Theorem opprdomnbg 14566

Description: A class is a domain if and only if its opposite is a domain, biconditional form of opprdomn 14567. (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 14475	. . 3 |- ( R e. V -> ( R e. NzRing <-> O e. NzRing ) )
3		eqid 2238	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
4	1, 3	opprbasg 14363	. . . . 5 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
5		vex 2824	. . . . . . . . . 10 |- y e. _V
6		vex 2824	. . . . . . . . . 10 |- x e. _V
7		eqid 2238	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
8		eqid 2238	. . . . . . . . . . 11 |- ( .r ` O ) = ( .r ` O )
9	3, 7, 1, 8	opprmulg 14359	. . . . . . . . . 10 |- ( ( R e. V /\ y e. _V /\ x e. _V ) -> ( y ( .r ` O ) x ) = ( x ( .r ` R ) y ) )
10	5, 6, 9	mp3an23 1370	. . . . . . . . 9 |- ( R e. V -> ( y ( .r ` O ) x ) = ( x ( .r ` R ) y ) )
11	10	eqcomd 2244	. . . . . . . 8 |- ( R e. V -> ( x ( .r ` R ) y ) = ( y ( .r ` O ) x ) )
12		eqid 2238	. . . . . . . . 9 |- ( 0g ` R ) = ( 0g ` R )
13	1, 12	oppr0g 14370	. . . . . . . 8 |- ( R e. V -> ( 0g ` R ) = ( 0g ` O ) )
14	11, 13	eqeq12d 2253	. . . . . . 7 |- ( R e. V -> ( ( x ( .r ` R ) y ) = ( 0g ` R ) <-> ( y ( .r ` O ) x ) = ( 0g ` O ) ) )
15	13	eqeq2d 2250	. . . . . . . . 9 |- ( R e. V -> ( x = ( 0g ` R ) <-> x = ( 0g ` O ) ) )
16	13	eqeq2d 2250	. . . . . . . . 9 |- ( R e. V -> ( y = ( 0g ` R ) <-> y = ( 0g ` O ) ) )
17	15, 16	orbi12d 805	. . . . . . . 8 |- ( R e. V -> ( ( x = ( 0g ` R ) \/ y = ( 0g ` R ) ) <-> ( x = ( 0g ` O ) \/ y = ( 0g ` O ) ) ) )
18		orcom 740	. . . . . . . 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 2765	. . . . 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 2765	. . . 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 2714	. . . 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 477	. 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 14561	. 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 2238	. . 3 |- ( Base ` O ) = ( Base ` O )
28		eqid 2238	. . 3 |- ( 0g ` O ) = ( 0g ` O )
29	27, 8, 28	isdomn 14561	. 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 720   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   ` cfv 5375  (class class class)co 6079   Basecbs 13335   .rcmulr 13415   0gc0g 13593  opp_rcoppr 14355  NzRingcnzr 14469  Domncdomn 14547

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-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-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-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-tpos 6510  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-plusg 13427  df-mulr 13428  df-0g 13595  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-grp 13791  df-mgp 14201  df-ur 14246  df-ring 14285  df-oppr 14356  df-nzr 14470  df-domn 14550

This theorem is referenced by:  opprdomn  14567