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Mirrors > Home > MPE Home > Th. List > gsummptcl | Structured version Visualization version GIF version |
Description: Closure of a finite group sum over a finite set as map. (Contributed by AV, 29-Dec-2018.) |
Ref | Expression |
---|---|
gsummptcl.b | ⊢ 𝐵 = (Base‘𝐺) |
gsummptcl.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsummptcl.n | ⊢ (𝜑 → 𝑁 ∈ Fin) |
gsummptcl.e | ⊢ (𝜑 → ∀𝑖 ∈ 𝑁 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
gsummptcl | ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ 𝑁 ↦ 𝑋)) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummptcl.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2799 | . 2 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
3 | gsummptcl.g | . 2 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | gsummptcl.n | . 2 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
5 | gsummptcl.e | . . 3 ⊢ (𝜑 → ∀𝑖 ∈ 𝑁 𝑋 ∈ 𝐵) | |
6 | eqid 2799 | . . . 4 ⊢ (𝑖 ∈ 𝑁 ↦ 𝑋) = (𝑖 ∈ 𝑁 ↦ 𝑋) | |
7 | 6 | fmpt 6606 | . . 3 ⊢ (∀𝑖 ∈ 𝑁 𝑋 ∈ 𝐵 ↔ (𝑖 ∈ 𝑁 ↦ 𝑋):𝑁⟶𝐵) |
8 | 5, 7 | sylib 210 | . 2 ⊢ (𝜑 → (𝑖 ∈ 𝑁 ↦ 𝑋):𝑁⟶𝐵) |
9 | 6 | fnmpt 6231 | . . . 4 ⊢ (∀𝑖 ∈ 𝑁 𝑋 ∈ 𝐵 → (𝑖 ∈ 𝑁 ↦ 𝑋) Fn 𝑁) |
10 | 5, 9 | syl 17 | . . 3 ⊢ (𝜑 → (𝑖 ∈ 𝑁 ↦ 𝑋) Fn 𝑁) |
11 | fvexd 6426 | . . 3 ⊢ (𝜑 → (0g‘𝐺) ∈ V) | |
12 | 10, 4, 11 | fndmfifsupp 8530 | . 2 ⊢ (𝜑 → (𝑖 ∈ 𝑁 ↦ 𝑋) finSupp (0g‘𝐺)) |
13 | 1, 2, 3, 4, 8, 12 | gsumcl 18631 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑖 ∈ 𝑁 ↦ 𝑋)) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 ∀wral 3089 Vcvv 3385 ↦ cmpt 4922 Fn wfn 6096 ⟶wf 6097 ‘cfv 6101 (class class class)co 6878 Fincfn 8195 Basecbs 16184 0gc0g 16415 Σg cgsu 16416 CMndccmn 18508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-supp 7533 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-fsupp 8518 df-oi 8657 df-card 9051 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-n0 11581 df-z 11667 df-uz 11931 df-fz 12581 df-fzo 12721 df-seq 13056 df-hash 13371 df-0g 16417 df-gsum 16418 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-cntz 18062 df-cmn 18510 |
This theorem is referenced by: srgbinomlem3 18858 srgbinomlem4 18859 gsummgp0 18924 coe1fzgsumdlem 19993 evl1gsumdlem 20042 mamucl 20532 matgsumcl 20592 madetsmelbas 20596 madetsmelbas2 20597 mat1dimmul 20608 mavmulcl 20679 mdetleib2 20720 mdetf 20727 mdetdiaglem 20730 mdetdiag 20731 mdetrlin 20734 mdetrsca 20735 mdetralt 20740 gsummatr01 20792 smadiadet 20803 m2pmfzgsumcl 20881 decpmatmul 20905 pmatcollpw3fi1lem1 20919 pm2mpmhmlem2 20952 chfacfscmulgsum 20993 chfacfpmmulgsum 20997 cpmadugsumlemF 21009 cpmadugsumfi 21010 gsummptres 30300 mdetpmtr1 30405 gsumesum 30637 esumlub 30638 esum2d 30671 mgpsumz 42940 mgpsumn 42941 ply1mulgsum 42977 |
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