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Mirrors > Home > MPE Home > Th. List > lspsnid | Structured version Visualization version GIF version |
Description: A vector belongs to the span of its singleton. (spansnid 30854 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lspsnid.v | β’ π = (Baseβπ) |
lspsnid.n | β’ π = (LSpanβπ) |
Ref | Expression |
---|---|
lspsnid | β’ ((π β LMod β§ π β π) β π β (πβ{π})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssi 4811 | . . 3 β’ (π β π β {π} β π) | |
2 | lspsnid.v | . . . 4 β’ π = (Baseβπ) | |
3 | lspsnid.n | . . . 4 β’ π = (LSpanβπ) | |
4 | 2, 3 | lspssid 20601 | . . 3 β’ ((π β LMod β§ {π} β π) β {π} β (πβ{π})) |
5 | 1, 4 | sylan2 593 | . 2 β’ ((π β LMod β§ π β π) β {π} β (πβ{π})) |
6 | snssg 4787 | . . 3 β’ (π β π β (π β (πβ{π}) β {π} β (πβ{π}))) | |
7 | 6 | adantl 482 | . 2 β’ ((π β LMod β§ π β π) β (π β (πβ{π}) β {π} β (πβ{π}))) |
8 | 5, 7 | mpbird 256 | 1 β’ ((π β LMod β§ π β π) β π β (πβ{π})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 = wceq 1541 β wcel 2106 β wss 3948 {csn 4628 βcfv 6543 Basecbs 17146 LModclmod 20475 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: lspsnel6 20610 lssats2 20616 lspsneli 20617 lspsn 20618 lspsneq0 20628 lsmelval2 20701 lspprabs 20711 lspabs3 20740 lspsnel4 20743 lspdisjb 20745 lspfixed 20747 lindsadd 36567 lshpnelb 37940 lsateln0 37951 lssats 37968 dia1dimid 40020 dochnel 40350 dihjat1lem 40385 dochsnkr2cl 40431 lcfrvalsnN 40498 lcfrlem15 40514 mapdpglem2 40630 mapdpglem9 40637 mapdpglem12 40640 mapdpglem14 40642 mapdindp0 40676 mapdindp3 40679 hdmap11lem2 40799 hdmaprnlem3N 40807 hdmaprnlem7N 40812 hdmaprnlem8N 40813 hdmaprnlem3eN 40815 hdmaplkr 40870 |
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