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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 6925 | . . . . 5 β’ (π΅ β (LBasisβπ) β π β dom LBasis) | |
2 | lbsss.j | . . . . 5 β’ π½ = (LBasisβπ) | |
3 | 1, 2 | eleq2s 2851 | . . . 4 β’ (π΅ β π½ β π β dom LBasis) |
4 | lbsss.v | . . . . 5 β’ π = (Baseβπ) | |
5 | eqid 2732 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
6 | eqid 2732 | . . . . 5 β’ ( Β·π βπ) = ( Β·π βπ) | |
7 | eqid 2732 | . . . . 5 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
8 | eqid 2732 | . . . . 5 β’ (LSpanβπ) = (LSpanβπ) | |
9 | eqid 2732 | . . . . 5 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
10 | 4, 5, 6, 7, 2, 8, 9 | islbs 20679 | . . . 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 266 | . 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 1142 | 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 1087 = wceq 1541 β wcel 2106 βwral 3061 β cdif 3944 β wss 3947 {csn 4627 dom cdm 5675 βcfv 6540 (class class class)co 7405 Basecbs 17140 Scalarcsca 17196 Β·π cvsca 17197 0gc0g 17381 LSpanclspn 20574 LBasisclbs 20677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-ov 7408 df-lbs 20678 |
This theorem is referenced by: lbsel 20681 lbspss 20685 islbs2 20759 islbs3 20760 lmimlbs 21382 lbslsp 32481 lmimdim 32677 lvecdim0 32679 lssdimle 32680 lbsdiflsp0 32699 dimkerim 32700 fedgmullem1 32702 fedgmullem2 32703 fedgmul 32704 extdg1id 32730 |
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