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Mirrors > Home > MPE Home > Th. List > gsumcl | Structured version Visualization version GIF version |
Description: Closure of a finite group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.) |
Ref | Expression |
---|---|
gsumcl.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumcl.z | ⊢ 0 = (0g‘𝐺) |
gsumcl.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsumcl.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsumcl.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
gsumcl.w | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
Ref | Expression |
---|---|
gsumcl | ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumcl.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsumcl.z | . 2 ⊢ 0 = (0g‘𝐺) | |
3 | gsumcl.g | . 2 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | gsumcl.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | gsumcl.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
6 | gsumcl.w | . . 3 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
7 | 6 | fsuppimpd 9319 | . 2 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin) |
8 | 1, 2, 3, 4, 5, 7 | gsumcl2 19698 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 class class class wbr 5110 ⟶wf 6497 ‘cfv 6501 (class class class)co 7362 finSupp cfsupp 9312 Basecbs 17090 0gc0g 17328 Σg cgsu 17329 CMndccmn 19569 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-n0 12421 df-z 12507 df-uz 12771 df-fz 13432 df-fzo 13575 df-seq 13914 df-hash 14238 df-0g 17330 df-gsum 17331 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-cntz 19104 df-cmn 19571 |
This theorem is referenced by: gsummhm2 19723 gsumsub 19732 gsummptcl 19751 prdsgsum 19765 gsumdixp 20040 frlmphl 21203 frlmup1 21220 islindf4 21260 psrass1lemOLD 21358 psrass1lem 21361 psrmulcllem 21371 psrbagev2 21503 psrbagev2OLD 21504 evlslem3 21506 evlslem1 21508 gsumsmonply1 21690 pmatcollpw1 22141 pm2mpcl 22162 mply1topmatcl 22170 mp2pm2mplem2 22172 mp2pm2mp 22176 pm2mpmhmlem2 22184 cayhamlem4 22253 tsmslem1 23496 tsmsgsum 23506 tsmsid 23507 tsmssubm 23510 tsmsxplem1 23520 tsmsxplem2 23521 imasdsf1olem 23742 xrge0gsumle 24212 xrge0tsms 24213 amgm 26356 lgseisenlem3 26741 lgseisenlem4 26742 xrge0tsmsd 31941 gsumle 31974 gsumvsca1 32103 gsumvsca2 32104 elrspunidl 32243 matunitlindflem1 36103 pwsgprod 40761 rhmmpllem2 40767 mnringmulrcld 42582 gsumge0cl 44686 ply1mulgsum 46545 lincfsuppcl 46568 linccl 46569 lincresunit3 46636 amgmlemALT 47324 |
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