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Mirrors > Home > MPE Home > Th. List > lspsnel5 | Structured version Visualization version GIF version |
Description: Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.) |
Ref | Expression |
---|---|
lspsnel5.v | β’ π = (Baseβπ) |
lspsnel5.s | β’ π = (LSubSpβπ) |
lspsnel5.n | β’ π = (LSpanβπ) |
lspsnel5.w | β’ (π β π β LMod) |
lspsnel5.a | β’ (π β π β π) |
lspsnel5.x | β’ (π β π β π) |
Ref | Expression |
---|---|
lspsnel5 | β’ (π β (π β π β (πβ{π}) β π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspsnel5.x | . 2 β’ (π β π β π) | |
2 | lspsnel5.v | . . 3 β’ π = (Baseβπ) | |
3 | lspsnel5.s | . . 3 β’ π = (LSubSpβπ) | |
4 | lspsnel5.n | . . 3 β’ π = (LSpanβπ) | |
5 | lspsnel5.w | . . 3 β’ (π β π β LMod) | |
6 | lspsnel5.a | . . 3 β’ (π β π β π) | |
7 | 2, 3, 4, 5, 6 | lspsnel6 20378 | . 2 β’ (π β (π β π β (π β π β§ (πβ{π}) β π))) |
8 | 1, 7 | mpbirand 706 | 1 β’ (π β (π β π β (πβ{π}) β π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1542 β wcel 2107 β wss 3909 {csn 4585 βcfv 6492 Basecbs 17018 LModclmod 20245 LSubSpclss 20315 LSpanclspn 20355 |
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 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-0g 17258 df-mgm 18432 df-sgrp 18481 df-mnd 18492 df-grp 18686 df-lmod 20247 df-lss 20316 df-lsp 20356 |
This theorem is referenced by: lspsnel5a 20380 lspprid1 20381 lspsnss2 20389 lsmelpr 20475 lspsncmp 20500 lspsnne1 20501 lspsnne2 20502 lspsneq 20506 lspindpi 20516 islbs2 20538 lindsadd 35957 lindsenlbs 35959 lsatelbN 37354 lsmsat 37356 lsatfixedN 37357 l1cvpat 37402 dia2dimlem5 39417 dochsncom 39731 dihjat1lem 39777 dvh4dimlem 39792 lclkrlem2a 39856 lcfrlem6 39896 lcfrlem20 39911 lcfrlem26 39917 lcfrlem36 39927 mapdval2N 39979 mapdrvallem2 39994 mapdindp 40020 mapdh6aN 40084 lspindp5 40119 mapdh8ab 40126 mapdh8e 40133 hdmap1l6a 40158 hdmaprnlem3eN 40207 hdmapoc 40280 |
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