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Theorem opprdomn 16290
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 16284 . . 3  |-  ( R  e. Domn  ->  R  e. NzRing )
2 opprdomn.1 . . . 4  |-  O  =  (oppr
`  R )
32opprnzr 16264 . . 3  |-  ( R  e. NzRing  ->  O  e. NzRing )
41, 3syl 16 . 2  |-  ( R  e. Domn  ->  O  e. NzRing )
5 eqid 2389 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
6 eqid 2389 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
7 eqid 2389 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
85, 6, 7domneq0 16286 . . . . . . 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 1159 . . . . . 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 2389 . . . . . . . 8  |-  ( .r
`  O )  =  ( .r `  O
)
115, 6, 2, 10opprmul 15660 . . . . . . 7  |-  ( x ( .r `  O
) y )  =  ( y ( .r
`  R ) x )
1211eqeq1i 2396 . . . . . 6  |-  ( ( x ( .r `  O ) y )  =  ( 0g `  R )  <->  ( y
( .r `  R
) x )  =  ( 0g `  R
) )
13 orcom 377 . . . . . 6  |-  ( ( x  =  ( 0g
`  R )  \/  y  =  ( 0g
`  R ) )  <-> 
( y  =  ( 0g `  R )  \/  x  =  ( 0g `  R ) ) )
149, 12, 133bitr4g 280 . . . . 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 199 . . . 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 1154 . . 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 2743 . 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 15663 . . 3  |-  ( Base `  R )  =  (
Base `  O )
192, 7oppr0 15667 . . 3  |-  ( 0g
`  R )  =  ( 0g `  O
)
2018, 10, 19isdomn 16283 . 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 646 1  |-  ( R  e. Domn  ->  O  e. Domn )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2651   ` cfv 5396  (class class class)co 6022   Basecbs 13398   .rcmulr 13459   0gc0g 13652  opprcoppr 15656  NzRingcnzr 16257  Domncdomn 16269
This theorem is referenced by:  fidomndrng  16296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-tpos 6417  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-2o 6663  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-3 9993  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-plusg 13471  df-mulr 13472  df-0g 13656  df-mnd 14619  df-grp 14741  df-minusg 14742  df-mgp 15578  df-rng 15592  df-ur 15594  df-oppr 15657  df-nzr 16258  df-domn 16273
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