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Theorem opprdomn 17506
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 17500 . . 3  |-  ( R  e. Domn  ->  R  e. NzRing )
2 opprdomn.1 . . . 4  |-  O  =  (oppr
`  R )
32opprnzr 17479 . . 3  |-  ( R  e. NzRing  ->  O  e. NzRing )
41, 3syl 16 . 2  |-  ( R  e. Domn  ->  O  e. NzRing )
5 eqid 2454 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
6 eqid 2454 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
7 eqid 2454 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
85, 6, 7domneq0 17502 . . . . . . 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 1194 . . . . . 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 2454 . . . . . . . 8  |-  ( .r
`  O )  =  ( .r `  O
)
115, 6, 2, 10opprmul 16851 . . . . . . 7  |-  ( x ( .r `  O
) y )  =  ( y ( .r
`  R ) x )
1211eqeq1i 2461 . . . . . 6  |-  ( ( x ( .r `  O ) y )  =  ( 0g `  R )  <->  ( y
( .r `  R
) x )  =  ( 0g `  R
) )
13 orcom 387 . . . . . 6  |-  ( ( x  =  ( 0g
`  R )  \/  y  =  ( 0g
`  R ) )  <-> 
( y  =  ( 0g `  R )  \/  x  =  ( 0g `  R ) ) )
149, 12, 133bitr4g 288 . . . . 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 207 . . . 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 1189 . . 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 2914 . 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 16854 . . 3  |-  ( Base `  R )  =  (
Base `  O )
192, 7oppr0 16858 . . 3  |-  ( 0g
`  R )  =  ( 0g `  O
)
2018, 10, 19isdomn 17499 . 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 664 1  |-  ( R  e. Domn  ->  O  e. Domn )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   ` cfv 5529  (class class class)co 6203   Basecbs 14296   .rcmulr 14362   0gc0g 14501  opprcoppr 16847  NzRingcnzr 17472  Domncdomn 17484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-tpos 6858  df-recs 6945  df-rdg 6979  df-1o 7033  df-2o 7034  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-3 10496  df-ndx 14299  df-slot 14300  df-base 14301  df-sets 14302  df-plusg 14374  df-mulr 14375  df-0g 14503  df-mnd 15538  df-grp 15668  df-minusg 15669  df-mgp 16724  df-ur 16736  df-rng 16780  df-oppr 16848  df-nzr 17473  df-domn 17488
This theorem is referenced by:  fidomndrng  17512
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