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Mirrors > Home > MPE Home > Th. List > lspcl | Structured version Visualization version GIF version |
Description: The span of a set of vectors is a subspace. (spancl 28909 analog.) (Contributed by NM, 9-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lspval.v | ⊢ 𝑉 = (Base‘𝑊) |
lspval.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lspval.n | ⊢ 𝑁 = (LSpan‘𝑊) |
Ref | Expression |
---|---|
lspcl | ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspval.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
2 | lspval.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | lspval.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
4 | 1, 2, 3 | lspf 19480 | . 2 ⊢ (𝑊 ∈ LMod → 𝑁:𝒫 𝑉⟶𝑆) |
5 | 1 | fvexi 6510 | . . . 4 ⊢ 𝑉 ∈ V |
6 | 5 | elpw2 5100 | . . 3 ⊢ (𝑈 ∈ 𝒫 𝑉 ↔ 𝑈 ⊆ 𝑉) |
7 | 6 | biimpri 220 | . 2 ⊢ (𝑈 ⊆ 𝑉 → 𝑈 ∈ 𝒫 𝑉) |
8 | ffvelrn 6672 | . 2 ⊢ ((𝑁:𝒫 𝑉⟶𝑆 ∧ 𝑈 ∈ 𝒫 𝑉) → (𝑁‘𝑈) ∈ 𝑆) | |
9 | 4, 7, 8 | syl2an 587 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ⊆ wss 3822 𝒫 cpw 4416 ⟶wf 6181 ‘cfv 6185 Basecbs 16337 LModclmod 19368 LSubSpclss 19437 LSpanclspn 19477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-plusg 16432 df-0g 16569 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-grp 17906 df-minusg 17907 df-sbg 17908 df-mgp 18975 df-ur 18987 df-ring 19034 df-lmod 19370 df-lss 19438 df-lsp 19478 |
This theorem is referenced by: lspsncl 19483 lspprcl 19484 lsptpcl 19485 lspssv 19489 lspidm 19492 lspsnvsi 19510 lsp0 19515 lspun0 19517 lsslsp 19521 lmhmlsp 19555 lsmsp 19592 lsmsp2 19593 lspvadd 19602 lspsolvlem 19648 lspsolv 19649 lsppratlem2 19654 lsppratlem3 19655 islbs2 19660 islbs3 19661 lbsextlem2 19665 rspcl 19728 obselocv 20589 frlmsslsp 20657 islinds3 20695 0ellsp 30639 lbslsat 30675 lsatdim 30676 drngdimgt0 30677 lindsunlem 30681 lbsdiflsp0 30683 dimkerim 30684 lindsadd 34363 lindsenlbs 34365 islshpsm 35598 lssats 35630 dvh4dimlem 38061 islssfgi 39106 lmhmfgsplit 39120 |
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