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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 30816 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 4812 | . . 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 20596 | . . 3 β’ ((π β LMod β§ {π} β π) β {π} β (πβ{π})) |
5 | 1, 4 | sylan2 594 | . 2 β’ ((π β LMod β§ π β π) β {π} β (πβ{π})) |
6 | snssg 4788 | . . 3 β’ (π β π β (π β (πβ{π}) β {π} β (πβ{π}))) | |
7 | 6 | adantl 483 | . 2 β’ ((π β LMod β§ π β π) β (π β (πβ{π}) β {π} β (πβ{π}))) |
8 | 5, 7 | mpbird 257 | 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 397 = wceq 1542 β wcel 2107 β wss 3949 {csn 4629 βcfv 6544 Basecbs 17144 LModclmod 20471 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: lspsnel6 20605 lssats2 20611 lspsneli 20612 lspsn 20613 lspsneq0 20623 lsmelval2 20696 lspprabs 20706 lspabs3 20734 lspsnel4 20737 lspdisjb 20739 lspfixed 20741 lindsadd 36481 lshpnelb 37854 lsateln0 37865 lssats 37882 dia1dimid 39934 dochnel 40264 dihjat1lem 40299 dochsnkr2cl 40345 lcfrvalsnN 40412 lcfrlem15 40428 mapdpglem2 40544 mapdpglem9 40551 mapdpglem12 40554 mapdpglem14 40556 mapdindp0 40590 mapdindp3 40593 hdmap11lem2 40713 hdmaprnlem3N 40721 hdmaprnlem7N 40726 hdmaprnlem8N 40727 hdmaprnlem3eN 40729 hdmaplkr 40784 |
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