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Theorem opprdomnbg 13754
Description: A class is a domain if and only if its opposite is a domain, biconditional form of opprdomn 13755. (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.
StepHypRef Expression
1 opprdomn.1 . . . 4  |-  O  =  (oppr
`  R )
21opprnzrbg 13665 . . 3  |-  ( R  e.  V  ->  ( R  e. NzRing  <->  O  e. NzRing ) )
3 eqid 2193 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
41, 3opprbasg 13555 . . . . 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
)
93, 7, 1, 8opprmulg 13551 . . . . . . . . . 10  |-  ( ( R  e.  V  /\  y  e.  _V  /\  x  e.  _V )  ->  (
y ( .r `  O ) x )  =  ( x ( .r `  R ) y ) )
105, 6, 9mp3an23 1340 . . . . . . . . 9  |-  ( R  e.  V  ->  (
y ( .r `  O ) x )  =  ( x ( .r `  R ) y ) )
1110eqcomd 2199 . . . . . . . 8  |-  ( R  e.  V  ->  (
x ( .r `  R ) y )  =  ( y ( .r `  O ) x ) )
12 eqid 2193 . . . . . . . . 9  |-  ( 0g
`  R )  =  ( 0g `  R
)
131, 12oppr0g 13561 . . . . . . . 8  |-  ( R  e.  V  ->  ( 0g `  R )  =  ( 0g `  O
) )
1411, 13eqeq12d 2208 . . . . . . 7  |-  ( R  e.  V  ->  (
( x ( .r
`  R ) y )  =  ( 0g
`  R )  <->  ( y
( .r `  O
) x )  =  ( 0g `  O
) ) )
1513eqeq2d 2205 . . . . . . . . 9  |-  ( R  e.  V  ->  (
x  =  ( 0g
`  R )  <->  x  =  ( 0g `  O ) ) )
1613eqeq2d 2205 . . . . . . . . 9  |-  ( R  e.  V  ->  (
y  =  ( 0g
`  R )  <->  y  =  ( 0g `  O ) ) )
1715, 16orbi12d 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 ) ) )
1917, 18bitrdi 196 . . . . . . 7  |-  ( R  e.  V  ->  (
( x  =  ( 0g `  R )  \/  y  =  ( 0g `  R ) )  <->  ( y  =  ( 0g `  O
)  \/  x  =  ( 0g `  O
) ) ) )
2014, 19imbi12d 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 ) ) ) ) )
214, 20raleqbidv 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 ) ) ) ) )
224, 21raleqbidv 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 ) ) ) )
2422, 23bitrdi 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 ) ) ) ) )
252, 24anbi12d 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 ) ) ) ) ) )
263, 7, 12isdomn 13749 . 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
)
2927, 8, 28isdomn 13749 . 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 ) ) ) ) )
3025, 26, 293bitr4g 223 1  |-  ( R  e.  V  ->  ( R  e. Domn  <->  O  e. Domn ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709    = wceq 1364    e. wcel 2164   A.wral 2472   _Vcvv 2760   ` cfv 5246  (class class class)co 5910   Basecbs 12608   .rcmulr 12686   0gc0g 12857  opprcoppr 13547  NzRingcnzr 13659  Domncdomn 13736
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 4462  ax-setind 4565  ax-cnex 7953  ax-resscn 7954  ax-1cn 7955  ax-1re 7956  ax-icn 7957  ax-addcl 7958  ax-addrcl 7959  ax-mulcl 7960  ax-addcom 7962  ax-addass 7964  ax-i2m1 7967  ax-0lt1 7968  ax-0id 7970  ax-rnegex 7971  ax-pre-ltirr 7974  ax-pre-lttrn 7976  ax-pre-ltadd 7978
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 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-iota 5207  df-fun 5248  df-fn 5249  df-fv 5254  df-riota 5865  df-ov 5913  df-oprab 5914  df-mpo 5915  df-tpos 6289  df-pnf 8046  df-mnf 8047  df-ltxr 8049  df-inn 8973  df-2 9031  df-3 9032  df-ndx 12611  df-slot 12612  df-base 12614  df-sets 12615  df-plusg 12698  df-mulr 12699  df-0g 12859  df-mgm 12929  df-sgrp 12975  df-mnd 12988  df-grp 13065  df-mgp 13401  df-ur 13440  df-ring 13478  df-oppr 13548  df-nzr 13660  df-domn 13739
This theorem is referenced by:  opprdomn  13755
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