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Theorem opprval 16714
Description: Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
Hypotheses
Ref Expression
opprval.1  |-  B  =  ( Base `  R
)
opprval.2  |-  .x.  =  ( .r `  R )
opprval.3  |-  O  =  (oppr
`  R )
Assertion
Ref Expression
opprval  |-  O  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )

Proof of Theorem opprval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 opprval.3 . 2  |-  O  =  (oppr
`  R )
2 id 22 . . . . 5  |-  ( x  =  R  ->  x  =  R )
3 fveq2 5689 . . . . . . . 8  |-  ( x  =  R  ->  ( .r `  x )  =  ( .r `  R
) )
4 opprval.2 . . . . . . . 8  |-  .x.  =  ( .r `  R )
53, 4syl6eqr 2491 . . . . . . 7  |-  ( x  =  R  ->  ( .r `  x )  = 
.x.  )
65tposeqd 6746 . . . . . 6  |-  ( x  =  R  -> tpos  ( .r
`  x )  = tpos  .x.  )
76opeq2d 4064 . . . . 5  |-  ( x  =  R  ->  <. ( .r `  ndx ) , tpos  ( .r `  x
) >.  =  <. ( .r `  ndx ) , tpos  .x.  >. )
82, 7oveq12d 6107 . . . 4  |-  ( x  =  R  ->  (
x sSet  <. ( .r `  ndx ) , tpos  ( .r
`  x ) >.
)  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
)
9 df-oppr 16713 . . . 4  |- oppr  =  ( x  e.  _V  |->  ( x sSet  <. ( .r `  ndx ) , tpos  ( .r `  x
) >. ) )
10 ovex 6114 . . . 4  |-  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )  e.  _V
118, 9, 10fvmpt 5772 . . 3  |-  ( R  e.  _V  ->  (oppr `  R
)  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
)
12 fvprc 5683 . . . 4  |-  ( -.  R  e.  _V  ->  (oppr `  R )  =  (/) )
13 reldmsets 14194 . . . . 5  |-  Rel  dom sSet
1413ovprc1 6117 . . . 4  |-  ( -.  R  e.  _V  ->  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )  =  (/) )
1512, 14eqtr4d 2476 . . 3  |-  ( -.  R  e.  _V  ->  (oppr `  R )  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
)
1611, 15pm2.61i 164 . 2  |-  (oppr `  R
)  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
171, 16eqtri 2461 1  |-  O  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1369    e. wcel 1756   _Vcvv 2970   (/)c0 3635   <.cop 3881   ` cfv 5416  (class class class)co 6089  tpos ctpos 6742   ndxcnx 14169   sSet csts 14170   Basecbs 14172   .rcmulr 14237  opprcoppr 16712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-res 4850  df-iota 5379  df-fun 5418  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-tpos 6743  df-sets 14178  df-oppr 16713
This theorem is referenced by:  opprmulfval  16715  opprlem  16718
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