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Mirrors > Home > MPE Home > Th. List > gsumz | Structured version Visualization version GIF version |
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 2733 | . 2 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | gsumz.z | . 2 ⊢ 0 = (0g‘𝐺) | |
3 | eqid 2733 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | eqid 2733 | . 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 484 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → 𝐺 ∈ Mnd) | |
6 | simpr 486 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
7 | 2 | fvexi 6857 | . . . . . 6 ⊢ 0 ∈ V |
8 | 7 | snid 4623 | . . . . 5 ⊢ 0 ∈ { 0 } |
9 | 1, 2, 3, 4 | gsumvallem2 18649 | . . . . 5 ⊢ (𝐺 ∈ Mnd → {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐺)𝑥) = 𝑦)} = { 0 }) |
10 | 8, 9 | eleqtrrid 2841 | . . . 4 ⊢ (𝐺 ∈ Mnd → 0 ∈ {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐺)𝑥) = 𝑦)}) |
11 | 10 | ad2antrr 725 | . . 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 7064 | . 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 18543 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3061 {crab 3406 {csn 4587 ↦ cmpt 5189 ‘cfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 0gc0g 17326 Σg cgsu 17327 Mndcmnd 18561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-seq 13913 df-0g 17328 df-gsum 17329 df-mgm 18502 df-sgrp 18551 df-mnd 18562 |
This theorem is referenced by: gsumval3 19689 gsumzres 19691 gsumzcl2 19692 gsumzf1o 19694 gsumzaddlem 19703 gsumzmhm 19719 gsumzoppg 19726 gsum2d 19754 dprdfeq0 19806 dprddisj2 19823 mplsubrglem 21426 evlslem1 21508 mhpsclcl 21553 mhpmulcl 21555 coe1tmmul2 21663 coe1tmmul 21664 cply1mul 21681 gsummoncoe1 21691 dmatmul 21862 smadiadetlem1a 22028 cpmatmcllem 22083 mp2pm2mplem4 22174 chfacfscmulgsum 22225 chfacfpmmulgsum 22229 tsms0 23509 tgptsmscls 23517 tdeglem4 25440 tdeglem4OLD 25441 mdegmullem 25459 dchrptlem3 26630 gsummptres 31943 gsummptres2 31944 freshmansdream 32116 elrspunidl 32251 lbsdiflsp0 32378 fedgmullem2 32382 esum0 32705 ply1mulgsumlem2 46554 lincvalsc0 46588 linc0scn0 46590 |
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