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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 2734 | . 2 β’ (π β LMod β (Scalarβπ) = (Scalarβπ)) | |
2 | eqidd 2734 | . 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 2734 | . 2 β’ (π β LMod β (+gβπ) = (+gβπ)) | |
6 | eqidd 2734 | . 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 4006 | . 2 β’ (π β LMod β π β π) | |
10 | 3 | lmodbn0 20482 | . 2 β’ (π β LMod β π β β ) |
11 | simpl 484 | . . 3 β’ ((π β LMod β§ (π₯ β (Baseβ(Scalarβπ)) β§ π β π β§ π β π)) β π β LMod) | |
12 | eqid 2733 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
13 | eqid 2733 | . . . . 5 β’ ( Β·π βπ) = ( Β·π βπ) | |
14 | eqid 2733 | . . . . 5 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
15 | 3, 12, 13, 14 | lmodvscl 20489 | . . . 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 1185 | . . 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 1197 | . . 3 β’ ((π β LMod β§ (π₯ β (Baseβ(Scalarβπ)) β§ π β π β§ π β π)) β π β π) | |
18 | eqid 2733 | . . . 4 β’ (+gβπ) = (+gβπ) | |
19 | 3, 18 | lmodvacl 20486 | . . 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 1372 | . 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 20546 | 1 β’ (π β LMod β π β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βcfv 6544 (class class class)co 7409 Basecbs 17144 +gcplusg 17197 Scalarcsca 17200 Β·π cvsca 17201 LModclmod 20471 LSubSpclss 20542 |
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-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-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-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-iota 6496 df-fun 6546 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 |
This theorem is referenced by: lssuni 20550 islss3 20570 lssmre 20577 lspf 20585 lspval 20586 lmhmrnlss 20661 lidl1 20845 isphld 21207 ocv1 21232 aspval 21427 islshpcv 37923 dochexmidlem8 40338 hdmaprnlem4N 40724 lnmfg 41824 |
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