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Mirrors > Home > MPE Home > Th. List > lsssssubg | Structured version Visualization version GIF version |
Description: All subspaces are subgroups. (Contributed by Mario Carneiro, 19-Apr-2016.) |
Ref | Expression |
---|---|
lsssubg.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
Ref | Expression |
---|---|
lsssssubg | ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsssubg.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑊) | |
2 | 1 | lsssubg 19005 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (SubGrp‘𝑊)) |
3 | 2 | ex 449 | . 2 ⊢ (𝑊 ∈ LMod → (𝑥 ∈ 𝑆 → 𝑥 ∈ (SubGrp‘𝑊))) |
4 | 3 | ssrdv 3642 | 1 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 ⊆ wss 3607 ‘cfv 5926 SubGrpcsubg 17635 LModclmod 18911 LSubSpclss 18980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-minusg 17473 df-sbg 17474 df-subg 17638 df-mgp 18536 df-ur 18548 df-ring 18595 df-lmod 18913 df-lss 18981 |
This theorem is referenced by: lsmsp 19134 lspprabs 19143 pj1lmhm 19148 pj1lmhm2 19149 lspindpi 19180 lvecindp 19186 lsmcv 19189 pjdm2 20103 pjf2 20106 pjfo 20107 ocvpj 20109 pjthlem2 23255 lshpnel 34588 lshpnelb 34589 lsmsat 34613 lrelat 34619 lsmcv2 34634 lcvexchlem1 34639 lcvexchlem2 34640 lcvexchlem3 34641 lcvexchlem4 34642 lcvexchlem5 34643 lcv1 34646 lcv2 34647 lsatexch 34648 lsatcv0eq 34652 lsatcvatlem 34654 lsatcvat 34655 lsatcvat3 34657 l1cvat 34660 lkrlsp 34707 lshpsmreu 34714 lshpkrlem5 34719 dia2dimlem5 36674 dia2dimlem9 36678 dvhopellsm 36723 diblsmopel 36777 cdlemn5pre 36806 cdlemn11c 36815 dihjustlem 36822 dihord1 36824 dihord2a 36825 dihord2b 36826 dihord11c 36830 dihord6apre 36862 dihord5b 36865 dihord5apre 36868 dihjatc3 36919 dihmeetlem9N 36921 dihjatcclem1 37024 dihjatcclem2 37025 dihjat 37029 dvh3dim3N 37055 dochexmidlem2 37067 dochexmidlem6 37071 dochexmidlem7 37072 lclkrlem2b 37114 lclkrlem2f 37118 lclkrlem2v 37134 lclkrslem2 37144 lcfrlem23 37171 lcfrlem25 37173 lcfrlem35 37183 mapdlsm 37270 mapdpglem3 37281 mapdindp0 37325 lspindp5 37376 hdmaprnlem3eN 37467 hdmapglem7a 37536 |
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