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Mathbox for Norm Megill |
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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 37860 | . . 3 β’ (π β π β π΄ = ran (π₯ β (π β { 0 }) β¦ (πβ{π₯}))) |
6 | 5 | eleq2d 2820 | . 2 β’ (π β π β (π β π΄ β π β ran (π₯ β (π β { 0 }) β¦ (πβ{π₯})))) |
7 | eqid 2733 | . . 3 β’ (π₯ β (π β { 0 }) β¦ (πβ{π₯})) = (π₯ β (π β { 0 }) β¦ (πβ{π₯})) | |
8 | fvex 6905 | . . 3 β’ (πβ{π₯}) β V | |
9 | 7, 8 | elrnmpti 5960 | . 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 3071 β cdif 3946 {csn 4629 β¦ cmpt 5232 ran crn 5678 βcfv 6544 Basecbs 17144 0gc0g 17385 LSpanclspn 20582 LSAtomsclsa 37844 |
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 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
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 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-lsatoms 37846 |
This theorem is referenced by: lsatlspsn2 37862 lsatlspsn 37863 islsati 37864 lsateln0 37865 lsatn0 37869 lsatcmp 37873 lsmsat 37878 lsatfixedN 37879 islshpat 37887 lsatcv0 37901 lsat0cv 37903 lcv1 37911 l1cvpat 37924 dih1dimatlem 40200 dihlatat 40208 dochsatshp 40322 |
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