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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > islsati | Structured version Visualization version GIF version |
Description: A 1-dim subspace (atom) (of a left module or left vector space) equals the span of some vector. (Contributed by NM, 1-Oct-2014.) |
Ref | Expression |
---|---|
islsati.v | β’ π = (Baseβπ) |
islsati.n | β’ π = (LSpanβπ) |
islsati.a | β’ π΄ = (LSAtomsβπ) |
Ref | Expression |
---|---|
islsati | β’ ((π β π β§ π β π΄) β βπ£ β π π = (πβ{π£})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss 4096 | . 2 β’ (π β {(0gβπ)}) β π | |
2 | islsati.v | . . . 4 β’ π = (Baseβπ) | |
3 | islsati.n | . . . 4 β’ π = (LSpanβπ) | |
4 | eqid 2737 | . . . 4 β’ (0gβπ) = (0gβπ) | |
5 | islsati.a | . . . 4 β’ π΄ = (LSAtomsβπ) | |
6 | 2, 3, 4, 5 | islsat 37482 | . . 3 β’ (π β π β (π β π΄ β βπ£ β (π β {(0gβπ)})π = (πβ{π£}))) |
7 | 6 | biimpa 478 | . 2 β’ ((π β π β§ π β π΄) β βπ£ β (π β {(0gβπ)})π = (πβ{π£})) |
8 | ssrexv 4016 | . 2 β’ ((π β {(0gβπ)}) β π β (βπ£ β (π β {(0gβπ)})π = (πβ{π£}) β βπ£ β π π = (πβ{π£}))) | |
9 | 1, 7, 8 | mpsyl 68 | 1 β’ ((π β π β§ π β π΄) β βπ£ β π π = (πβ{π£})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwrex 3074 β cdif 3912 β wss 3915 {csn 4591 βcfv 6501 Basecbs 17090 0gc0g 17328 LSpanclspn 20448 LSAtomsclsa 37465 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-fv 6509 df-lsatoms 37467 |
This theorem is referenced by: lsmsatcv 37501 dihjat2 39923 dvh4dimlem 39935 lcfl8 39994 mapdval2N 40122 mapdspex 40160 hdmaprnlem16N 40354 |
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