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Theorem gsumz 14710
Description: Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.)
Hypothesis
Ref Expression
gsum0.z  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
gsumz  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
Distinct variable groups:    A, k    k, G    k, V
Allowed substitution hint:    .0. ( k)

Proof of Theorem gsumz
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2389 . 2  |-  ( Base `  G )  =  (
Base `  G )
2 gsum0.z . 2  |-  .0.  =  ( 0g `  G )
3 eqid 2389 . 2  |-  ( +g  `  G )  =  ( +g  `  G )
4 eqid 2389 . 2  |-  { x  e.  ( Base `  G
)  |  A. y  e.  ( Base `  G
) ( ( x ( +g  `  G
) y )  =  y  /\  ( y ( +g  `  G
) x )  =  y ) }  =  { x  e.  ( Base `  G )  | 
A. y  e.  (
Base `  G )
( ( x ( +g  `  G ) y )  =  y  /\  ( y ( +g  `  G ) x )  =  y ) }
5 simpl 444 . 2  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  G  e.  Mnd )
6 simpr 448 . 2  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  A  e.  V )
7 fvex 5684 . . . . . . 7  |-  ( 0g
`  G )  e. 
_V
82, 7eqeltri 2459 . . . . . 6  |-  .0.  e.  _V
98snid 3786 . . . . 5  |-  .0.  e.  {  .0.  }
101, 2, 3, 4gsumvallem2 14701 . . . . 5  |-  ( G  e.  Mnd  ->  { x  e.  ( Base `  G
)  |  A. y  e.  ( Base `  G
) ( ( x ( +g  `  G
) y )  =  y  /\  ( y ( +g  `  G
) x )  =  y ) }  =  {  .0.  } )
119, 10syl5eleqr 2476 . . . 4  |-  ( G  e.  Mnd  ->  .0.  e.  { x  e.  (
Base `  G )  |  A. y  e.  (
Base `  G )
( ( x ( +g  `  G ) y )  =  y  /\  ( y ( +g  `  G ) x )  =  y ) } )
1211ad2antrr 707 . . 3  |-  ( ( ( G  e.  Mnd  /\  A  e.  V )  /\  k  e.  A
)  ->  .0.  e.  { x  e.  ( Base `  G )  |  A. y  e.  ( Base `  G ) ( ( x ( +g  `  G
) y )  =  y  /\  ( y ( +g  `  G
) x )  =  y ) } )
13 eqid 2389 . . 3  |-  ( k  e.  A  |->  .0.  )  =  ( k  e.  A  |->  .0.  )
1412, 13fmptd 5834 . 2  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( k  e.  A  |->  .0.  ) : A --> { x  e.  ( Base `  G )  | 
A. y  e.  (
Base `  G )
( ( x ( +g  `  G ) y )  =  y  /\  ( y ( +g  `  G ) x )  =  y ) } )
151, 2, 3, 4, 5, 6, 14gsumval1 14708 1  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651   {crab 2655   _Vcvv 2901   {csn 3759    e. cmpt 4209   ` cfv 5396  (class class class)co 6022   Basecbs 13398   +g cplusg 13458   0gc0g 13652    gsumg cgsu 13653   Mndcmnd 14613
This theorem is referenced by:  gsumval3  15443  gsumzres  15446  gsumzcl  15447  gsumzf1o  15448  gsumzaddlem  15455  gsumzmhm  15462  gsumzoppg  15468  gsum2d  15475  dprdfeq0  15509  dprddisj2  15526  mplsubrglem  16431  coe1tmmul2  16597  coe1tmmul  16598  tsms0  18094  tgptsmscls  18102  evlslem1  19805  tdeglem4  19852  mdegmullem  19870  dchrptlem3  20919  esum0  24242
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 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-riota 6487  df-recs 6571  df-rdg 6606  df-seq 11253  df-0g 13656  df-gsum 13657  df-mnd 14619
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