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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatlspsn2 | Structured version Visualization version GIF version |
Description: The span of a nonzero singleton is an atom. TODO: make this obsolete and use lsatlspsn 37484 instead? (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 |
---|---|
lsatlspsn2 | β’ ((π β LMod β§ π β π β§ π β 0 ) β (πβ{π}) β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpc 1151 | . . . 4 β’ ((π β LMod β§ π β π β§ π β 0 ) β (π β π β§ π β 0 )) | |
2 | eldifsn 4752 | . . . 4 β’ (π β (π β { 0 }) β (π β π β§ π β 0 )) | |
3 | 1, 2 | sylibr 233 | . . 3 β’ ((π β LMod β§ π β π β§ π β 0 ) β π β (π β { 0 })) |
4 | eqid 2737 | . . 3 β’ (πβ{π}) = (πβ{π}) | |
5 | sneq 4601 | . . . . 5 β’ (π£ = π β {π£} = {π}) | |
6 | 5 | fveq2d 6851 | . . . 4 β’ (π£ = π β (πβ{π£}) = (πβ{π})) |
7 | 6 | rspceeqv 3600 | . . 3 β’ ((π β (π β { 0 }) β§ (πβ{π}) = (πβ{π})) β βπ£ β (π β { 0 })(πβ{π}) = (πβ{π£})) |
8 | 3, 4, 7 | sylancl 587 | . 2 β’ ((π β LMod β§ π β π β§ π β 0 ) β βπ£ β (π β { 0 })(πβ{π}) = (πβ{π£})) |
9 | lsatset.v | . . . 4 β’ π = (Baseβπ) | |
10 | lsatset.n | . . . 4 β’ π = (LSpanβπ) | |
11 | lsatset.z | . . . 4 β’ 0 = (0gβπ) | |
12 | lsatset.a | . . . 4 β’ π΄ = (LSAtomsβπ) | |
13 | 9, 10, 11, 12 | islsat 37482 | . . 3 β’ (π β LMod β ((πβ{π}) β π΄ β βπ£ β (π β { 0 })(πβ{π}) = (πβ{π£}))) |
14 | 13 | 3ad2ant1 1134 | . 2 β’ ((π β LMod β§ π β π β§ π β 0 ) β ((πβ{π}) β π΄ β βπ£ β (π β { 0 })(πβ{π}) = (πβ{π£}))) |
15 | 8, 14 | mpbird 257 | 1 β’ ((π β LMod β§ π β π β§ π β 0 ) β (πβ{π}) β π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2944 βwrex 3074 β cdif 3912 {csn 4591 βcfv 6501 Basecbs 17090 0gc0g 17328 LModclmod 20338 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: lsatel 37496 lsmsat 37499 lssatomic 37502 lssats 37503 dihlsprn 39823 dihatlat 39826 dihatexv 39830 dochsatshpb 39944 |
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