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Theorem mgpval 15571
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 20 . . . . 5  |-  ( r  =  R  ->  r  =  R )
3 fveq2 5661 . . . . . . 7  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
4 mgpval.2 . . . . . . 7  |-  .x.  =  ( .r `  R )
53, 4syl6eqr 2430 . . . . . 6  |-  ( r  =  R  ->  ( .r `  r )  = 
.x.  )
65opeq2d 3926 . . . . 5  |-  ( r  =  R  ->  <. ( +g  `  ndx ) ,  ( .r `  r
) >.  =  <. ( +g  `  ndx ) , 
.x.  >. )
72, 6oveq12d 6031 . . . 4  |-  ( r  =  R  ->  (
r sSet  <. ( +g  `  ndx ) ,  ( .r `  r ) >. )  =  ( R sSet  <. ( +g  `  ndx ) ,  .x.  >. ) )
8 df-mgp 15569 . . . 4  |- mulGrp  =  ( r  e.  _V  |->  ( r sSet  <. ( +g  `  ndx ) ,  ( .r `  r ) >. )
)
9 ovex 6038 . . . 4  |-  ( R sSet  <. ( +g  `  ndx ) ,  .x.  >. )  e.  _V
107, 8, 9fvmpt 5738 . . 3  |-  ( R  e.  _V  ->  (mulGrp `  R )  =  ( R sSet  <. ( +g  `  ndx ) ,  .x.  >. )
)
11 fvprc 5655 . . . 4  |-  ( -.  R  e.  _V  ->  (mulGrp `  R )  =  (/) )
12 reldmsets 13411 . . . . 5  |-  Rel  dom sSet
1312ovprc1 6041 . . . 4  |-  ( -.  R  e.  _V  ->  ( R sSet  <. ( +g  `  ndx ) ,  .x.  >. )  =  (/) )
1411, 13eqtr4d 2415 . . 3  |-  ( -.  R  e.  _V  ->  (mulGrp `  R )  =  ( R sSet  <. ( +g  `  ndx ) ,  .x.  >. )
)
1510, 14pm2.61i 158 . 2  |-  (mulGrp `  R )  =  ( R sSet  <. ( +g  `  ndx ) ,  .x.  >. )
161, 15eqtri 2400 1  |-  M  =  ( R sSet  <. ( +g  `  ndx ) , 
.x.  >. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1717   _Vcvv 2892   (/)c0 3564   <.cop 3753   ` cfv 5387  (class class class)co 6013   ndxcnx 13386   sSet csts 13387   +g cplusg 13449   .rcmulr 13450  mulGrpcmgp 15568
This theorem is referenced by:  mgpplusg  15572  mgplem  15573  mgpress  15579
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5351  df-fun 5389  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-sets 13395  df-mgp 15569
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