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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 19352 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (SubGrp‘𝑊)) |
3 | 2 | ex 403 | . 2 ⊢ (𝑊 ∈ LMod → (𝑥 ∈ 𝑆 → 𝑥 ∈ (SubGrp‘𝑊))) |
4 | 3 | ssrdv 3827 | 1 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ⊆ wss 3792 ‘cfv 6135 SubGrpcsubg 17972 LModclmod 19255 LSubSpclss 19324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-mgp 18877 df-ur 18889 df-ring 18936 df-lmod 19257 df-lss 19325 |
This theorem is referenced by: lsmsp 19481 lspprabs 19490 pj1lmhm 19495 pj1lmhm2 19496 lspindpi 19528 lvecindp 19534 lsmcv 19537 pjdm2 20454 pjf2 20457 pjfo 20458 ocvpj 20460 pjthlem2 23644 lshpnel 35137 lshpnelb 35138 lsmsat 35162 lrelat 35168 lsmcv2 35183 lcvexchlem1 35188 lcvexchlem2 35189 lcvexchlem3 35190 lcvexchlem4 35191 lcvexchlem5 35192 lcv1 35195 lcv2 35196 lsatexch 35197 lsatcv0eq 35201 lsatcvatlem 35203 lsatcvat 35204 lsatcvat3 35206 l1cvat 35209 lkrlsp 35256 lshpsmreu 35263 lshpkrlem5 35268 dia2dimlem5 37222 dia2dimlem9 37226 dvhopellsm 37271 diblsmopel 37325 cdlemn5pre 37354 cdlemn11c 37363 dihjustlem 37370 dihord1 37372 dihord2a 37373 dihord2b 37374 dihord11c 37378 dihord6apre 37410 dihord5b 37413 dihord5apre 37416 dihjatc3 37467 dihmeetlem9N 37469 dihjatcclem1 37572 dihjatcclem2 37573 dihjat 37577 dvh3dim3N 37603 dochexmidlem2 37615 dochexmidlem6 37619 dochexmidlem7 37620 lclkrlem2b 37662 lclkrlem2f 37666 lclkrlem2v 37682 lclkrslem2 37692 lcfrlem23 37719 lcfrlem25 37721 lcfrlem35 37731 mapdlsm 37818 mapdpglem3 37829 mapdindp0 37873 lspindp5 37924 hdmaprnlem3eN 38012 hdmapglem7a 38081 |
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