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Mirrors > Home > MPE Home > Th. List > lss1 | Structured version Visualization version GIF version |
Description: The set of vectors in a left module is a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lssss.v | β’ π = (Baseβπ) |
lssss.s | β’ π = (LSubSpβπ) |
Ref | Expression |
---|---|
lss1 | β’ (π β LMod β π β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2733 | . 2 β’ (π β LMod β (Scalarβπ) = (Scalarβπ)) | |
2 | eqidd 2733 | . 2 β’ (π β LMod β (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ))) | |
3 | lssss.v | . . 3 β’ π = (Baseβπ) | |
4 | 3 | a1i 11 | . 2 β’ (π β LMod β π = (Baseβπ)) |
5 | eqidd 2733 | . 2 β’ (π β LMod β (+gβπ) = (+gβπ)) | |
6 | eqidd 2733 | . 2 β’ (π β LMod β ( Β·π βπ) = ( Β·π βπ)) | |
7 | lssss.s | . . 3 β’ π = (LSubSpβπ) | |
8 | 7 | a1i 11 | . 2 β’ (π β LMod β π = (LSubSpβπ)) |
9 | ssidd 4005 | . 2 β’ (π β LMod β π β π) | |
10 | 3 | lmodbn0 20486 | . 2 β’ (π β LMod β π β β ) |
11 | simpl 483 | . . 3 β’ ((π β LMod β§ (π₯ β (Baseβ(Scalarβπ)) β§ π β π β§ π β π)) β π β LMod) | |
12 | eqid 2732 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
13 | eqid 2732 | . . . . 5 β’ ( Β·π βπ) = ( Β·π βπ) | |
14 | eqid 2732 | . . . . 5 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
15 | 3, 12, 13, 14 | lmodvscl 20493 | . . . 4 β’ ((π β LMod β§ π₯ β (Baseβ(Scalarβπ)) β§ π β π) β (π₯( Β·π βπ)π) β π) |
16 | 15 | 3adant3r3 1184 | . . 3 β’ ((π β LMod β§ (π₯ β (Baseβ(Scalarβπ)) β§ π β π β§ π β π)) β (π₯( Β·π βπ)π) β π) |
17 | simpr3 1196 | . . 3 β’ ((π β LMod β§ (π₯ β (Baseβ(Scalarβπ)) β§ π β π β§ π β π)) β π β π) | |
18 | eqid 2732 | . . . 4 β’ (+gβπ) = (+gβπ) | |
19 | 3, 18 | lmodvacl 20490 | . . 3 β’ ((π β LMod β§ (π₯( Β·π βπ)π) β π β§ π β π) β ((π₯( Β·π βπ)π)(+gβπ)π) β π) |
20 | 11, 16, 17, 19 | syl3anc 1371 | . 2 β’ ((π β LMod β§ (π₯ β (Baseβ(Scalarβπ)) β§ π β π β§ π β π)) β ((π₯( Β·π βπ)π)(+gβπ)π) β π) |
21 | 1, 2, 4, 5, 6, 8, 9, 10, 20 | islssd 20551 | 1 β’ (π β LMod β π β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βcfv 6543 (class class class)co 7411 Basecbs 17146 +gcplusg 17199 Scalarcsca 17202 Β·π cvsca 17203 LModclmod 20475 LSubSpclss 20547 |
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-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-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-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-iota 6495 df-fun 6545 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 |
This theorem is referenced by: lssuni 20555 islss3 20575 lssmre 20582 lspf 20590 lspval 20591 lmhmrnlss 20666 lidl1 20851 isphld 21213 ocv1 21238 aspval 21433 islshpcv 38015 dochexmidlem8 40430 hdmaprnlem4N 40816 lnmfg 41912 |
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