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Theorem opprval 15729
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 20 . . . . 5  |-  ( x  =  R  ->  x  =  R )
3 fveq2 5728 . . . . . . . 8  |-  ( x  =  R  ->  ( .r `  x )  =  ( .r `  R
) )
4 opprval.2 . . . . . . . 8  |-  .x.  =  ( .r `  R )
53, 4syl6eqr 2486 . . . . . . 7  |-  ( x  =  R  ->  ( .r `  x )  = 
.x.  )
65tposeqd 6482 . . . . . 6  |-  ( x  =  R  -> tpos  ( .r
`  x )  = tpos  .x.  )
76opeq2d 3991 . . . . 5  |-  ( x  =  R  ->  <. ( .r `  ndx ) , tpos  ( .r `  x
) >.  =  <. ( .r `  ndx ) , tpos  .x.  >. )
82, 7oveq12d 6099 . . . 4  |-  ( x  =  R  ->  (
x sSet  <. ( .r `  ndx ) , tpos  ( .r
`  x ) >.
)  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
)
9 df-oppr 15728 . . . 4  |- oppr  =  ( x  e.  _V  |->  ( x sSet  <. ( .r `  ndx ) , tpos  ( .r `  x
) >. ) )
10 ovex 6106 . . . 4  |-  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )  e.  _V
118, 9, 10fvmpt 5806 . . 3  |-  ( R  e.  _V  ->  (oppr `  R
)  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
)
12 fvprc 5722 . . . 4  |-  ( -.  R  e.  _V  ->  (oppr `  R )  =  (/) )
13 reldmsets 13491 . . . . 5  |-  Rel  dom sSet
1413ovprc1 6109 . . . 4  |-  ( -.  R  e.  _V  ->  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )  =  (/) )
1512, 14eqtr4d 2471 . . 3  |-  ( -.  R  e.  _V  ->  (oppr `  R )  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
)
1611, 15pm2.61i 158 . 2  |-  (oppr `  R
)  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
171, 16eqtri 2456 1  |-  O  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   _Vcvv 2956   (/)c0 3628   <.cop 3817   ` cfv 5454  (class class class)co 6081  tpos ctpos 6478   ndxcnx 13466   sSet csts 13467   Basecbs 13469   .rcmulr 13530  opprcoppr 15727
This theorem is referenced by:  opprmulfval  15730  opprlem  15733
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-res 4890  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-tpos 6479  df-sets 13475  df-oppr 15728
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