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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 6880 | . . . 4 β’ (π β (LIndSβπ) β π β dom LIndS) | |
2 | islinds.b | . . . . 5 β’ π΅ = (Baseβπ) | |
3 | 2 | islinds 21231 | . . . 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 267 | . 2 β’ (π β (LIndSβπ) β (π β π΅ β§ ( I βΎ π) LIndF π)) |
6 | 5 | simpld 496 | 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 397 = wceq 1542 β wcel 2107 β wss 3911 class class class wbr 5106 I cid 5531 dom cdm 5634 βΎ cres 5636 βcfv 6497 Basecbs 17088 LIndF clindf 21226 LIndSclinds 21227 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-res 5646 df-iota 6449 df-fun 6499 df-fv 6505 df-linds 21229 |
This theorem is referenced by: lindsss 21246 lindsmm2 21251 islinds3 21256 islinds4 21257 0nellinds 32206 linds2eq 32216 lindsunlem 32376 lindsun 32377 dimkerim 32379 lindsadd 36117 lindsdom 36118 lindsenlbs 36119 |
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