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Theorem oppgval 15862
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 5691 . . . . . . . 8  |-  ( x  =  R  ->  ( +g  `  x )  =  ( +g  `  R
) )
4 oppgval.2 . . . . . . . 8  |-  .+  =  ( +g  `  R )
53, 4syl6eqr 2493 . . . . . . 7  |-  ( x  =  R  ->  ( +g  `  x )  = 
.+  )
65tposeqd 6748 . . . . . 6  |-  ( x  =  R  -> tpos  ( +g  `  x )  = tpos  .+  )
76opeq2d 4066 . . . . 5  |-  ( x  =  R  ->  <. ( +g  `  ndx ) , tpos  ( +g  `  x
) >.  =  <. ( +g  `  ndx ) , tpos  .+  >. )
82, 7oveq12d 6109 . . . 4  |-  ( x  =  R  ->  (
x sSet  <. ( +g  `  ndx ) , tpos  ( +g  `  x
) >. )  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
)
9 df-oppg 15861 . . . 4  |- oppg  =  ( x  e.  _V  |->  ( x sSet  <. ( +g  `  ndx ) , tpos  ( +g  `  x
) >. ) )
10 ovex 6116 . . . 4  |-  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )  e.  _V
118, 9, 10fvmpt 5774 . . 3  |-  ( R  e.  _V  ->  (oppg `  R
)  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
)
12 fvprc 5685 . . . 4  |-  ( -.  R  e.  _V  ->  (oppg `  R )  =  (/) )
13 reldmsets 14196 . . . . 5  |-  Rel  dom sSet
1413ovprc1 6119 . . . 4  |-  ( -.  R  e.  _V  ->  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )  =  (/) )
1512, 14eqtr4d 2478 . . 3  |-  ( -.  R  e.  _V  ->  (oppg `  R )  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
)
1611, 15pm2.61i 164 . 2  |-  (oppg `  R
)  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
171, 16eqtri 2463 1  |-  O  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1369    e. wcel 1756   _Vcvv 2972   (/)c0 3637   <.cop 3883   ` cfv 5418  (class class class)co 6091  tpos ctpos 6744   ndxcnx 14171   sSet csts 14172   +g cplusg 14238  oppgcoppg 15860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-res 4852  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-tpos 6745  df-sets 14180  df-oppg 15861
This theorem is referenced by:  oppgplusfval  15863  oppglem  15865
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