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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 2733 | . . 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 20548 | . . . 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 20606 | . 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 3949 {csn 4629 βcfv 6544 Basecbs 17144 LModclmod 20471 LSubSpclss 20542 LSpanclspn 20582 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 |
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 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-0g 17387 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-lmod 20473 df-lss 20543 df-lsp 20583 |
This theorem is referenced by: lssats2 20611 lspsn 20613 lspsnvsi 20615 lsmelval2 20696 lspprabs 20706 lspvadd 20707 lspabs3 20734 lsmcv 20754 lspsnat 20758 lsppratlem6 20765 issubassa2 21446 lshpnel 37853 lsatel 37875 lsmsat 37878 lssatomic 37881 lssats 37882 lsat0cv 37903 dia2dimlem10 39944 dochsatshpb 40323 lclkrlem2f 40383 lcfrlem25 40438 lcfrlem35 40448 mapdval2N 40501 mapdrvallem2 40516 mapdpglem8 40550 mapdpglem13 40555 mapdindp0 40590 mapdh6aN 40606 mapdh8e 40655 mapdh9a 40660 hdmap1l6a 40680 hdmapval0 40704 hdmapval3lemN 40708 hdmap10lem 40710 hdmap11lem1 40712 hdmap11lem2 40713 hdmaprnlem4N 40724 hdmaprnlem3eN 40729 |
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