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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 20934 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (SubGrp‘𝑊)) |
3 | 2 | ex 411 | . 2 ⊢ (𝑊 ∈ LMod → (𝑥 ∈ 𝑆 → 𝑥 ∈ (SubGrp‘𝑊))) |
4 | 3 | ssrdv 3985 | 1 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 ‘cfv 6554 SubGrpcsubg 19114 LModclmod 20836 LSubSpclss 20908 |
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 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-0g 17456 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-grp 18931 df-minusg 18932 df-sbg 18933 df-subg 19117 df-mgp 20118 df-ur 20165 df-ring 20218 df-lmod 20838 df-lss 20909 |
This theorem is referenced by: lsmsp 21064 lspprabs 21073 pj1lmhm 21078 pj1lmhm2 21079 lspindpi 21113 lvecindp 21119 lsmcv 21122 pjdm2 21709 pjf2 21712 pjfo 21713 ocvpj 21715 pjthlem2 25457 lshpnel 38681 lshpnelb 38682 lsmsat 38706 lrelat 38712 lsmcv2 38727 lcvexchlem1 38732 lcvexchlem2 38733 lcvexchlem3 38734 lcvexchlem4 38735 lcvexchlem5 38736 lcv1 38739 lcv2 38740 lsatexch 38741 lsatcv0eq 38745 lsatcvatlem 38747 lsatcvat 38748 lsatcvat3 38750 l1cvat 38753 lkrlsp 38800 lshpsmreu 38807 lshpkrlem5 38812 dia2dimlem5 40767 dia2dimlem9 40771 dvhopellsm 40816 diblsmopel 40870 cdlemn5pre 40899 cdlemn11c 40908 dihjustlem 40915 dihord1 40917 dihord2a 40918 dihord2b 40919 dihord11c 40923 dihord6apre 40955 dihord5b 40958 dihord5apre 40961 dihjatc3 41012 dihmeetlem9N 41014 dihjatcclem1 41117 dihjatcclem2 41118 dihjat 41122 dvh3dim3N 41148 dochexmidlem2 41160 dochexmidlem6 41164 dochexmidlem7 41165 lclkrlem2b 41207 lclkrlem2f 41211 lclkrlem2v 41227 lclkrslem2 41237 lcfrlem23 41264 lcfrlem25 41266 lcfrlem35 41276 mapdlsm 41363 mapdpglem3 41374 mapdindp0 41418 lspindp5 41469 hdmaprnlem3eN 41557 hdmapglem7a 41626 |
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