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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 8676 | . 2 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin) |
8 | 1, 2, 3, 4, 5, 7 | gsumcl2 18743 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1520 ∈ wcel 2079 class class class wbr 4956 ⟶wf 6213 ‘cfv 6217 (class class class)co 7007 finSupp cfsupp 8669 Basecbs 16300 0gc0g 16530 Σg cgsu 16531 CMndccmn 18621 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-se 5395 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-isom 6226 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-supp 7673 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-oadd 7948 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-fsupp 8670 df-oi 8810 df-card 9203 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-nn 11476 df-n0 11735 df-z 11819 df-uz 12083 df-fz 12732 df-fzo 12873 df-seq 13208 df-hash 13529 df-0g 16532 df-gsum 16533 df-mgm 17669 df-sgrp 17711 df-mnd 17722 df-cntz 18176 df-cmn 18623 |
This theorem is referenced by: gsummhm2 18767 gsumsub 18776 gsummptcl 18795 prdsgsum 18806 gsumdixp 19037 psrass1lem 19833 psrmulcllem 19843 psrbagev2 19966 evlslem3 19969 evlslem1 19970 gsumsmonply1 20142 frlmphl 20595 frlmup1 20612 islindf4 20652 pmatcollpw1 21056 pm2mpcl 21077 mply1topmatcl 21085 mp2pm2mplem2 21087 mp2pm2mp 21091 pm2mpmhmlem2 21099 cayhamlem4 21168 tsmslem1 22408 tsmsgsum 22418 tsmsid 22419 tsmssubm 22422 tsmsxplem1 22432 tsmsxplem2 22433 imasdsf1olem 22654 xrge0gsumle 23112 xrge0tsms 23113 amgm 25238 lgseisenlem3 25623 lgseisenlem4 25624 gsumle 30452 gsumvsca1 30455 gsumvsca2 30456 xrge0tsmsd 30460 matunitlindflem1 34365 gsumge0cl 42149 ply1mulgsum 43878 lincfsuppcl 43902 linccl 43903 lincresunit3 43970 amgmlemALT 44338 |
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