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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 2733 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
2 | lspssp.s | . . . 4 β’ π = (LSubSpβπ) | |
3 | 1, 2 | lssss 20412 | . . 3 β’ (π β π β π β (Baseβπ)) |
4 | lspssp.n | . . . 4 β’ π = (LSpanβπ) | |
5 | 1, 4 | lspss 20460 | . . 3 β’ ((π β LMod β§ π β (Baseβπ) β§ π β π) β (πβπ) β (πβπ)) |
6 | 3, 5 | syl3an2 1165 | . 2 β’ ((π β LMod β§ π β π β§ π β π) β (πβπ) β (πβπ)) |
7 | 2, 4 | lspid 20458 | . . 3 β’ ((π β LMod β§ π β π) β (πβπ) = π) |
8 | 7 | 3adant3 1133 | . 2 β’ ((π β LMod β§ π β π β§ π β π) β (πβπ) = π) |
9 | 6, 8 | sseqtrd 3985 | 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 1088 = wceq 1542 β wcel 2107 β wss 3911 βcfv 6497 Basecbs 17088 LModclmod 20336 LSubSpclss 20407 LSpanclspn 20447 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 |
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 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-lmod 20338 df-lss 20408 df-lsp 20448 |
This theorem is referenced by: lspsnss 20466 lspprss 20468 lsp0 20485 lsslsp 20491 lmhmlsp 20525 lspextmo 20532 lsmsp 20562 lsppratlem3 20626 lsppratlem4 20627 islbs3 20632 rspssp 20712 ocvlsp 21096 frlmsslsp 21218 ply1degltdimlem 32374 lspsslco 46604 |
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