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Mirrors > Home > MPE Home > Th. List > lbssp | Structured version Visualization version GIF version |
Description: The span of a basis is the whole space. (Contributed by Mario Carneiro, 24-Jun-2014.) |
Ref | Expression |
---|---|
lbsss.v | β’ π = (Baseβπ) |
lbsss.j | β’ π½ = (LBasisβπ) |
lbssp.n | β’ π = (LSpanβπ) |
Ref | Expression |
---|---|
lbssp | β’ (π΅ β π½ β (πβπ΅) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6929 | . . . . 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 | lbssp.n | . . . . 5 β’ π = (LSpanβπ) | |
9 | eqid 2733 | . . . . 5 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
10 | 4, 5, 6, 7, 2, 8, 9 | islbs 20687 | . . . 4 β’ (π β dom LBasis β (π΅ β π½ β (π΅ β π β§ (πβπ΅) = π β§ βπ₯ β π΅ βπ¦ β ((Baseβ(Scalarβπ)) β {(0gβ(Scalarβπ))}) Β¬ (π¦( Β·π βπ)π₯) β (πβ(π΅ β {π₯}))))) |
11 | 3, 10 | syl 17 | . . 3 β’ (π΅ β π½ β (π΅ β π½ β (π΅ β π β§ (πβπ΅) = π β§ βπ₯ β π΅ βπ¦ β ((Baseβ(Scalarβπ)) β {(0gβ(Scalarβπ))}) Β¬ (π¦( Β·π βπ)π₯) β (πβ(π΅ β {π₯}))))) |
12 | 11 | ibi 267 | . 2 β’ (π΅ β π½ β (π΅ β π β§ (πβπ΅) = π β§ βπ₯ β π΅ βπ¦ β ((Baseβ(Scalarβπ)) β {(0gβ(Scalarβπ))}) Β¬ (π¦( Β·π βπ)π₯) β (πβ(π΅ β {π₯})))) |
13 | 12 | simp2d 1144 | 1 β’ (π΅ β π½ β (πβπ΅) = 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 3062 β cdif 3946 β wss 3949 {csn 4629 dom cdm 5677 βcfv 6544 (class class class)co 7409 Basecbs 17144 Scalarcsca 17200 Β·π cvsca 17201 0gc0g 17385 LSpanclspn 20582 LBasisclbs 20685 |
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 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 |
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 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7412 df-lbs 20686 |
This theorem is referenced by: islbs2 20767 islbs3 20768 frlmup3 21355 frlmup4 21356 lmimlbs 21391 lbslcic 21396 lbslsp 32493 lvecdim0i 32690 dimkerim 32712 lindsdom 36482 matunitlindflem2 36485 aacllem 47848 |
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