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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 9394 | . 2 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin) |
8 | 1, 2, 3, 4, 5, 7 | gsumcl2 19869 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 class class class wbr 5148 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 finSupp cfsupp 9386 Basecbs 17180 0gc0g 17421 Σg cgsu 17422 CMndccmn 19735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-fzo 13661 df-seq 14000 df-hash 14323 df-0g 17423 df-gsum 17424 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-cntz 19268 df-cmn 19737 |
This theorem is referenced by: gsummhm2 19894 gsumsub 19903 gsummptcl 19922 prdsgsum 19936 gsumdixp 20255 frlmphl 21715 frlmup1 21732 islindf4 21772 psrass1lemOLD 21874 psrass1lem 21877 psrmulcllem 21888 psrbagev2 22023 psrbagev2OLD 22024 evlslem3 22026 evlslem1 22028 psdmul 22090 gsumsmonply1 22226 pmatcollpw1 22691 pm2mpcl 22712 mply1topmatcl 22720 mp2pm2mplem2 22722 mp2pm2mp 22726 pm2mpmhmlem2 22734 cayhamlem4 22803 tsmslem1 24046 tsmsgsum 24056 tsmsid 24057 tsmssubm 24060 tsmsxplem1 24070 tsmsxplem2 24071 imasdsf1olem 24292 xrge0gsumle 24762 xrge0tsms 24763 amgm 26936 lgseisenlem3 27323 lgseisenlem4 27324 xrge0tsmsd 32784 gsumle 32817 gsumvsca1 32946 gsumvsca2 32947 elrspunidl 33157 matunitlindflem1 37089 pwsgprod 41774 rhmmpllem2 41783 evlsvvvallem 41794 selvvvval 41818 evlselv 41820 mnringmulrcld 43665 gsumge0cl 45759 ply1mulgsum 47458 lincfsuppcl 47481 linccl 47482 lincresunit3 47549 amgmlemALT 48236 |
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