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Theorem gsumz 14458
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 2283 . 2  |-  ( Base `  G )  =  (
Base `  G )
2 gsum0.z . 2  |-  .0.  =  ( 0g `  G )
3 eqid 2283 . 2  |-  ( +g  `  G )  =  ( +g  `  G )
4 eqid 2283 . 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 443 . 2  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  G  e.  Mnd )
6 simpr 447 . 2  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  A  e.  V )
7 fvex 5539 . . . . . . 7  |-  ( 0g
`  G )  e. 
_V
82, 7eqeltri 2353 . . . . . 6  |-  .0.  e.  _V
98snid 3667 . . . . 5  |-  .0.  e.  {  .0.  }
101, 2, 3, 4gsumvallem2 14449 . . . . 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 2370 . . . 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 706 . . 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 2283 . . 3  |-  ( k  e.  A  |->  .0.  )  =  ( k  e.  A  |->  .0.  )
1412, 13fmptd 5684 . 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 14456 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 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788   {csn 3640    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400    gsumg cgsu 13401   Mndcmnd 14361
This theorem is referenced by:  gsumval3  15191  gsumzres  15194  gsumzcl  15195  gsumzf1o  15196  gsumzaddlem  15203  gsumzmhm  15210  gsumzoppg  15216  gsum2d  15223  dprdfeq0  15257  dprddisj2  15274  mplsubrglem  16183  coe1tmmul2  16352  coe1tmmul  16353  tsms0  17824  tgptsmscls  17832  evlslem1  19399  tdeglem4  19446  mdegmullem  19464  dchrptlem3  20505  esum0  23428
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  ax-un 4512
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-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  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-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-seq 11047  df-0g 13404  df-gsum 13405  df-mnd 14367
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