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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 6928 | . . . . 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 | lbssp.n | . . . . 5 β’ π = (LSpanβπ) | |
9 | eqid 2732 | . . . . 5 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
10 | 4, 5, 6, 7, 2, 8, 9 | islbs 20686 | . . . 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 266 | . 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 1143 | 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 1087 = wceq 1541 β wcel 2106 βwral 3061 β cdif 3945 β wss 3948 {csn 4628 dom cdm 5676 βcfv 6543 (class class class)co 7408 Basecbs 17143 Scalarcsca 17199 Β·π cvsca 17200 0gc0g 17384 LSpanclspn 20581 LBasisclbs 20684 |
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 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 |
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 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7411 df-lbs 20685 |
This theorem is referenced by: islbs2 20766 islbs3 20767 frlmup3 21354 frlmup4 21355 lmimlbs 21390 lbslcic 21395 lbslsp 32488 lvecdim0i 32685 dimkerim 32707 lindsdom 36477 matunitlindflem2 36480 aacllem 47838 |
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