df-base 13024  df-sets 13025  df-plusg 13109  df-mulr 13110  df-0g 13277  df-mgm 13375  df-sgrp 13421  df-mnd 13436  df-grp 13522  df-mgp 13870  df-ur 13909  df-ring 13947  df-oppr 14017  df-nzr 14129  df-domn 14208

This theorem is referenced by:  opprdomn  14224