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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 2726 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
2 | lbslinds.j | . . . 4 β’ π½ = (LBasisβπ) | |
3 | eqid 2726 | . . . 4 β’ (LSpanβπ) = (LSpanβπ) | |
4 | 1, 2, 3 | islbs4 21727 | . . 3 β’ (π β π½ β (π β (LIndSβπ) β§ ((LSpanβπ)βπ) = (Baseβπ))) |
5 | 4 | simplbi 497 | . 2 β’ (π β π½ β π β (LIndSβπ)) |
6 | 5 | ssriv 3981 | 1 β’ π½ β (LIndSβπ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 β wcel 2098 β wss 3943 βcfv 6537 Basecbs 17153 LSpanclspn 20818 LBasisclbs 20922 LIndSclinds 21700 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-lbs 20923 df-lindf 21701 df-linds 21702 |
This theorem is referenced by: islinds4 21730 lmimlbs 21731 lbslcic 21736 lvecdim0 33209 lssdimle 33210 lbsdiflsp0 33229 dimkerim 33230 fedgmullem2 33233 fedgmul 33234 extdg1id 33260 |
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