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Mirrors > Home > MPE Home > Th. List > lspssid | Structured version Visualization version GIF version |
Description: A set of vectors is a subset of its span. (spanss2 31175 analog.) (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lspss.v | β’ π = (Baseβπ) |
lspss.n | β’ π = (LSpanβπ) |
Ref | Expression |
---|---|
lspssid | β’ ((π β LMod β§ π β π) β π β (πβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssintub 4973 | . 2 β’ π β β© {π‘ β (LSubSpβπ) β£ π β π‘} | |
2 | lspss.v | . . 3 β’ π = (Baseβπ) | |
3 | eqid 2728 | . . 3 β’ (LSubSpβπ) = (LSubSpβπ) | |
4 | lspss.n | . . 3 β’ π = (LSpanβπ) | |
5 | 2, 3, 4 | lspval 20866 | . 2 β’ ((π β LMod β§ π β π) β (πβπ) = β© {π‘ β (LSubSpβπ) β£ π β π‘}) |
6 | 1, 5 | sseqtrrid 4035 | 1 β’ ((π β LMod β§ π β π) β π β (πβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 {crab 3430 β wss 3949 β© cint 4953 βcfv 6553 Basecbs 17187 LModclmod 20750 LSubSpclss 20822 LSpanclspn 20862 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-lmod 20752 df-lss 20823 df-lsp 20863 |
This theorem is referenced by: lspun 20878 lspsnid 20884 lsslsp 20906 lsslspOLD 20907 lmhmlsp 20941 lsmsp 20978 lsmssspx 20980 lspvadd 20988 lspsolvlem 21037 lspsolv 21038 lsppratlem3 21044 lsppratlem4 21045 islbs3 21050 lbsextlem2 21054 lbsextlem4 21056 rspssid 21139 ocvlsp 21615 obselocv 21669 frlmsslsp 21737 lindff1 21761 islinds3 21775 mxidlprm 33208 lbslsat 33347 lindsunlem 33355 dimkerim 33358 lindsenlbs 37121 dochocsp 40884 djhunssN 40914 islssfg2 42526 |
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