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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 2610 | . 2 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | gsumz.z | . 2 ⊢ 0 = (0g‘𝐺) | |
3 | eqid 2610 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | eqid 2610 | . 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 6113 | . . . . . . 7 ⊢ (0g‘𝐺) ∈ V | |
8 | 2, 7 | eqeltri 2684 | . . . . . 6 ⊢ 0 ∈ V |
9 | 8 | snid 4155 | . . . . 5 ⊢ 0 ∈ { 0 } |
10 | 1, 2, 3, 4 | gsumvallem2 17195 | . . . . 5 ⊢ (𝐺 ∈ Mnd → {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐺)𝑥) = 𝑦)} = { 0 }) |
11 | 9, 10 | syl5eleqr 2695 | . . . 4 ⊢ (𝐺 ∈ Mnd → 0 ∈ {𝑥 ∈ (Base‘𝐺) ∣ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐺)𝑥) = 𝑦)}) |
12 | 11 | ad2antrr 758 | . . 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 2610 | . . 3 ⊢ (𝑘 ∈ 𝐴 ↦ 0 ) = (𝑘 ∈ 𝐴 ↦ 0 ) | |
14 | 12, 13 | fmptd 6292 | . 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 17100 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 Vcvv 3173 {csn 4125 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Σg cgsu 15924 Mndcmnd 17117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seq 12664 df-0g 15925 df-gsum 15926 df-mgm 17065 df-sgrp 17107 df-mnd 17118 |
This theorem is referenced by: gsumval3 18131 gsumzres 18133 gsumzcl2 18134 gsumzf1o 18136 gsumzaddlem 18144 gsumzmhm 18160 gsumzoppg 18167 gsum2d 18194 dprdfeq0 18244 dprddisj2 18261 mplsubrglem 19260 evlslem1 19336 coe1tmmul2 19467 coe1tmmul 19468 cply1mul 19485 gsummoncoe1 19495 dmatmul 20122 smadiadetlem1a 20288 cpmatmcllem 20342 mp2pm2mplem4 20433 chfacfscmulgsum 20484 chfacfpmmulgsum 20488 tsms0 21755 tgptsmscls 21763 tdeglem4 23624 mdegmullem 23642 dchrptlem3 24791 gsummptres 29115 esum0 29438 ply1mulgsumlem2 41969 lincvalsc0 42004 linc0scn0 42006 |
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