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| Description: Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.) | 
| Ref | Expression | 
|---|---|
| gsumz.z | ⊢ 0 = (0g‘𝐺) | 
| Ref | Expression | 
|---|---|
| gsumz | ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2736 | . 2 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 2 | gsumz.z | . 2 ⊢ 0 = (0g‘𝐺) | |
| 3 | eqid 2736 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 4 | eqid 2736 | . 2 ⊢ {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐺)𝑥) = 𝑦)} = {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐺)𝑥) = 𝑦)} | |
| 5 | simpl 482 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → 𝐺 ∈ Mnd) | |
| 6 | simpr 484 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
| 7 | 2 | fvexi 6919 | . . . . . 6 ⊢ 0 ∈ V | 
| 8 | 7 | snid 4661 | . . . . 5 ⊢ 0 ∈ { 0 } | 
| 9 | 1, 2, 3, 4 | gsumvallem2 18848 | . . . . 5 ⊢ (𝐺 ∈ Mnd → {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐺)𝑥) = 𝑦)} = { 0 }) | 
| 10 | 8, 9 | eleqtrrid 2847 | . . . 4 ⊢ (𝐺 ∈ Mnd → 0 ∈ {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐺)𝑥) = 𝑦)}) | 
| 11 | 10 | ad2antrr 726 | . . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) ∧ 𝑘 ∈ 𝐴) → 0 ∈ {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐺)𝑥) = 𝑦)}) | 
| 12 | 11 | fmpttd 7134 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝑘 ∈ 𝐴 ↦ 0 ):𝐴⟶{𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐺)𝑥) = 𝑦)}) | 
| 13 | 1, 2, 3, 4, 5, 6, 12 | gsumval1 18697 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 {crab 3435 {csn 4625 ↦ cmpt 5224 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 0gc0g 17485 Σg cgsu 17486 Mndcmnd 18748 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-seq 14044 df-0g 17487 df-gsum 17488 df-mgm 18654 df-sgrp 18733 df-mnd 18749 | 
| This theorem is referenced by: gsumval3 19926 gsumzres 19928 gsumzcl2 19929 gsumzf1o 19931 gsumzaddlem 19940 gsumzmhm 19956 gsumzoppg 19963 gsum2d 19991 dprdfeq0 20043 dprddisj2 20060 freshmansdream 21594 mplsubrglem 22025 evlslem1 22107 mhpsclcl 22152 mhpmulcl 22154 coe1tmmul2 22280 coe1tmmul 22281 cply1mul 22301 gsummoncoe1 22313 dmatmul 22504 smadiadetlem1a 22670 cpmatmcllem 22725 mp2pm2mplem4 22816 chfacfscmulgsum 22867 chfacfpmmulgsum 22871 tsms0 24151 tgptsmscls 24159 tdeglem4 26100 mdegmullem 26118 dchrptlem3 27311 gsummptres 33056 gsummptres2 33057 gsumfs2d 33059 elrgspnlem1 33247 elrgspnsubrunlem2 33253 elrspunidl 33457 rprmdvdsprod 33563 evl1deg1 33602 evl1deg2 33603 evl1deg3 33604 lbsdiflsp0 33678 fedgmullem2 33682 esum0 34051 ply1mulgsumlem2 48309 lincvalsc0 48343 linc0scn0 48345 | 
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