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Theorem gsumz 17988
Description: Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.)
Hypothesis
Ref Expression
gsumz.z 0 = (0g𝐺)
Assertion
Ref Expression
gsumz ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴0 )) = 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 2818 . 2 (Base‘𝐺) = (Base‘𝐺)
2 gsumz.z . 2 0 = (0g𝐺)
3 eqid 2818 . 2 (+g𝐺) = (+g𝐺)
4 eqid 2818 . 2 {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦) = 𝑦 ∧ (𝑦(+g𝐺)𝑥) = 𝑦)} = {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦) = 𝑦 ∧ (𝑦(+g𝐺)𝑥) = 𝑦)}
5 simpl 483 . 2 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → 𝐺 ∈ Mnd)
6 simpr 485 . 2 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → 𝐴𝑉)
72fvexi 6677 . . . . . 6 0 ∈ V
87snid 4591 . . . . 5 0 ∈ { 0 }
91, 2, 3, 4gsumvallem2 17986 . . . . 5 (𝐺 ∈ Mnd → {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦) = 𝑦 ∧ (𝑦(+g𝐺)𝑥) = 𝑦)} = { 0 })
108, 9eleqtrrid 2917 . . . 4 (𝐺 ∈ Mnd → 0 ∈ {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦) = 𝑦 ∧ (𝑦(+g𝐺)𝑥) = 𝑦)})
1110ad2antrr 722 . . 3 (((𝐺 ∈ Mnd ∧ 𝐴𝑉) ∧ 𝑘𝐴) → 0 ∈ {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦) = 𝑦 ∧ (𝑦(+g𝐺)𝑥) = 𝑦)})
1211fmpttd 6871 . 2 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝑘𝐴0 ):𝐴⟶{𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦) = 𝑦 ∧ (𝑦(+g𝐺)𝑥) = 𝑦)})
131, 2, 3, 4, 5, 6, 12gsumval1 17881 1 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴0 )) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  {crab 3139  {csn 4557  cmpt 5137  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  0gc0g 16701   Σg cgsu 16702  Mndcmnd 17899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-seq 13358  df-0g 16703  df-gsum 16704  df-mgm 17840  df-sgrp 17889  df-mnd 17900
This theorem is referenced by:  gsumval3  18956  gsumzres  18958  gsumzcl2  18959  gsumzf1o  18961  gsumzaddlem  18970  gsumzmhm  18986  gsumzoppg  18993  gsum2d  19021  dprdfeq0  19073  dprddisj2  19090  mplsubrglem  20147  evlslem1  20223  coe1tmmul2  20372  coe1tmmul  20373  cply1mul  20390  gsummoncoe1  20400  dmatmul  21034  smadiadetlem1a  21200  cpmatmcllem  21254  mp2pm2mplem4  21345  chfacfscmulgsum  21396  chfacfpmmulgsum  21400  tsms0  22677  tgptsmscls  22685  tdeglem4  24581  mdegmullem  24599  dchrptlem3  25769  gsummptres  30617  freshmansdream  30786  lbsdiflsp0  30921  fedgmullem2  30925  esum0  31207  ply1mulgsumlem2  44369  lincvalsc0  44404  linc0scn0  44406
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