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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 9409 | . 2 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin) | 
| 8 | 1, 2, 3, 4, 5, 7 | gsumcl2 19932 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 finSupp cfsupp 9401 Basecbs 17247 0gc0g 17484 Σg cgsu 17485 CMndccmn 19798 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-0g 17486 df-gsum 17487 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-cntz 19335 df-cmn 19800 | 
| This theorem is referenced by: gsummhm2 19957 gsumsub 19966 gsummptcl 19985 prdsgsum 19999 gsumdixp 20316 frlmphl 21801 frlmup1 21818 islindf4 21858 psrass1lem 21952 rhmpsrlem2 21961 psrbagev2 22102 evlslem3 22104 evlslem1 22106 psdmul 22170 gsumsmonply1 22311 pmatcollpw1 22782 pm2mpcl 22803 mply1topmatcl 22811 mp2pm2mplem2 22813 mp2pm2mp 22817 pm2mpmhmlem2 22825 cayhamlem4 22894 tsmslem1 24137 tsmsgsum 24147 tsmsid 24148 tsmssubm 24151 tsmsxplem1 24161 tsmsxplem2 24162 imasdsf1olem 24383 xrge0gsumle 24855 xrge0tsms 24856 amgm 27034 lgseisenlem3 27421 lgseisenlem4 27422 gsumfs2d 33058 xrge0tsmsd 33065 gsumle 33101 gsumvsca1 33232 gsumvsca2 33233 elrgspnlem1 33246 elrgspn 33250 unitprodclb 33417 elrspunidl 33456 rprmdvdsprod 33562 1arithidomlem1 33563 1arithidom 33565 1arithufdlem3 33574 dfufd2lem 33577 evl1deg2 33602 matunitlindflem1 37623 pwsgprod 42554 evlsvvvallem 42571 selvvvval 42595 evlselv 42597 mnringmulrcld 44247 gsumge0cl 46386 ply1mulgsum 48307 lincfsuppcl 48330 linccl 48331 lincresunit3 48398 amgmlemALT 49322 | 
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