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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 9366 | . 2 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin) |
8 | 1, 2, 3, 4, 5, 7 | gsumcl2 19830 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 class class class wbr 5139 ⟶wf 6530 ‘cfv 6534 (class class class)co 7402 finSupp cfsupp 9358 Basecbs 17149 0gc0g 17390 Σg cgsu 17391 CMndccmn 19696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-oi 9502 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13486 df-fzo 13629 df-seq 13968 df-hash 14292 df-0g 17392 df-gsum 17393 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-cntz 19229 df-cmn 19698 |
This theorem is referenced by: gsummhm2 19855 gsumsub 19864 gsummptcl 19883 prdsgsum 19897 gsumdixp 20214 frlmphl 21665 frlmup1 21682 islindf4 21722 psrass1lemOLD 21823 psrass1lem 21826 psrmulcllem 21837 psrbagev2 21971 psrbagev2OLD 21972 evlslem3 21974 evlslem1 21976 gsumsmonply1 22170 pmatcollpw1 22622 pm2mpcl 22643 mply1topmatcl 22651 mp2pm2mplem2 22653 mp2pm2mp 22657 pm2mpmhmlem2 22665 cayhamlem4 22734 tsmslem1 23977 tsmsgsum 23987 tsmsid 23988 tsmssubm 23991 tsmsxplem1 24001 tsmsxplem2 24002 imasdsf1olem 24223 xrge0gsumle 24693 xrge0tsms 24694 amgm 26864 lgseisenlem3 27251 lgseisenlem4 27252 xrge0tsmsd 32703 gsumle 32736 gsumvsca1 32865 gsumvsca2 32866 elrspunidl 33042 matunitlindflem1 36988 pwsgprod 41645 rhmmpllem2 41653 evlsvvvallem 41664 selvvvval 41688 evlselv 41690 mnringmulrcld 43537 gsumge0cl 45633 ply1mulgsum 47320 lincfsuppcl 47343 linccl 47344 lincresunit3 47411 amgmlemALT 48098 |
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