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Mirrors > Home > MPE Home > Th. List > lbslinds | Structured version Visualization version GIF version |
Description: A basis is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
Ref | Expression |
---|---|
lbslinds.j | β’ π½ = (LBasisβπ) |
Ref | Expression |
---|---|
lbslinds | β’ π½ β (LIndSβπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2732 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
2 | lbslinds.j | . . . 4 β’ π½ = (LBasisβπ) | |
3 | eqid 2732 | . . . 4 β’ (LSpanβπ) = (LSpanβπ) | |
4 | 1, 2, 3 | islbs4 21378 | . . 3 β’ (π β π½ β (π β (LIndSβπ) β§ ((LSpanβπ)βπ) = (Baseβπ))) |
5 | 4 | simplbi 498 | . 2 β’ (π β π½ β π β (LIndSβπ)) |
6 | 5 | ssriv 3985 | 1 β’ π½ β (LIndSβπ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 β wcel 2106 β wss 3947 βcfv 6540 Basecbs 17140 LSpanclspn 20574 LBasisclbs 20677 LIndSclinds 21351 |
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 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
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 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-lbs 20678 df-lindf 21352 df-linds 21353 |
This theorem is referenced by: islinds4 21381 lmimlbs 21382 lbslcic 21387 lvecdim0 32679 lssdimle 32680 lbsdiflsp0 32699 dimkerim 32700 fedgmullem2 32703 fedgmul 32704 extdg1id 32730 |
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