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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 6880 | . . . . 5 β’ (π΅ β (LBasisβπ) β π β dom LBasis) | |
2 | lbsss.j | . . . . 5 β’ π½ = (LBasisβπ) | |
3 | 1, 2 | eleq2s 2852 | . . . 4 β’ (π΅ β π½ β π β dom LBasis) |
4 | lbsss.v | . . . . 5 β’ π = (Baseβπ) | |
5 | eqid 2733 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
6 | eqid 2733 | . . . . 5 β’ ( Β·π βπ) = ( Β·π βπ) | |
7 | eqid 2733 | . . . . 5 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
8 | eqid 2733 | . . . . 5 β’ (LSpanβπ) = (LSpanβπ) | |
9 | eqid 2733 | . . . . 5 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
10 | 4, 5, 6, 7, 2, 8, 9 | islbs 20552 | . . . 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 1143 | 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 1088 = wceq 1542 β wcel 2107 βwral 3061 β cdif 3908 β wss 3911 {csn 4587 dom cdm 5634 βcfv 6497 (class class class)co 7358 Basecbs 17088 Scalarcsca 17141 Β·π cvsca 17142 0gc0g 17326 LSpanclspn 20447 LBasisclbs 20550 |
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-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-ov 7361 df-lbs 20551 |
This theorem is referenced by: lbsel 20554 lbspss 20558 islbs2 20631 islbs3 20632 lmimlbs 21258 lbslsp 32212 lvecdim0 32359 lssdimle 32360 lbsdiflsp0 32378 dimkerim 32379 fedgmullem1 32381 fedgmullem2 32382 fedgmul 32383 extdg1id 32409 |
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