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Theorem gsumz 14773
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 2435 . 2  |-  ( Base `  G )  =  (
Base `  G )
2 gsum0.z . 2  |-  .0.  =  ( 0g `  G )
3 eqid 2435 . 2  |-  ( +g  `  G )  =  ( +g  `  G )
4 eqid 2435 . 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 5734 . . . . . . 7  |-  ( 0g
`  G )  e. 
_V
82, 7eqeltri 2505 . . . . . 6  |-  .0.  e.  _V
98snid 3833 . . . . 5  |-  .0.  e.  {  .0.  }
101, 2, 3, 4gsumvallem2 14764 . . . . 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 2522 . . . 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 2435 . . 3  |-  ( k  e.  A  |->  .0.  )  =  ( k  e.  A  |->  .0.  )
1412, 13fmptd 5885 . 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 14771 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 1652    e. wcel 1725   A.wral 2697   {crab 2701   _Vcvv 2948   {csn 3806    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   0gc0g 13715    gsumg cgsu 13716   Mndcmnd 14676
This theorem is referenced by:  gsumval3  15506  gsumzres  15509  gsumzcl  15510  gsumzf1o  15511  gsumzaddlem  15518  gsumzmhm  15525  gsumzoppg  15531  gsum2d  15538  dprdfeq0  15572  dprddisj2  15589  mplsubrglem  16494  coe1tmmul2  16660  coe1tmmul  16661  tsms0  18163  tgptsmscls  18171  evlslem1  19928  tdeglem4  19975  mdegmullem  19993  dchrptlem3  21042  esum0  24436
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  ax-un 4693
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-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  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-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-seq 11316  df-0g 13719  df-gsum 13720  df-mnd 14682
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