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Mirrors > Home > MPE Home > Th. List > lssss | Structured version Visualization version GIF version |
Description: A subspace is a set of vectors. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.) |
Ref | Expression |
---|---|
lssss.v | ⊢ 𝑉 = (Base‘𝑊) |
lssss.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
Ref | Expression |
---|---|
lssss | ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
2 | eqid 2818 | . . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
3 | lssss.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
4 | eqid 2818 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
5 | eqid 2818 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
6 | lssss.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
7 | 1, 2, 3, 4, 5, 6 | islss 19635 | . 2 ⊢ (𝑈 ∈ 𝑆 ↔ (𝑈 ⊆ 𝑉 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ 𝑈)) |
8 | 7 | simp1bi 1137 | 1 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ⊆ wss 3933 ∅c0 4288 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 Scalarcsca 16556 ·𝑠 cvsca 16557 LSubSpclss 19632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-lss 19633 |
This theorem is referenced by: lssel 19638 lssuni 19640 00lss 19642 lsssubg 19658 islss3 19660 lsslss 19662 lssintcl 19665 lssmre 19667 lssacs 19668 lspid 19683 lspssv 19684 lspssp 19689 lsslsp 19716 lmhmima 19748 reslmhm 19753 lsmsp 19787 pj1lmhm 19801 lsppratlem2 19849 lsppratlem3 19850 lsppratlem4 19851 lspprat 19854 lbsextlem3 19861 lidlss 19911 ocvin 20746 pjdm2 20783 pjff 20784 pjf2 20786 pjfo 20787 pjcss 20788 frlmgsum 20844 frlmsplit2 20845 lsslindf 20902 lsslinds 20903 cphsscph 23781 lssbn 23882 minveclem1 23954 minveclem2 23956 minveclem3a 23957 minveclem3b 23958 minveclem3 23959 minveclem4a 23960 minveclem4b 23961 minveclem4 23962 minveclem6 23964 minveclem7 23965 pjthlem1 23967 pjthlem2 23968 pjth 23969 lssdimle 30905 islshpsm 35996 lshpnelb 36000 lshpnel2N 36001 lshpcmp 36004 lsatssv 36014 lssats 36028 lpssat 36029 lssatle 36031 lssat 36032 islshpcv 36069 lkrssv 36112 lkrlsp 36118 dvhopellsm 38133 dvadiaN 38144 dihss 38267 dihrnss 38294 dochord2N 38387 dochord3 38388 dihoml4 38393 dochsat 38399 dochshpncl 38400 dochnoncon 38407 djhlsmcl 38430 dihjat1lem 38444 dochsatshp 38467 dochsatshpb 38468 dochshpsat 38470 dochexmidlem2 38477 dochexmidlem5 38480 dochexmidlem6 38481 dochexmidlem7 38482 dochexmidlem8 38483 lclkrlem2p 38538 lclkrlem2v 38544 lcfrlem5 38562 lcfr 38601 mapdpglem17N 38704 mapdpglem18 38705 mapdpglem21 38708 islssfg 39548 islssfg2 39549 lnmlsslnm 39559 kercvrlsm 39561 lnmepi 39563 filnm 39568 gsumlsscl 44359 lincellss 44409 ellcoellss 44418 |
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