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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 3968 | . 2 β’ (π β LMod β π β π) | |
10 | 3 | lmodbn0 20347 | . 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 20354 | . . . 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 20351 | . . 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 20411 | 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 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 Scalarcsca 17141 Β·π cvsca 17142 LModclmod 20336 LSubSpclss 20407 |
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 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-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-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-iota 6449 df-fun 6499 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 |
This theorem is referenced by: lssuni 20415 islss3 20435 lssmre 20442 lspf 20450 lspval 20451 lmhmrnlss 20526 lidl1 20706 isphld 21074 ocv1 21099 aspval 21292 islshpcv 37561 dochexmidlem8 39976 hdmaprnlem4N 40362 lnmfg 41452 |
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