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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 2732 | . . 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 20553 | . . . 4 β’ ((π β π β§ π β π) β π β (Baseβπ)) |
8 | 6, 1, 7 | syl2anc 584 | . . 3 β’ (π β π β (Baseβπ)) |
9 | 2, 3, 4, 5, 6, 8 | lspsnel5 20611 | . 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 1541 β wcel 2106 β wss 3948 {csn 4628 βcfv 6543 Basecbs 17146 LModclmod 20475 LSubSpclss 20547 LSpanclspn 20587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-0g 17389 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-grp 18824 df-lmod 20477 df-lss 20548 df-lsp 20588 |
This theorem is referenced by: lssats2 20616 lspsn 20618 lspsnvsi 20620 lsmelval2 20701 lspprabs 20711 lspvadd 20712 lspabs3 20740 lsmcv 20760 lspsnat 20764 lsppratlem6 20771 issubassa2 21452 lshpnel 37939 lsatel 37961 lsmsat 37964 lssatomic 37967 lssats 37968 lsat0cv 37989 dia2dimlem10 40030 dochsatshpb 40409 lclkrlem2f 40469 lcfrlem25 40524 lcfrlem35 40534 mapdval2N 40587 mapdrvallem2 40602 mapdpglem8 40636 mapdpglem13 40641 mapdindp0 40676 mapdh6aN 40692 mapdh8e 40741 mapdh9a 40746 hdmap1l6a 40766 hdmapval0 40790 hdmapval3lemN 40794 hdmap10lem 40796 hdmap11lem1 40798 hdmap11lem2 40799 hdmaprnlem4N 40810 hdmaprnlem3eN 40815 |
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