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Mirrors > Home > MPE Home > Th. List > lspssp | Structured version Visualization version GIF version |
Description: If a set of vectors is a subset of a subspace, then the span of those vectors is also contained in the subspace. (Contributed by Mario Carneiro, 4-Sep-2014.) |
Ref | Expression |
---|---|
lspssp.s | β’ π = (LSubSpβπ) |
lspssp.n | β’ π = (LSpanβπ) |
Ref | Expression |
---|---|
lspssp | β’ ((π β LMod β§ π β π β§ π β π) β (πβπ) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
2 | lspssp.s | . . . 4 β’ π = (LSubSpβπ) | |
3 | 1, 2 | lssss 20820 | . . 3 β’ (π β π β π β (Baseβπ)) |
4 | lspssp.n | . . . 4 β’ π = (LSpanβπ) | |
5 | 1, 4 | lspss 20868 | . . 3 β’ ((π β LMod β§ π β (Baseβπ) β§ π β π) β (πβπ) β (πβπ)) |
6 | 3, 5 | syl3an2 1162 | . 2 β’ ((π β LMod β§ π β π β§ π β π) β (πβπ) β (πβπ)) |
7 | 2, 4 | lspid 20866 | . . 3 β’ ((π β LMod β§ π β π) β (πβπ) = π) |
8 | 7 | 3adant3 1130 | . 2 β’ ((π β LMod β§ π β π β§ π β π) β (πβπ) = π) |
9 | 6, 8 | sseqtrd 4020 | 1 β’ ((π β LMod β§ π β π β§ π β π) β (πβπ) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1085 = wceq 1534 β wcel 2099 β wss 3947 βcfv 6548 Basecbs 17180 LModclmod 20743 LSubSpclss 20815 LSpanclspn 20855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-lmod 20745 df-lss 20816 df-lsp 20856 |
This theorem is referenced by: lspsnss 20874 lspprss 20876 lsp0 20893 lsslsp 20899 lsslspOLD 20900 lmhmlsp 20934 lspextmo 20941 lsmsp 20971 lsppratlem3 21037 lsppratlem4 21038 islbs3 21043 rspssp 21135 ocvlsp 21608 frlmsslsp 21730 ply1degltdimlem 33320 lspsslco 47505 |
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