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Mirrors > Home > MPE Home > Th. List > lsssssubg | Structured version Visualization version Unicode version |
Description: All subspaces are subgroups. (Contributed by Mario Carneiro, 19-Apr-2016.) |
Ref | Expression |
---|---|
lsssubg.s |
Ref | Expression |
---|---|
lsssssubg | SubGrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsssubg.s | . . . 4 | |
2 | 1 | lsssubg 18957 | . . 3 SubGrp |
3 | 2 | ex 450 | . 2 SubGrp |
4 | 3 | ssrdv 3609 | 1 SubGrp |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 wss 3574 cfv 5888 SubGrpcsubg 17588 clmod 18863 clss 18932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-mgp 18490 df-ur 18502 df-ring 18549 df-lmod 18865 df-lss 18933 |
This theorem is referenced by: lsmsp 19086 lspprabs 19095 pj1lmhm 19100 pj1lmhm2 19101 lspindpi 19132 lvecindp 19138 lsmcv 19141 pjdm2 20055 pjf2 20058 pjfo 20059 ocvpj 20061 pjthlem2 23209 lshpnel 34270 lshpnelb 34271 lsmsat 34295 lrelat 34301 lsmcv2 34316 lcvexchlem1 34321 lcvexchlem2 34322 lcvexchlem3 34323 lcvexchlem4 34324 lcvexchlem5 34325 lcv1 34328 lcv2 34329 lsatexch 34330 lsatcv0eq 34334 lsatcvatlem 34336 lsatcvat 34337 lsatcvat3 34339 l1cvat 34342 lkrlsp 34389 lshpsmreu 34396 lshpkrlem5 34401 dia2dimlem5 36357 dia2dimlem9 36361 dvhopellsm 36406 diblsmopel 36460 cdlemn5pre 36489 cdlemn11c 36498 dihjustlem 36505 dihord1 36507 dihord2a 36508 dihord2b 36509 dihord11c 36513 dihord6apre 36545 dihord5b 36548 dihord5apre 36551 dihjatc3 36602 dihmeetlem9N 36604 dihjatcclem1 36707 dihjatcclem2 36708 dihjat 36712 dvh3dim3N 36738 dochexmidlem2 36750 dochexmidlem6 36754 dochexmidlem7 36755 lclkrlem2b 36797 lclkrlem2f 36801 lclkrlem2v 36817 lclkrslem2 36827 lcfrlem23 36854 lcfrlem25 36856 lcfrlem35 36866 mapdlsm 36953 mapdpglem3 36964 mapdindp0 37008 lspindp5 37059 hdmaprnlem3eN 37150 hdmapglem7a 37219 |
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