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Theorem mgpval 15643
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 5720 . . . . . . 7  |-  ( r  =  R  ->  ( .r `  r )  =  ( .r `  R
) )
4 mgpval.2 . . . . . . 7  |-  .x.  =  ( .r `  R )
53, 4syl6eqr 2485 . . . . . 6  |-  ( r  =  R  ->  ( .r `  r )  = 
.x.  )
65opeq2d 3983 . . . . 5  |-  ( r  =  R  ->  <. ( +g  `  ndx ) ,  ( .r `  r
) >.  =  <. ( +g  `  ndx ) , 
.x.  >. )
72, 6oveq12d 6091 . . . 4  |-  ( r  =  R  ->  (
r sSet  <. ( +g  `  ndx ) ,  ( .r `  r ) >. )  =  ( R sSet  <. ( +g  `  ndx ) ,  .x.  >. ) )
8 df-mgp 15641 . . . 4  |- mulGrp  =  ( r  e.  _V  |->  ( r sSet  <. ( +g  `  ndx ) ,  ( .r `  r ) >. )
)
9 ovex 6098 . . . 4  |-  ( R sSet  <. ( +g  `  ndx ) ,  .x.  >. )  e.  _V
107, 8, 9fvmpt 5798 . . 3  |-  ( R  e.  _V  ->  (mulGrp `  R )  =  ( R sSet  <. ( +g  `  ndx ) ,  .x.  >. )
)
11 fvprc 5714 . . . 4  |-  ( -.  R  e.  _V  ->  (mulGrp `  R )  =  (/) )
12 reldmsets 13483 . . . . 5  |-  Rel  dom sSet
1312ovprc1 6101 . . . 4  |-  ( -.  R  e.  _V  ->  ( R sSet  <. ( +g  `  ndx ) ,  .x.  >. )  =  (/) )
1411, 13eqtr4d 2470 . . 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 2455 1  |-  M  =  ( R sSet  <. ( +g  `  ndx ) , 
.x.  >. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   _Vcvv 2948   (/)c0 3620   <.cop 3809   ` cfv 5446  (class class class)co 6073   ndxcnx 13458   sSet csts 13459   +g cplusg 13521   .rcmulr 13522  mulGrpcmgp 15640
This theorem is referenced by:  mgpplusg  15644  mgplem  15645  mgpress  15651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-sets 13467  df-mgp 15641
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