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Theorem opprval 15406
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 19 . . . . 5  |-  ( x  =  R  ->  x  =  R )
3 fveq2 5525 . . . . . . . 8  |-  ( x  =  R  ->  ( .r `  x )  =  ( .r `  R
) )
4 opprval.2 . . . . . . . 8  |-  .x.  =  ( .r `  R )
53, 4syl6eqr 2333 . . . . . . 7  |-  ( x  =  R  ->  ( .r `  x )  = 
.x.  )
6 tposeq 6236 . . . . . . 7  |-  ( ( .r `  x )  =  .x.  -> tpos  ( .r
`  x )  = tpos  .x.  )
75, 6syl 15 . . . . . 6  |-  ( x  =  R  -> tpos  ( .r
`  x )  = tpos  .x.  )
87opeq2d 3803 . . . . 5  |-  ( x  =  R  ->  <. ( .r `  ndx ) , tpos  ( .r `  x
) >.  =  <. ( .r `  ndx ) , tpos  .x.  >. )
92, 8oveq12d 5876 . . . 4  |-  ( x  =  R  ->  (
x sSet  <. ( .r `  ndx ) , tpos  ( .r
`  x ) >.
)  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
)
10 df-oppr 15405 . . . 4  |- oppr  =  ( x  e.  _V  |->  ( x sSet  <. ( .r `  ndx ) , tpos  ( .r `  x
) >. ) )
11 ovex 5883 . . . 4  |-  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )  e.  _V
129, 10, 11fvmpt 5602 . . 3  |-  ( R  e.  _V  ->  (oppr `  R
)  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
)
13 fvprc 5519 . . . 4  |-  ( -.  R  e.  _V  ->  (oppr `  R )  =  (/) )
14 reldmsets 13170 . . . . 5  |-  Rel  dom sSet
1514ovprc1 5886 . . . 4  |-  ( -.  R  e.  _V  ->  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )  =  (/) )
1613, 15eqtr4d 2318 . . 3  |-  ( -.  R  e.  _V  ->  (oppr `  R )  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
)
1712, 16pm2.61i 156 . 2  |-  (oppr `  R
)  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
181, 17eqtri 2303 1  |-  O  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   <.cop 3643   ` cfv 5255  (class class class)co 5858  tpos ctpos 6233   ndxcnx 13145   sSet csts 13146   Basecbs 13148   .rcmulr 13209  opprcoppr 15404
This theorem is referenced by:  opprmulfval  15407  opprlem  15410
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-tpos 6234  df-sets 13154  df-oppr 15405
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