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Theorem gsumz 17368
 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 2621 . 2 (Base‘𝐺) = (Base‘𝐺)
2 gsumz.z . 2 0 = (0g𝐺)
3 eqid 2621 . 2 (+g𝐺) = (+g𝐺)
4 eqid 2621 . 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 473 . 2 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → 𝐺 ∈ Mnd)
6 simpr 477 . 2 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → 𝐴𝑉)
7 fvex 6199 . . . . . . 7 (0g𝐺) ∈ V
82, 7eqeltri 2696 . . . . . 6 0 ∈ V
98snid 4206 . . . . 5 0 ∈ { 0 }
101, 2, 3, 4gsumvallem2 17366 . . . . 5 (𝐺 ∈ Mnd → {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦) = 𝑦 ∧ (𝑦(+g𝐺)𝑥) = 𝑦)} = { 0 })
119, 10syl5eleqr 2707 . . . 4 (𝐺 ∈ Mnd → 0 ∈ {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦) = 𝑦 ∧ (𝑦(+g𝐺)𝑥) = 𝑦)})
1211ad2antrr 762 . . 3 (((𝐺 ∈ Mnd ∧ 𝐴𝑉) ∧ 𝑘𝐴) → 0 ∈ {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦) = 𝑦 ∧ (𝑦(+g𝐺)𝑥) = 𝑦)})
13 eqid 2621 . . 3 (𝑘𝐴0 ) = (𝑘𝐴0 )
1412, 13fmptd 6383 . 2 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝑘𝐴0 ):𝐴⟶{𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦) = 𝑦 ∧ (𝑦(+g𝐺)𝑥) = 𝑦)})
151, 2, 3, 4, 5, 6, 14gsumval1 17271 1 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴0 )) = 0 )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1482   ∈ wcel 1989  ∀wral 2911  {crab 2915  Vcvv 3198  {csn 4175   ↦ cmpt 4727  ‘cfv 5886  (class class class)co 6647  Basecbs 15851  +gcplusg 15935  0gc0g 16094   Σg cgsu 16095  Mndcmnd 17288 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-seq 12797  df-0g 16096  df-gsum 16097  df-mgm 17236  df-sgrp 17278  df-mnd 17289 This theorem is referenced by:  gsumval3  18302  gsumzres  18304  gsumzcl2  18305  gsumzf1o  18307  gsumzaddlem  18315  gsumzmhm  18331  gsumzoppg  18338  gsum2d  18365  dprdfeq0  18415  dprddisj2  18432  mplsubrglem  19433  evlslem1  19509  coe1tmmul2  19640  coe1tmmul  19641  cply1mul  19658  gsummoncoe1  19668  dmatmul  20297  smadiadetlem1a  20463  cpmatmcllem  20517  mp2pm2mplem4  20608  chfacfscmulgsum  20659  chfacfpmmulgsum  20663  tsms0  21939  tgptsmscls  21947  tdeglem4  23814  mdegmullem  23832  dchrptlem3  24985  gsummptres  29769  esum0  30096  ply1mulgsumlem2  41946  lincvalsc0  41981  linc0scn0  41983
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