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Theorem oppgval 17076
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 22 . . . . 5  |-  ( x  =  R  ->  x  =  R )
3 fveq2 5879 . . . . . . . 8  |-  ( x  =  R  ->  ( +g  `  x )  =  ( +g  `  R
) )
4 oppgval.2 . . . . . . . 8  |-  .+  =  ( +g  `  R )
53, 4syl6eqr 2523 . . . . . . 7  |-  ( x  =  R  ->  ( +g  `  x )  = 
.+  )
65tposeqd 6994 . . . . . 6  |-  ( x  =  R  -> tpos  ( +g  `  x )  = tpos  .+  )
76opeq2d 4165 . . . . 5  |-  ( x  =  R  ->  <. ( +g  `  ndx ) , tpos  ( +g  `  x
) >.  =  <. ( +g  `  ndx ) , tpos  .+  >. )
82, 7oveq12d 6326 . . . 4  |-  ( x  =  R  ->  (
x sSet  <. ( +g  `  ndx ) , tpos  ( +g  `  x
) >. )  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
)
9 df-oppg 17075 . . . 4  |- oppg  =  ( x  e.  _V  |->  ( x sSet  <. ( +g  `  ndx ) , tpos  ( +g  `  x
) >. ) )
10 ovex 6336 . . . 4  |-  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )  e.  _V
118, 9, 10fvmpt 5963 . . 3  |-  ( R  e.  _V  ->  (oppg `  R
)  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
)
12 fvprc 5873 . . . 4  |-  ( -.  R  e.  _V  ->  (oppg `  R )  =  (/) )
13 reldmsets 15222 . . . . 5  |-  Rel  dom sSet
1413ovprc1 6339 . . . 4  |-  ( -.  R  e.  _V  ->  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )  =  (/) )
1512, 14eqtr4d 2508 . . 3  |-  ( -.  R  e.  _V  ->  (oppg `  R )  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
)
1611, 15pm2.61i 169 . 2  |-  (oppg `  R
)  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
171, 16eqtri 2493 1  |-  O  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1452    e. wcel 1904   _Vcvv 3031   (/)c0 3722   <.cop 3965   ` cfv 5589  (class class class)co 6308  tpos ctpos 6990   ndxcnx 15196   sSet csts 15197   +g cplusg 15268  oppgcoppg 17074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-res 4851  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-tpos 6991  df-sets 15205  df-oppg 17075
This theorem is referenced by:  oppgplusfval  17077  oppglem  17079
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