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Mirrors > Home > MPE Home > Th. List > linds1 | Structured version Visualization version GIF version |
Description: An independent set of vectors is a set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
Ref | Expression |
---|---|
islinds.b | β’ π΅ = (Baseβπ) |
Ref | Expression |
---|---|
linds1 | β’ (π β (LIndSβπ) β π β π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6939 | . . . 4 β’ (π β (LIndSβπ) β π β dom LIndS) | |
2 | islinds.b | . . . . 5 β’ π΅ = (Baseβπ) | |
3 | 2 | islinds 21750 | . . . 4 β’ (π β dom LIndS β (π β (LIndSβπ) β (π β π΅ β§ ( I βΎ π) LIndF π))) |
4 | 1, 3 | syl 17 | . . 3 β’ (π β (LIndSβπ) β (π β (LIndSβπ) β (π β π΅ β§ ( I βΎ π) LIndF π))) |
5 | 4 | ibi 266 | . 2 β’ (π β (LIndSβπ) β (π β π΅ β§ ( I βΎ π) LIndF π)) |
6 | 5 | simpld 493 | 1 β’ (π β (LIndSβπ) β π β π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 β wss 3949 class class class wbr 5152 I cid 5579 dom cdm 5682 βΎ cres 5684 βcfv 6553 Basecbs 17187 LIndF clindf 21745 LIndSclinds 21746 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-res 5694 df-iota 6505 df-fun 6555 df-fv 6561 df-linds 21748 |
This theorem is referenced by: lindsss 21765 lindsmm2 21770 islinds3 21775 islinds4 21776 0nellinds 33106 linds2eq 33121 lindsunlem 33355 lindsun 33356 dimkerim 33358 lindsadd 37119 lindsdom 37120 lindsenlbs 37121 |
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