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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 19438). (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 4648 | . 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 19438 | . 2 ⊢ ((𝑊 ∈ LMod ∧ {𝑋} ⊆ 𝑉) → (𝑁‘{𝑋}) ∈ 𝑆) |
6 | 1, 5 | sylan2 592 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ⊆ wss 3859 {csn 4472 ‘cfv 6225 Basecbs 16312 LModclmod 19324 LSubSpclss 19393 LSpanclspn 19433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-plusg 16407 df-0g 16544 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-grp 17864 df-minusg 17865 df-sbg 17866 df-mgp 18930 df-ur 18942 df-ring 18989 df-lmod 19326 df-lss 19394 df-lsp 19434 |
This theorem is referenced by: lspsnsubg 19442 lspsneli 19463 lspsn 19464 lspsnss2 19467 lsmelval2 19547 lsmpr 19551 lsppr 19555 lspprabs 19557 lspsncmp 19578 lspsnne1 19579 lspsnne2 19580 lspabs3 19583 lspsneq 19584 lspdisj 19587 lspdisj2 19589 lspfixed 19590 lspexchn1 19592 lspindpi 19594 lsmcv 19603 rnasclassa 19811 lshpnel 35650 lshpnelb 35651 lshpnel2N 35652 lshpdisj 35654 lsatlss 35663 lsmsat 35675 lsatfixedN 35676 lssats 35679 lsmcv2 35696 lsat0cv 35700 lkrlsp 35769 lkrlsp3 35771 lshpsmreu 35776 lshpkrlem5 35781 dochnel 38060 djhlsmat 38094 dihjat1lem 38095 dvh3dim3N 38116 lclkrlem2b 38175 lclkrlem2f 38179 lclkrlem2p 38189 lcfrvalsnN 38208 lcfrlem23 38232 mapdsn 38308 mapdn0 38336 mapdncol 38337 mapdindp 38338 mapdpglem1 38339 mapdpglem2a 38341 mapdpglem3 38342 mapdpglem6 38345 mapdpglem8 38346 mapdpglem9 38347 mapdpglem12 38350 mapdpglem13 38351 mapdpglem14 38352 mapdpglem17N 38355 mapdpglem18 38356 mapdpglem19 38357 mapdpglem21 38359 mapdpglem23 38361 mapdpglem29 38367 mapdindp0 38386 mapdheq4lem 38398 mapdh6lem1N 38400 mapdh6lem2N 38401 mapdh6dN 38406 lspindp5 38437 hdmaplem3 38440 mapdh9a 38456 hdmap1l6lem1 38474 hdmap1l6lem2 38475 hdmap1l6d 38480 hdmap1eulem 38489 hdmap11lem2 38509 hdmapeq0 38511 hdmaprnlem1N 38516 hdmaprnlem3N 38517 hdmaprnlem3uN 38518 hdmaprnlem4N 38520 hdmaprnlem7N 38522 hdmaprnlem8N 38523 hdmaprnlem9N 38524 hdmaprnlem3eN 38525 hdmaprnlem16N 38529 hdmap14lem9 38543 hgmaprnlem2N 38564 hdmapglem7a 38594 |
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