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Mirrors > Home > MPE Home > Th. List > lssn0 | Structured version Visualization version GIF version |
Description: A subspace is not empty. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.) |
Ref | Expression |
---|---|
lssn0.s | β’ π = (LSubSpβπ) |
Ref | Expression |
---|---|
lssn0 | β’ (π β π β π β β ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . 3 β’ (Scalarβπ) = (Scalarβπ) | |
2 | eqid 2728 | . . 3 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
3 | eqid 2728 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
4 | eqid 2728 | . . 3 β’ (+gβπ) = (+gβπ) | |
5 | eqid 2728 | . . 3 β’ ( Β·π βπ) = ( Β·π βπ) | |
6 | lssn0.s | . . 3 β’ π = (LSubSpβπ) | |
7 | 1, 2, 3, 4, 5, 6 | islss 20811 | . 2 β’ (π β π β (π β (Baseβπ) β§ π β β β§ βπ₯ β (Baseβ(Scalarβπ))βπ β π βπ β π ((π₯( Β·π βπ)π)(+gβπ)π) β π)) |
8 | 7 | simp2bi 1144 | 1 β’ (π β π β π β β ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 β wne 2936 βwral 3057 β wss 3945 β c0 4318 βcfv 6542 (class class class)co 7414 Basecbs 17173 +gcplusg 17226 Scalarcsca 17229 Β·π cvsca 17230 LSubSpclss 20808 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7417 df-lss 20809 |
This theorem is referenced by: 00lss 20818 lss0cl 20824 lssne0 20828 lsssubg 20834 lbsextlem2 21040 minveclem1 25345 |
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