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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 38166 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 1148 | . . . 4 β’ ((π β LMod β§ π β π β§ π β 0 ) β (π β π β§ π β 0 )) | |
2 | eldifsn 4789 | . . . 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 2730 | . . 3 β’ (πβ{π}) = (πβ{π}) | |
5 | sneq 4637 | . . . . 5 β’ (π£ = π β {π£} = {π}) | |
6 | 5 | fveq2d 6894 | . . . 4 β’ (π£ = π β (πβ{π£}) = (πβ{π})) |
7 | 6 | rspceeqv 3632 | . . 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 584 | . 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 38164 | . . 3 β’ (π β LMod β ((πβ{π}) β π΄ β βπ£ β (π β { 0 })(πβ{π}) = (πβ{π£}))) |
14 | 13 | 3ad2ant1 1131 | . 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 394 β§ w3a 1085 = wceq 1539 β wcel 2104 β wne 2938 βwrex 3068 β cdif 3944 {csn 4627 βcfv 6542 Basecbs 17148 0gc0g 17389 LModclmod 20614 LSpanclspn 20726 LSAtomsclsa 38147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-lsatoms 38149 |
This theorem is referenced by: lsatel 38178 lsmsat 38181 lssatomic 38184 lssats 38185 dihlsprn 40505 dihatlat 40508 dihatexv 40512 dochsatshpb 40626 |
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