![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > lsmub1 | Structured version Visualization version GIF version |
Description: Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
Ref | Expression |
---|---|
lsmub1.p | β’ β = (LSSumβπΊ) |
Ref | Expression |
---|---|
lsmub1 | β’ ((π β (SubGrpβπΊ) β§ π β (SubGrpβπΊ)) β π β (π β π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . 3 β’ (BaseβπΊ) = (BaseβπΊ) | |
2 | 1 | subgss 19082 | . 2 β’ (π β (SubGrpβπΊ) β π β (BaseβπΊ)) |
3 | subgsubm 19103 | . 2 β’ (π β (SubGrpβπΊ) β π β (SubMndβπΊ)) | |
4 | lsmub1.p | . . 3 β’ β = (LSSumβπΊ) | |
5 | 1, 4 | lsmub1x 19601 | . 2 β’ ((π β (BaseβπΊ) β§ π β (SubMndβπΊ)) β π β (π β π)) |
6 | 2, 3, 5 | syl2an 595 | 1 β’ ((π β (SubGrpβπΊ) β§ π β (SubGrpβπΊ)) β π β (π β π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β wss 3947 βcfv 6548 (class class class)co 7420 Basecbs 17180 SubMndcsubmnd 18739 SubGrpcsubg 19075 LSSumclsm 19589 |
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 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-grp 18893 df-minusg 18894 df-subg 19078 df-lsm 19591 |
This theorem is referenced by: lsmunss 19614 lsmlub 19619 lsmss1b 19621 lsmss2 19622 lssnle 19629 lsmdisj 19636 lsmdisj2 19637 pj1lid 19656 dmdprdsplit2lem 20002 dprdsplit 20005 pgpfac1lem1 20031 pgpfac1lem3 20034 lspprabs 20980 lspindpi 21020 lshpnelb 38456 lcvexchlem1 38506 lsatexch 38515 lsatcvatlem 38521 dia2dimlem9 40545 dihord2a 40692 dihord6apre 40729 dihord5apre 40735 dochexmidlem6 40938 dochexmidlem7 40939 mapdlsm 41137 |
Copyright terms: Public domain | W3C validator |