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Theorem oppgval 14913
Description: Value of the opposite group. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
Hypotheses
Ref Expression
oppgval.2  |-  .+  =  ( +g  `  R )
oppgval.3  |-  O  =  (oppg
`  R )
Assertion
Ref Expression
oppgval  |-  O  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )

Proof of Theorem oppgval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 oppgval.3 . 2  |-  O  =  (oppg
`  R )
2 id 19 . . . . 5  |-  ( x  =  R  ->  x  =  R )
3 fveq2 5605 . . . . . . . 8  |-  ( x  =  R  ->  ( +g  `  x )  =  ( +g  `  R
) )
4 oppgval.2 . . . . . . . 8  |-  .+  =  ( +g  `  R )
53, 4syl6eqr 2408 . . . . . . 7  |-  ( x  =  R  ->  ( +g  `  x )  = 
.+  )
6 tposeq 6320 . . . . . . 7  |-  ( ( +g  `  x )  =  .+  -> tpos  ( +g  `  x )  = tpos  .+  )
75, 6syl 15 . . . . . 6  |-  ( x  =  R  -> tpos  ( +g  `  x )  = tpos  .+  )
87opeq2d 3882 . . . . 5  |-  ( x  =  R  ->  <. ( +g  `  ndx ) , tpos  ( +g  `  x
) >.  =  <. ( +g  `  ndx ) , tpos  .+  >. )
92, 8oveq12d 5960 . . . 4  |-  ( x  =  R  ->  (
x sSet  <. ( +g  `  ndx ) , tpos  ( +g  `  x
) >. )  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
)
10 df-oppg 14912 . . . 4  |- oppg  =  ( x  e.  _V  |->  ( x sSet  <. ( +g  `  ndx ) , tpos  ( +g  `  x
) >. ) )
11 ovex 5967 . . . 4  |-  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )  e.  _V
129, 10, 11fvmpt 5682 . . 3  |-  ( R  e.  _V  ->  (oppg `  R
)  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
)
13 fvprc 5599 . . . 4  |-  ( -.  R  e.  _V  ->  (oppg `  R )  =  (/) )
14 reldmsets 13261 . . . . 5  |-  Rel  dom sSet
1514ovprc1 5970 . . . 4  |-  ( -.  R  e.  _V  ->  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )  =  (/) )
1613, 15eqtr4d 2393 . . 3  |-  ( -.  R  e.  _V  ->  (oppg `  R )  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
)
1712, 16pm2.61i 156 . 2  |-  (oppg `  R
)  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
181, 17eqtri 2378 1  |-  O  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1642    e. wcel 1710   _Vcvv 2864   (/)c0 3531   <.cop 3719   ` cfv 5334  (class class class)co 5942  tpos ctpos 6317   ndxcnx 13236   sSet csts 13237   +g cplusg 13299  oppgcoppg 14911
This theorem is referenced by:  oppgplusfval  14914  oppglem  14916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-res 4780  df-iota 5298  df-fun 5336  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-tpos 6318  df-sets 13245  df-oppg 14912
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