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Theorem opprdomn 16058
Description: The opposite of a domain is also a domain. (Contributed by Mario Carneiro, 15-Jun-2015.)
Hypothesis
Ref Expression
opprdomn.1  |-  O  =  (oppr
`  R )
Assertion
Ref Expression
opprdomn  |-  ( R  e. Domn  ->  O  e. Domn )

Proof of Theorem opprdomn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 domnnzr 16052 . . 3  |-  ( R  e. Domn  ->  R  e. NzRing )
2 opprdomn.1 . . . 4  |-  O  =  (oppr
`  R )
32opprnzr 16032 . . 3  |-  ( R  e. NzRing  ->  O  e. NzRing )
41, 3syl 15 . 2  |-  ( R  e. Domn  ->  O  e. NzRing )
5 eqid 2296 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
6 eqid 2296 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
7 eqid 2296 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
85, 6, 7domneq0 16054 . . . . . . 7  |-  ( ( R  e. Domn  /\  y  e.  ( Base `  R
)  /\  x  e.  ( Base `  R )
)  ->  ( (
y ( .r `  R ) x )  =  ( 0g `  R )  <->  ( y  =  ( 0g `  R )  \/  x  =  ( 0g `  R ) ) ) )
983com23 1157 . . . . . 6  |-  ( ( R  e. Domn  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( (
y ( .r `  R ) x )  =  ( 0g `  R )  <->  ( y  =  ( 0g `  R )  \/  x  =  ( 0g `  R ) ) ) )
10 eqid 2296 . . . . . . . 8  |-  ( .r
`  O )  =  ( .r `  O
)
115, 6, 2, 10opprmul 15424 . . . . . . 7  |-  ( x ( .r `  O
) y )  =  ( y ( .r
`  R ) x )
1211eqeq1i 2303 . . . . . 6  |-  ( ( x ( .r `  O ) y )  =  ( 0g `  R )  <->  ( y
( .r `  R
) x )  =  ( 0g `  R
) )
13 orcom 376 . . . . . 6  |-  ( ( x  =  ( 0g
`  R )  \/  y  =  ( 0g
`  R ) )  <-> 
( y  =  ( 0g `  R )  \/  x  =  ( 0g `  R ) ) )
149, 12, 133bitr4g 279 . . . . 5  |-  ( ( R  e. Domn  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( (
x ( .r `  O ) y )  =  ( 0g `  R )  <->  ( x  =  ( 0g `  R )  \/  y  =  ( 0g `  R ) ) ) )
1514biimpd 198 . . . 4  |-  ( ( R  e. Domn  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( (
x ( .r `  O ) y )  =  ( 0g `  R )  ->  (
x  =  ( 0g
`  R )  \/  y  =  ( 0g
`  R ) ) ) )
16153expb 1152 . . 3  |-  ( ( R  e. Domn  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) ) )  -> 
( ( x ( .r `  O ) y )  =  ( 0g `  R )  ->  ( x  =  ( 0g `  R
)  \/  y  =  ( 0g `  R
) ) ) )
1716ralrimivva 2648 . 2  |-  ( R  e. Domn  ->  A. x  e.  (
Base `  R ) A. y  e.  ( Base `  R ) ( ( x ( .r
`  O ) y )  =  ( 0g
`  R )  -> 
( x  =  ( 0g `  R )  \/  y  =  ( 0g `  R ) ) ) )
182, 5opprbas 15427 . . 3  |-  ( Base `  R )  =  (
Base `  O )
192, 7oppr0 15431 . . 3  |-  ( 0g
`  R )  =  ( 0g `  O
)
2018, 10, 19isdomn 16051 . 2  |-  ( O  e. Domn 
<->  ( O  e. NzRing  /\  A. x  e.  ( Base `  R ) A. y  e.  ( Base `  R
) ( ( x ( .r `  O
) y )  =  ( 0g `  R
)  ->  ( x  =  ( 0g `  R )  \/  y  =  ( 0g `  R ) ) ) ) )
214, 17, 20sylanbrc 645 1  |-  ( R  e. Domn  ->  O  e. Domn )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   ` cfv 5271  (class class class)co 5874   Basecbs 13164   .rcmulr 13225   0gc0g 13416  opprcoppr 15420  NzRingcnzr 16025  Domncdomn 16037
This theorem is referenced by:  fidomndrng  16064
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-nzr 16026  df-domn 16041
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