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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 2610 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
2 | eqid 2610 | . . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
3 | lssss.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
4 | eqid 2610 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
5 | eqid 2610 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
6 | lssss.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
7 | 1, 2, 3, 4, 5, 6 | islss 18756 | . 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 1069 | 1 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ⊆ wss 3540 ∅c0 3874 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 Scalarcsca 15771 ·𝑠 cvsca 15772 LSubSpclss 18753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-lss 18754 |
This theorem is referenced by: lssel 18759 lssuni 18761 00lss 18763 lsssubg 18778 islss3 18780 lsslss 18782 lssintcl 18785 lssmre 18787 lssacs 18788 lspid 18803 lspssv 18804 lspssp 18809 lsslsp 18836 lmhmima 18868 reslmhm 18873 lsmsp 18907 pj1lmhm 18921 lsppratlem2 18969 lsppratlem3 18970 lsppratlem4 18971 lspprat 18974 lbsextlem3 18981 lidlss 19031 ocvin 19837 pjdm2 19874 pjff 19875 pjf2 19877 pjfo 19878 pjcss 19879 frlmgsum 19930 frlmsplit2 19931 lsslindf 19988 lsslinds 19989 lssbn 22956 minveclem1 23003 minveclem2 23005 minveclem3a 23006 minveclem3b 23007 minveclem3 23008 minveclem4a 23009 minveclem4b 23010 minveclem4 23011 minveclem6 23013 minveclem7 23014 pjthlem1 23016 pjthlem2 23017 pjth 23018 islshpsm 33285 lshpnelb 33289 lshpnel2N 33290 lshpcmp 33293 lsatssv 33303 lssats 33317 lpssat 33318 lssatle 33320 lssat 33321 islshpcv 33358 lkrssv 33401 lkrlsp 33407 dvhopellsm 35424 dvadiaN 35435 dihss 35558 dihrnss 35585 dochord2N 35678 dochord3 35679 dihoml4 35684 dochsat 35690 dochshpncl 35691 dochnoncon 35698 djhlsmcl 35721 dihjat1lem 35735 dochsatshp 35758 dochsatshpb 35759 dochshpsat 35761 dochexmidlem2 35768 dochexmidlem5 35771 dochexmidlem6 35772 dochexmidlem7 35773 dochexmidlem8 35774 lclkrlem2p 35829 lclkrlem2v 35835 lcfrlem5 35853 lcfr 35892 mapdpglem17N 35995 mapdpglem18 35996 mapdpglem21 35999 islssfg 36658 islssfg2 36659 lnmlsslnm 36669 kercvrlsm 36671 lnmepi 36673 filnm 36678 gsumlsscl 41958 lincellss 42009 ellcoellss 42018 |
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