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Mirrors > Home > MPE Home > Th. List > lspsnel5a | Structured version Visualization version GIF version |
Description: Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 20-Feb-2015.) |
Ref | Expression |
---|---|
lspsnel5a.s | β’ π = (LSubSpβπ) |
lspsnel5a.n | β’ π = (LSpanβπ) |
lspsnel5a.w | β’ (π β π β LMod) |
lspsnel5a.a | β’ (π β π β π) |
lspsnel5a.x | β’ (π β π β π) |
Ref | Expression |
---|---|
lspsnel5a | β’ (π β (πβ{π}) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspsnel5a.x | . 2 β’ (π β π β π) | |
2 | eqid 2737 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
3 | lspsnel5a.s | . . 3 β’ π = (LSubSpβπ) | |
4 | lspsnel5a.n | . . 3 β’ π = (LSpanβπ) | |
5 | lspsnel5a.w | . . 3 β’ (π β π β LMod) | |
6 | lspsnel5a.a | . . 3 β’ (π β π β π) | |
7 | 2, 3 | lssel 20414 | . . . 4 β’ ((π β π β§ π β π) β π β (Baseβπ)) |
8 | 6, 1, 7 | syl2anc 585 | . . 3 β’ (π β π β (Baseβπ)) |
9 | 2, 3, 4, 5, 6, 8 | lspsnel5 20472 | . 2 β’ (π β (π β π β (πβ{π}) β π)) |
10 | 1, 9 | mpbid 231 | 1 β’ (π β (πβ{π}) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wss 3915 {csn 4591 βcfv 6501 Basecbs 17090 LModclmod 20338 LSubSpclss 20408 LSpanclspn 20448 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-0g 17330 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-grp 18758 df-lmod 20340 df-lss 20409 df-lsp 20449 |
This theorem is referenced by: lssats2 20477 lspsn 20479 lspsnvsi 20481 lsmelval2 20562 lspprabs 20572 lspvadd 20573 lspabs3 20598 lsmcv 20618 lspsnat 20622 lsppratlem6 20629 issubassa2 21311 lshpnel 37474 lsatel 37496 lsmsat 37499 lssatomic 37502 lssats 37503 lsat0cv 37524 dia2dimlem10 39565 dochsatshpb 39944 lclkrlem2f 40004 lcfrlem25 40059 lcfrlem35 40069 mapdval2N 40122 mapdrvallem2 40137 mapdpglem8 40171 mapdpglem13 40176 mapdindp0 40211 mapdh6aN 40227 mapdh8e 40276 mapdh9a 40281 hdmap1l6a 40301 hdmapval0 40325 hdmapval3lemN 40329 hdmap10lem 40331 hdmap11lem1 40333 hdmap11lem2 40334 hdmaprnlem4N 40345 hdmaprnlem3eN 40350 |
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