Theorem opprdomnbg 13770

Description: A class is a domain if and only if its opposite is a domain, biconditional form of opprdomn 13771. (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 13681	. . 3 |- ( R e. V -> ( R e. NzRing <-> O e. NzRing ) )
3		eqid 2193	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
4	1, 3	opprbasg 13571	. . . . 5 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
5		vex 2763	. . . . . . . . . 10 |- y e. _V
6		vex 2763	. . . . . . . . . 10 |- x e. _V
7		eqid 2193	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
8		eqid 2193	. . . . . . . . . . 11 |- ( .r ` O ) = ( .r ` O )
9	3, 7, 1, 8	opprmulg 13567	. . . . . . . . . 10 |- ( ( R e. V /\ y e. _V /\ x e. _V ) -> ( y ( .r ` O ) x ) = ( x ( .r ` R ) y ) )
10	5, 6, 9	mp3an23 1340	. . . . . . . . 9 |- ( R e. V -> ( y ( .r ` O ) x ) = ( x ( .r ` R ) y ) )
11	10	eqcomd 2199	. . . . . . . 8 |- ( R e. V -> ( x ( .r ` R ) y ) = ( y ( .r ` O ) x ) )
12		eqid 2193	. . . . . . . . 9 |- ( 0g ` R ) = ( 0g ` R )
13	1, 12	oppr0g 13577	. . . . . . . 8 |- ( R e. V -> ( 0g ` R ) = ( 0g ` O ) )
14	11, 13	eqeq12d 2208	. . . . . . 7 |- ( R e. V -> ( ( x ( .r ` R ) y ) = ( 0g ` R ) <-> ( y ( .r ` O ) x ) = ( 0g ` O ) ) )
15	13	eqeq2d 2205	. . . . . . . . 9 |- ( R e. V -> ( x = ( 0g ` R ) <-> x = ( 0g ` O ) ) )
16	13	eqeq2d 2205	. . . . . . . . 9 |- ( R e. V -> ( y = ( 0g ` R ) <-> y = ( 0g ` O ) ) )
17	15, 16	orbi12d 794	. . . . . . . 8 |- ( R e. V -> ( ( x = ( 0g ` R ) \/ y = ( 0g ` R ) ) <-> ( x = ( 0g ` O ) \/ y = ( 0g ` O ) ) ) )
18		orcom 729	. . . . . . . 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 2706	. . . . 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 2706	. . . 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 2657	. . . 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 13765	. 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 2193	. . 3 |- ( Base ` O ) = ( Base ` O )
28		eqid 2193	. . 3 |- ( 0g ` O ) = ( 0g ` O )
29	27, 8, 28	isdomn 13765	. 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 709   = wceq 1364   e. wcel 2164   A.wral 2472   _Vcvv 2760   ` cfv 5254  (class class class)co 5918   Basecbs 12618   .rcmulr 12696   0gc0g 12867  opp_rcoppr 13563  NzRingcnzr 13675  Domncdomn 13752

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-tpos 6298  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-mgp 13417  df-ur 13456  df-ring 13494  df-oppr 13564  df-nzr 13676  df-domn 13755

This theorem is referenced by:  opprdomn  13771