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Theorem opprdomn 18145
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 18139 . . 3  |-  ( R  e. Domn  ->  R  e. NzRing )
2 opprdomn.1 . . . 4  |-  O  =  (oppr
`  R )
32opprnzr 18108 . . 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 18141 . . . . . . 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 1200 . . . . . 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 17470 . . . . . . 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 385 . . . . . 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 1195 . . 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 2875 . 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 17473 . . 3  |-  ( Base `  R )  =  (
Base `  O )
192, 7oppr0 17477 . . 3  |-  ( 0g
`  R )  =  ( 0g `  O
)
2018, 10, 19isdomn 18138 . 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 662 1  |-  ( R  e. Domn  ->  O  e. Domn )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   ` cfv 5570  (class class class)co 6270   Basecbs 14716   .rcmulr 14785   0gc0g 14929  opprcoppr 17466  NzRingcnzr 18100  Domncdomn 18123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-tpos 6947  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-plusg 14797  df-mulr 14798  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-minusg 16257  df-mgp 17337  df-ur 17349  df-ring 17395  df-oppr 17467  df-nzr 18101  df-domn 18127
This theorem is referenced by:  fidomndrng  18151
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