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Theorem opprval 17399
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 5872 . . . . . . . 8  |-  ( x  =  R  ->  ( .r `  x )  =  ( .r `  R
) )
4 opprval.2 . . . . . . . 8  |-  .x.  =  ( .r `  R )
53, 4syl6eqr 2516 . . . . . . 7  |-  ( x  =  R  ->  ( .r `  x )  = 
.x.  )
65tposeqd 6976 . . . . . 6  |-  ( x  =  R  -> tpos  ( .r
`  x )  = tpos  .x.  )
76opeq2d 4226 . . . . 5  |-  ( x  =  R  ->  <. ( .r `  ndx ) , tpos  ( .r `  x
) >.  =  <. ( .r `  ndx ) , tpos  .x.  >. )
82, 7oveq12d 6314 . . . 4  |-  ( x  =  R  ->  (
x sSet  <. ( .r `  ndx ) , tpos  ( .r
`  x ) >.
)  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
)
9 df-oppr 17398 . . . 4  |- oppr  =  ( x  e.  _V  |->  ( x sSet  <. ( .r `  ndx ) , tpos  ( .r `  x
) >. ) )
10 ovex 6324 . . . 4  |-  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )  e.  _V
118, 9, 10fvmpt 5956 . . 3  |-  ( R  e.  _V  ->  (oppr `  R
)  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
)
12 fvprc 5866 . . . 4  |-  ( -.  R  e.  _V  ->  (oppr `  R )  =  (/) )
13 reldmsets 14666 . . . . 5  |-  Rel  dom sSet
1413ovprc1 6327 . . . 4  |-  ( -.  R  e.  _V  ->  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )  =  (/) )
1512, 14eqtr4d 2501 . . 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 2486 1  |-  O  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1395    e. wcel 1819   _Vcvv 3109   (/)c0 3793   <.cop 4038   ` cfv 5594  (class class class)co 6296  tpos ctpos 6972   ndxcnx 14640   sSet csts 14641   Basecbs 14643   .rcmulr 14712  opprcoppr 17397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-res 5020  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-tpos 6973  df-sets 14649  df-oppr 17398
This theorem is referenced by:  opprmulfval  17400  opprlem  17403
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