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Theorem opprval 15422
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 5541 . . . . . . . 8  |-  ( x  =  R  ->  ( .r `  x )  =  ( .r `  R
) )
4 opprval.2 . . . . . . . 8  |-  .x.  =  ( .r `  R )
53, 4syl6eqr 2346 . . . . . . 7  |-  ( x  =  R  ->  ( .r `  x )  = 
.x.  )
6 tposeq 6252 . . . . . . 7  |-  ( ( .r `  x )  =  .x.  -> tpos  ( .r
`  x )  = tpos  .x.  )
75, 6syl 15 . . . . . 6  |-  ( x  =  R  -> tpos  ( .r
`  x )  = tpos  .x.  )
87opeq2d 3819 . . . . 5  |-  ( x  =  R  ->  <. ( .r `  ndx ) , tpos  ( .r `  x
) >.  =  <. ( .r `  ndx ) , tpos  .x.  >. )
92, 8oveq12d 5892 . . . 4  |-  ( x  =  R  ->  (
x sSet  <. ( .r `  ndx ) , tpos  ( .r
`  x ) >.
)  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
)
10 df-oppr 15421 . . . 4  |- oppr  =  ( x  e.  _V  |->  ( x sSet  <. ( .r `  ndx ) , tpos  ( .r `  x
) >. ) )
11 ovex 5899 . . . 4  |-  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )  e.  _V
129, 10, 11fvmpt 5618 . . 3  |-  ( R  e.  _V  ->  (oppr `  R
)  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
)
13 fvprc 5535 . . . 4  |-  ( -.  R  e.  _V  ->  (oppr `  R )  =  (/) )
14 reldmsets 13186 . . . . 5  |-  Rel  dom sSet
1514ovprc1 5902 . . . 4  |-  ( -.  R  e.  _V  ->  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )  =  (/) )
1613, 15eqtr4d 2331 . . 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 2316 1  |-  O  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   <.cop 3656   ` cfv 5271  (class class class)co 5874  tpos ctpos 6249   ndxcnx 13161   sSet csts 13162   Basecbs 13164   .rcmulr 13225  opprcoppr 15420
This theorem is referenced by:  opprmulfval  15423  opprlem  15426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-tpos 6250  df-sets 13170  df-oppr 15421
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