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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 2651 | . 2 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | gsumz.z | . 2 ⊢ 0 = (0g‘𝐺) | |
3 | eqid 2651 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | eqid 2651 | . 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 472 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → 𝐺 ∈ Mnd) | |
6 | simpr 476 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
7 | fvex 6239 | . . . . . . 7 ⊢ (0g‘𝐺) ∈ V | |
8 | 2, 7 | eqeltri 2726 | . . . . . 6 ⊢ 0 ∈ V |
9 | 8 | snid 4241 | . . . . 5 ⊢ 0 ∈ { 0 } |
10 | 1, 2, 3, 4 | gsumvallem2 17419 | . . . . 5 ⊢ (𝐺 ∈ Mnd → {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐺)𝑥) = 𝑦)} = { 0 }) |
11 | 9, 10 | syl5eleqr 2737 | . . . 4 ⊢ (𝐺 ∈ Mnd → 0 ∈ {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐺)𝑥) = 𝑦)}) |
12 | 11 | ad2antrr 762 | . . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) ∧ 𝑘 ∈ 𝐴) → 0 ∈ {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐺)𝑥) = 𝑦)}) |
13 | eqid 2651 | . . 3 ⊢ (𝑘 ∈ 𝐴 ↦ 0 ) = (𝑘 ∈ 𝐴 ↦ 0 ) | |
14 | 12, 13 | fmptd 6425 | . 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)}) |
15 | 1, 2, 3, 4, 5, 6, 14 | gsumval1 17324 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∀wral 2941 {crab 2945 Vcvv 3231 {csn 4210 ↦ cmpt 4762 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 +gcplusg 15988 0gc0g 16147 Σg cgsu 16148 Mndcmnd 17341 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-seq 12842 df-0g 16149 df-gsum 16150 df-mgm 17289 df-sgrp 17331 df-mnd 17342 |
This theorem is referenced by: gsumval3 18354 gsumzres 18356 gsumzcl2 18357 gsumzf1o 18359 gsumzaddlem 18367 gsumzmhm 18383 gsumzoppg 18390 gsum2d 18417 dprdfeq0 18467 dprddisj2 18484 mplsubrglem 19487 evlslem1 19563 coe1tmmul2 19694 coe1tmmul 19695 cply1mul 19712 gsummoncoe1 19722 dmatmul 20351 smadiadetlem1a 20517 cpmatmcllem 20571 mp2pm2mplem4 20662 chfacfscmulgsum 20713 chfacfpmmulgsum 20717 tsms0 21992 tgptsmscls 22000 tdeglem4 23865 mdegmullem 23883 dchrptlem3 25036 gsummptres 29912 esum0 30239 ply1mulgsumlem2 42500 lincvalsc0 42535 linc0scn0 42537 |
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