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Mirrors > Home > MPE Home > Th. List > lbsss | Structured version Visualization version GIF version |
Description: A basis is a set of vectors. (Contributed by Mario Carneiro, 24-Jun-2014.) |
Ref | Expression |
---|---|
lbsss.v | β’ π = (Baseβπ) |
lbsss.j | β’ π½ = (LBasisβπ) |
Ref | Expression |
---|---|
lbsss | β’ (π΅ β π½ β π΅ β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6934 | . . . . 5 β’ (π΅ β (LBasisβπ) β π β dom LBasis) | |
2 | lbsss.j | . . . . 5 β’ π½ = (LBasisβπ) | |
3 | 1, 2 | eleq2s 2847 | . . . 4 β’ (π΅ β π½ β π β dom LBasis) |
4 | lbsss.v | . . . . 5 β’ π = (Baseβπ) | |
5 | eqid 2728 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
6 | eqid 2728 | . . . . 5 β’ ( Β·π βπ) = ( Β·π βπ) | |
7 | eqid 2728 | . . . . 5 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
8 | eqid 2728 | . . . . 5 β’ (LSpanβπ) = (LSpanβπ) | |
9 | eqid 2728 | . . . . 5 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
10 | 4, 5, 6, 7, 2, 8, 9 | islbs 20960 | . . . 4 β’ (π β dom LBasis β (π΅ β π½ β (π΅ β π β§ ((LSpanβπ)βπ΅) = π β§ βπ₯ β π΅ βπ¦ β ((Baseβ(Scalarβπ)) β {(0gβ(Scalarβπ))}) Β¬ (π¦( Β·π βπ)π₯) β ((LSpanβπ)β(π΅ β {π₯}))))) |
11 | 3, 10 | syl 17 | . . 3 β’ (π΅ β π½ β (π΅ β π½ β (π΅ β π β§ ((LSpanβπ)βπ΅) = π β§ βπ₯ β π΅ βπ¦ β ((Baseβ(Scalarβπ)) β {(0gβ(Scalarβπ))}) Β¬ (π¦( Β·π βπ)π₯) β ((LSpanβπ)β(π΅ β {π₯}))))) |
12 | 11 | ibi 267 | . 2 β’ (π΅ β π½ β (π΅ β π β§ ((LSpanβπ)βπ΅) = π β§ βπ₯ β π΅ βπ¦ β ((Baseβ(Scalarβπ)) β {(0gβ(Scalarβπ))}) Β¬ (π¦( Β·π βπ)π₯) β ((LSpanβπ)β(π΅ β {π₯})))) |
13 | 12 | simp1d 1140 | 1 β’ (π΅ β π½ β π΅ β π) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ w3a 1085 = wceq 1534 β wcel 2099 βwral 3058 β cdif 3944 β wss 3947 {csn 4629 dom cdm 5678 βcfv 6548 (class class class)co 7420 Basecbs 17179 Scalarcsca 17235 Β·π cvsca 17236 0gc0g 17420 LSpanclspn 20854 LBasisclbs 20958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fv 6556 df-ov 7423 df-lbs 20959 |
This theorem is referenced by: lbsel 20962 lbspss 20966 islbs2 21041 islbs3 21042 lmimlbs 21769 lbslsp 33092 lmimdim 33297 lvecdim0 33300 lssdimle 33301 lbsdiflsp0 33320 dimkerim 33321 fedgmullem1 33323 fedgmullem2 33324 fedgmul 33325 extdg1id 33351 |
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