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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 38167 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 1149 | . . . 4 β’ ((π β LMod β§ π β π β§ π β 0 ) β (π β π β§ π β 0 )) | |
2 | eldifsn 4791 | . . . 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 2731 | . . 3 β’ (πβ{π}) = (πβ{π}) | |
5 | sneq 4639 | . . . . 5 β’ (π£ = π β {π£} = {π}) | |
6 | 5 | fveq2d 6896 | . . . 4 β’ (π£ = π β (πβ{π£}) = (πβ{π})) |
7 | 6 | rspceeqv 3634 | . . 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 585 | . 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 38165 | . . 3 β’ (π β LMod β ((πβ{π}) β π΄ β βπ£ β (π β { 0 })(πβ{π}) = (πβ{π£}))) |
14 | 13 | 3ad2ant1 1132 | . 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 256 | 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 395 β§ w3a 1086 = wceq 1540 β wcel 2105 β wne 2939 βwrex 3069 β cdif 3946 {csn 4629 βcfv 6544 Basecbs 17149 0gc0g 17390 LModclmod 20615 LSpanclspn 20727 LSAtomsclsa 38148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 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 38150 |
This theorem is referenced by: lsatel 38179 lsmsat 38182 lssatomic 38185 lssats 38186 dihlsprn 40506 dihatlat 40509 dihatexv 40513 dochsatshpb 40627 |
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