![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > islsat | Structured version Visualization version GIF version |
Description: The predicate "is a 1-dim subspace (atom)" (of a left module or left vector space). (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.) |
Ref | Expression |
---|---|
lsatset.v | β’ π = (Baseβπ) |
lsatset.n | β’ π = (LSpanβπ) |
lsatset.z | β’ 0 = (0gβπ) |
lsatset.a | β’ π΄ = (LSAtomsβπ) |
Ref | Expression |
---|---|
islsat | β’ (π β π β (π β π΄ β βπ₯ β (π β { 0 })π = (πβ{π₯}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatset.v | . . . 4 β’ π = (Baseβπ) | |
2 | lsatset.n | . . . 4 β’ π = (LSpanβπ) | |
3 | lsatset.z | . . . 4 β’ 0 = (0gβπ) | |
4 | lsatset.a | . . . 4 β’ π΄ = (LSAtomsβπ) | |
5 | 1, 2, 3, 4 | lsatset 37481 | . . 3 β’ (π β π β π΄ = ran (π₯ β (π β { 0 }) β¦ (πβ{π₯}))) |
6 | 5 | eleq2d 2824 | . 2 β’ (π β π β (π β π΄ β π β ran (π₯ β (π β { 0 }) β¦ (πβ{π₯})))) |
7 | eqid 2737 | . . 3 β’ (π₯ β (π β { 0 }) β¦ (πβ{π₯})) = (π₯ β (π β { 0 }) β¦ (πβ{π₯})) | |
8 | fvex 6860 | . . 3 β’ (πβ{π₯}) β V | |
9 | 7, 8 | elrnmpti 5920 | . 2 β’ (π β ran (π₯ β (π β { 0 }) β¦ (πβ{π₯})) β βπ₯ β (π β { 0 })π = (πβ{π₯})) |
10 | 6, 9 | bitrdi 287 | 1 β’ (π β π β (π β π΄ β βπ₯ β (π β { 0 })π = (πβ{π₯}))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1542 β wcel 2107 βwrex 3074 β cdif 3912 {csn 4591 β¦ cmpt 5193 ran crn 5639 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: lsatlspsn2 37483 lsatlspsn 37484 islsati 37485 lsateln0 37486 lsatn0 37490 lsatcmp 37494 lsmsat 37499 lsatfixedN 37500 islshpat 37508 lsatcv0 37522 lsat0cv 37524 lcv1 37532 l1cvpat 37545 dih1dimatlem 39821 dihlatat 39829 dochsatshp 39943 |
Copyright terms: Public domain | W3C validator |