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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 4132 | . 2 β’ (π β {(0gβπ)}) β π | |
2 | islsati.v | . . . 4 β’ π = (Baseβπ) | |
3 | islsati.n | . . . 4 β’ π = (LSpanβπ) | |
4 | eqid 2728 | . . . 4 β’ (0gβπ) = (0gβπ) | |
5 | islsati.a | . . . 4 β’ π΄ = (LSAtomsβπ) | |
6 | 2, 3, 4, 5 | islsat 38495 | . . 3 β’ (π β π β (π β π΄ β βπ£ β (π β {(0gβπ)})π = (πβ{π£}))) |
7 | 6 | biimpa 475 | . 2 β’ ((π β π β§ π β π΄) β βπ£ β (π β {(0gβπ)})π = (πβ{π£})) |
8 | ssrexv 4051 | . 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 394 = wceq 1533 β wcel 2098 βwrex 3067 β cdif 3946 β wss 3949 {csn 4632 βcfv 6553 Basecbs 17187 0gc0g 17428 LSpanclspn 20862 LSAtomsclsa 38478 |
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 ax-un 7746 |
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-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-lsatoms 38480 |
This theorem is referenced by: lsmsatcv 38514 dihjat2 40936 dvh4dimlem 40948 lcfl8 41007 mapdval2N 41135 mapdspex 41173 hdmaprnlem16N 41367 |
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