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Theorem mgpval 15328
Description: Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.)
Hypotheses
Ref Expression
mgpval.1  |-  M  =  (mulGrp `  R )
mgpval.2  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
mgpval  |-  M  =  ( R sSet  <. ( +g  `  ndx ) , 
.x.  >. )

Proof of Theorem mgpval
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 mgpval.1 . 2  |-  M  =  (mulGrp `  R )
2 id 19 . . . . 5  |-  ( r  =  R  ->  r  =  R )
3 fveq2 5525 . . . . . . 7  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
4 mgpval.2 . . . . . . 7  |-  .x.  =  ( .r `  R )
53, 4syl6eqr 2333 . . . . . 6  |-  ( r  =  R  ->  ( .r `  r )  = 
.x.  )
65opeq2d 3803 . . . . 5  |-  ( r  =  R  ->  <. ( +g  `  ndx ) ,  ( .r `  r
) >.  =  <. ( +g  `  ndx ) , 
.x.  >. )
72, 6oveq12d 5876 . . . 4  |-  ( r  =  R  ->  (
r sSet  <. ( +g  `  ndx ) ,  ( .r `  r ) >. )  =  ( R sSet  <. ( +g  `  ndx ) ,  .x.  >. ) )
8 df-mgp 15326 . . . 4  |- mulGrp  =  ( r  e.  _V  |->  ( r sSet  <. ( +g  `  ndx ) ,  ( .r `  r ) >. )
)
9 ovex 5883 . . . 4  |-  ( R sSet  <. ( +g  `  ndx ) ,  .x.  >. )  e.  _V
107, 8, 9fvmpt 5602 . . 3  |-  ( R  e.  _V  ->  (mulGrp `  R )  =  ( R sSet  <. ( +g  `  ndx ) ,  .x.  >. )
)
11 fvprc 5519 . . . 4  |-  ( -.  R  e.  _V  ->  (mulGrp `  R )  =  (/) )
12 reldmsets 13170 . . . . 5  |-  Rel  dom sSet
1312ovprc1 5886 . . . 4  |-  ( -.  R  e.  _V  ->  ( R sSet  <. ( +g  `  ndx ) ,  .x.  >. )  =  (/) )
1411, 13eqtr4d 2318 . . 3  |-  ( -.  R  e.  _V  ->  (mulGrp `  R )  =  ( R sSet  <. ( +g  `  ndx ) ,  .x.  >. )
)
1510, 14pm2.61i 156 . 2  |-  (mulGrp `  R )  =  ( R sSet  <. ( +g  `  ndx ) ,  .x.  >. )
161, 15eqtri 2303 1  |-  M  =  ( R sSet  <. ( +g  `  ndx ) , 
.x.  >. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   <.cop 3643   ` cfv 5255  (class class class)co 5858   ndxcnx 13145   sSet csts 13146   +g cplusg 13208   .rcmulr 13209  mulGrpcmgp 15325
This theorem is referenced by:  mgpplusg  15329  mgplem  15330  mgpress  15336
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-sets 13154  df-mgp 15326
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