![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > lspsncl | Structured version Visualization version GIF version |
Description: The span of a singleton is a subspace (frequently used special case of lspcl 19297). (Contributed by NM, 17-Jul-2014.) |
Ref | Expression |
---|---|
lspval.v | ⊢ 𝑉 = (Base‘𝑊) |
lspval.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lspval.n | ⊢ 𝑁 = (LSpan‘𝑊) |
Ref | Expression |
---|---|
lspsncl | ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssi 4527 | . 2 ⊢ (𝑋 ∈ 𝑉 → {𝑋} ⊆ 𝑉) | |
2 | lspval.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | lspval.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
4 | lspval.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
5 | 2, 3, 4 | lspcl 19297 | . 2 ⊢ ((𝑊 ∈ LMod ∧ {𝑋} ⊆ 𝑉) → (𝑁‘{𝑋}) ∈ 𝑆) |
6 | 1, 5 | sylan2 587 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ⊆ wss 3769 {csn 4368 ‘cfv 6101 Basecbs 16184 LModclmod 19181 LSubSpclss 19250 LSpanclspn 19292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-plusg 16280 df-0g 16417 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-grp 17741 df-minusg 17742 df-sbg 17743 df-mgp 18806 df-ur 18818 df-ring 18865 df-lmod 19183 df-lss 19251 df-lsp 19293 |
This theorem is referenced by: lspsnsubg 19301 lspsneli 19322 lspsn 19323 lspsnss2 19326 lsmelval2 19406 lsmpr 19410 lsppr 19414 lspprabs 19416 lspsncmp 19437 lspsnne1 19438 lspsnne2 19439 lspabs3 19442 lspsneq 19443 lspdisj 19446 lspdisj2 19448 lspfixed 19449 lspfixedOLD 19450 lspexchn1 19452 lspindpi 19454 lsmcv 19463 lshpnel 35004 lshpnelb 35005 lshpnel2N 35006 lshpdisj 35008 lsatlss 35017 lsmsat 35029 lsatfixedN 35030 lssats 35033 lsmcv2 35050 lsat0cv 35054 lkrlsp 35123 lkrlsp3 35125 lshpsmreu 35130 lshpkrlem5 35135 dochnel 37414 djhlsmat 37448 dihjat1lem 37449 dvh3dim3N 37470 lclkrlem2b 37529 lclkrlem2f 37533 lclkrlem2p 37543 lcfrvalsnN 37562 lcfrlem23 37586 mapdsn 37662 mapdn0 37690 mapdncol 37691 mapdindp 37692 mapdpglem1 37693 mapdpglem2a 37695 mapdpglem3 37696 mapdpglem6 37699 mapdpglem8 37700 mapdpglem9 37701 mapdpglem12 37704 mapdpglem13 37705 mapdpglem14 37706 mapdpglem17N 37709 mapdpglem18 37710 mapdpglem19 37711 mapdpglem21 37713 mapdpglem23 37715 mapdpglem29 37721 mapdindp0 37740 mapdheq4lem 37752 mapdh6lem1N 37754 mapdh6lem2N 37755 mapdh6dN 37760 lspindp5 37791 hdmaplem3 37794 mapdh9a 37810 hdmap1l6lem1 37828 hdmap1l6lem2 37829 hdmap1l6d 37834 hdmap1eulem 37843 hdmap11lem2 37863 hdmapeq0 37865 hdmaprnlem1N 37870 hdmaprnlem3N 37871 hdmaprnlem3uN 37872 hdmaprnlem4N 37874 hdmaprnlem7N 37876 hdmaprnlem8N 37877 hdmaprnlem9N 37878 hdmaprnlem3eN 37879 hdmaprnlem16N 37883 hdmap14lem9 37897 hgmaprnlem2N 37918 hdmapglem7a 37948 |
Copyright terms: Public domain | W3C validator |