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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 6919 | . . . . 5 β’ (π΅ β (LBasisβπ) β π β dom LBasis) | |
2 | lbsss.j | . . . . 5 β’ π½ = (LBasisβπ) | |
3 | 1, 2 | eleq2s 2843 | . . . 4 β’ (π΅ β π½ β π β dom LBasis) |
4 | lbsss.v | . . . . 5 β’ π = (Baseβπ) | |
5 | eqid 2724 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
6 | eqid 2724 | . . . . 5 β’ ( Β·π βπ) = ( Β·π βπ) | |
7 | eqid 2724 | . . . . 5 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
8 | eqid 2724 | . . . . 5 β’ (LSpanβπ) = (LSpanβπ) | |
9 | eqid 2724 | . . . . 5 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
10 | 4, 5, 6, 7, 2, 8, 9 | islbs 20920 | . . . 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 1139 | 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 1084 = wceq 1533 β wcel 2098 βwral 3053 β cdif 3938 β wss 3941 {csn 4621 dom cdm 5667 βcfv 6534 (class class class)co 7402 Basecbs 17149 Scalarcsca 17205 Β·π cvsca 17206 0gc0g 17390 LSpanclspn 20814 LBasisclbs 20918 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-iota 6486 df-fun 6536 df-fv 6542 df-ov 7405 df-lbs 20919 |
This theorem is referenced by: lbsel 20922 lbspss 20926 islbs2 21001 islbs3 21002 lmimlbs 21720 lbslsp 32988 lmimdim 33195 lvecdim0 33198 lssdimle 33199 lbsdiflsp0 33218 dimkerim 33219 fedgmullem1 33221 fedgmullem2 33222 fedgmul 33223 extdg1id 33249 |
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