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Theorem oppgval 15148
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 21 . . . . 5  |-  ( x  =  R  ->  x  =  R )
3 fveq2 5731 . . . . . . . 8  |-  ( x  =  R  ->  ( +g  `  x )  =  ( +g  `  R
) )
4 oppgval.2 . . . . . . . 8  |-  .+  =  ( +g  `  R )
53, 4syl6eqr 2488 . . . . . . 7  |-  ( x  =  R  ->  ( +g  `  x )  = 
.+  )
65tposeqd 6485 . . . . . 6  |-  ( x  =  R  -> tpos  ( +g  `  x )  = tpos  .+  )
76opeq2d 3993 . . . . 5  |-  ( x  =  R  ->  <. ( +g  `  ndx ) , tpos  ( +g  `  x
) >.  =  <. ( +g  `  ndx ) , tpos  .+  >. )
82, 7oveq12d 6102 . . . 4  |-  ( x  =  R  ->  (
x sSet  <. ( +g  `  ndx ) , tpos  ( +g  `  x
) >. )  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
)
9 df-oppg 15147 . . . 4  |- oppg  =  ( x  e.  _V  |->  ( x sSet  <. ( +g  `  ndx ) , tpos  ( +g  `  x
) >. ) )
10 ovex 6109 . . . 4  |-  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )  e.  _V
118, 9, 10fvmpt 5809 . . 3  |-  ( R  e.  _V  ->  (oppg `  R
)  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
)
12 fvprc 5725 . . . 4  |-  ( -.  R  e.  _V  ->  (oppg `  R )  =  (/) )
13 reldmsets 13496 . . . . 5  |-  Rel  dom sSet
1413ovprc1 6112 . . . 4  |-  ( -.  R  e.  _V  ->  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )  =  (/) )
1512, 14eqtr4d 2473 . . 3  |-  ( -.  R  e.  _V  ->  (oppg `  R )  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
)
1611, 15pm2.61i 159 . 2  |-  (oppg `  R
)  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
171, 16eqtri 2458 1  |-  O  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1653    e. wcel 1726   _Vcvv 2958   (/)c0 3630   <.cop 3819   ` cfv 5457  (class class class)co 6084  tpos ctpos 6481   ndxcnx 13471   sSet csts 13472   +g cplusg 13534  oppgcoppg 15146
This theorem is referenced by:  oppgplusfval  15149  oppglem  15151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-res 4893  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-tpos 6482  df-sets 13480  df-oppg 15147
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