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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 30329 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 4928 | . 2 β’ π β β© {π‘ β (LSubSpβπ) β£ π β π‘} | |
2 | lspss.v | . . 3 β’ π = (Baseβπ) | |
3 | eqid 2733 | . . 3 β’ (LSubSpβπ) = (LSubSpβπ) | |
4 | lspss.n | . . 3 β’ π = (LSpanβπ) | |
5 | 2, 3, 4 | lspval 20451 | . 2 β’ ((π β LMod β§ π β π) β (πβπ) = β© {π‘ β (LSubSpβπ) β£ π β π‘}) |
6 | 1, 5 | sseqtrrid 3998 | 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 397 = wceq 1542 β wcel 2107 {crab 3406 β wss 3911 β© cint 4908 βcfv 6497 Basecbs 17088 LModclmod 20336 LSubSpclss 20407 LSpanclspn 20447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-lmod 20338 df-lss 20408 df-lsp 20448 |
This theorem is referenced by: lspun 20463 lspsnid 20469 lsslsp 20491 lmhmlsp 20525 lsmsp 20562 lsmssspx 20564 lspvadd 20572 lspsolvlem 20619 lspsolv 20620 lsppratlem3 20626 lsppratlem4 20627 islbs3 20632 lbsextlem2 20636 lbsextlem4 20638 rspssid 20709 ocvlsp 21096 obselocv 21150 frlmsslsp 21218 lindff1 21242 islinds3 21256 mxidlprm 32285 lbslsat 32368 lindsunlem 32376 dimkerim 32379 lindsenlbs 36119 dochocsp 39888 djhunssN 39918 islssfg2 41441 |
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