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Theorem oppgval 15102
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 20 . . . . 5  |-  ( x  =  R  ->  x  =  R )
3 fveq2 5691 . . . . . . . 8  |-  ( x  =  R  ->  ( +g  `  x )  =  ( +g  `  R
) )
4 oppgval.2 . . . . . . . 8  |-  .+  =  ( +g  `  R )
53, 4syl6eqr 2458 . . . . . . 7  |-  ( x  =  R  ->  ( +g  `  x )  = 
.+  )
65tposeqd 6445 . . . . . 6  |-  ( x  =  R  -> tpos  ( +g  `  x )  = tpos  .+  )
76opeq2d 3955 . . . . 5  |-  ( x  =  R  ->  <. ( +g  `  ndx ) , tpos  ( +g  `  x
) >.  =  <. ( +g  `  ndx ) , tpos  .+  >. )
82, 7oveq12d 6062 . . . 4  |-  ( x  =  R  ->  (
x sSet  <. ( +g  `  ndx ) , tpos  ( +g  `  x
) >. )  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
)
9 df-oppg 15101 . . . 4  |- oppg  =  ( x  e.  _V  |->  ( x sSet  <. ( +g  `  ndx ) , tpos  ( +g  `  x
) >. ) )
10 ovex 6069 . . . 4  |-  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )  e.  _V
118, 9, 10fvmpt 5769 . . 3  |-  ( R  e.  _V  ->  (oppg `  R
)  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
)
12 fvprc 5685 . . . 4  |-  ( -.  R  e.  _V  ->  (oppg `  R )  =  (/) )
13 reldmsets 13450 . . . . 5  |-  Rel  dom sSet
1413ovprc1 6072 . . . 4  |-  ( -.  R  e.  _V  ->  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )  =  (/) )
1512, 14eqtr4d 2443 . . 3  |-  ( -.  R  e.  _V  ->  (oppg `  R )  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
)
1611, 15pm2.61i 158 . 2  |-  (oppg `  R
)  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
171, 16eqtri 2428 1  |-  O  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1721   _Vcvv 2920   (/)c0 3592   <.cop 3781   ` cfv 5417  (class class class)co 6044  tpos ctpos 6441   ndxcnx 13425   sSet csts 13426   +g cplusg 13488  oppgcoppg 15100
This theorem is referenced by:  oppgplusfval  15103  oppglem  15105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-res 4853  df-iota 5381  df-fun 5419  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-tpos 6442  df-sets 13434  df-oppg 15101
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