Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2821 | . 2 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | gsumz.z | . 2 ⊢ 0 = (0g‘𝐺) | |
3 | eqid 2821 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | eqid 2821 | . 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 485 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → 𝐺 ∈ Mnd) | |
6 | simpr 487 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
7 | 2 | fvexi 6679 | . . . . . 6 ⊢ 0 ∈ V |
8 | 7 | snid 4595 | . . . . 5 ⊢ 0 ∈ { 0 } |
9 | 1, 2, 3, 4 | gsumvallem2 17992 | . . . . 5 ⊢ (𝐺 ∈ Mnd → {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐺)𝑥) = 𝑦)} = { 0 }) |
10 | 8, 9 | eleqtrrid 2920 | . . . 4 ⊢ (𝐺 ∈ Mnd → 0 ∈ {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐺)𝑥) = 𝑦)}) |
11 | 10 | ad2antrr 724 | . . 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 6874 | . 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 17887 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 {crab 3142 {csn 4561 ↦ cmpt 5139 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 0gc0g 16707 Σg cgsu 16708 Mndcmnd 17905 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-seq 13364 df-0g 16709 df-gsum 16710 df-mgm 17846 df-sgrp 17895 df-mnd 17906 |
This theorem is referenced by: gsumval3 19021 gsumzres 19023 gsumzcl2 19024 gsumzf1o 19026 gsumzaddlem 19035 gsumzmhm 19051 gsumzoppg 19058 gsum2d 19086 dprdfeq0 19138 dprddisj2 19155 mplsubrglem 20213 evlslem1 20289 coe1tmmul2 20438 coe1tmmul 20439 cply1mul 20456 gsummoncoe1 20466 dmatmul 21100 smadiadetlem1a 21266 cpmatmcllem 21320 mp2pm2mplem4 21411 chfacfscmulgsum 21462 chfacfpmmulgsum 21466 tsms0 22744 tgptsmscls 22752 tdeglem4 24648 mdegmullem 24666 dchrptlem3 25836 gsummptres 30685 freshmansdream 30854 lbsdiflsp0 31017 fedgmullem2 31021 esum0 31303 ply1mulgsumlem2 44434 lincvalsc0 44469 linc0scn0 44471 |
Copyright terms: Public domain | W3C validator |