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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatlspsn | Structured version Visualization version GIF version |
Description: The span of a nonzero singleton is an atom. (Contributed by NM, 16-Jan-2015.) |
Ref | Expression |
---|---|
lsatset.v | β’ π = (Baseβπ) |
lsatset.n | β’ π = (LSpanβπ) |
lsatset.z | β’ 0 = (0gβπ) |
lsatset.a | β’ π΄ = (LSAtomsβπ) |
lsatlspsn.w | β’ (π β π β LMod) |
lsatlspsn.x | β’ (π β π β (π β { 0 })) |
Ref | Expression |
---|---|
lsatlspsn | β’ (π β (πβ{π}) β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatlspsn.x | . . 3 β’ (π β π β (π β { 0 })) | |
2 | eqid 2725 | . . 3 β’ (πβ{π}) = (πβ{π}) | |
3 | sneq 4634 | . . . . 5 β’ (π£ = π β {π£} = {π}) | |
4 | 3 | fveq2d 6896 | . . . 4 β’ (π£ = π β (πβ{π£}) = (πβ{π})) |
5 | 4 | rspceeqv 3623 | . . 3 β’ ((π β (π β { 0 }) β§ (πβ{π}) = (πβ{π})) β βπ£ β (π β { 0 })(πβ{π}) = (πβ{π£})) |
6 | 1, 2, 5 | sylancl 584 | . 2 β’ (π β βπ£ β (π β { 0 })(πβ{π}) = (πβ{π£})) |
7 | lsatlspsn.w | . . 3 β’ (π β π β LMod) | |
8 | lsatset.v | . . . 4 β’ π = (Baseβπ) | |
9 | lsatset.n | . . . 4 β’ π = (LSpanβπ) | |
10 | lsatset.z | . . . 4 β’ 0 = (0gβπ) | |
11 | lsatset.a | . . . 4 β’ π΄ = (LSAtomsβπ) | |
12 | 8, 9, 10, 11 | islsat 38519 | . . 3 β’ (π β LMod β ((πβ{π}) β π΄ β βπ£ β (π β { 0 })(πβ{π}) = (πβ{π£}))) |
13 | 7, 12 | syl 17 | . 2 β’ (π β ((πβ{π}) β π΄ β βπ£ β (π β { 0 })(πβ{π}) = (πβ{π£}))) |
14 | 6, 13 | mpbird 256 | 1 β’ (π β (πβ{π}) β π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1533 β wcel 2098 βwrex 3060 β cdif 3936 {csn 4624 βcfv 6543 Basecbs 17179 0gc0g 17420 LModclmod 20747 LSpanclspn 20859 LSAtomsclsa 38502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-lsatoms 38504 |
This theorem is referenced by: lsatspn0 38528 dvh4dimlem 40972 dochsnshp 40982 lclkrlem2a 41036 lclkrlem2c 41038 lclkrlem2e 41040 lcfrlem20 41091 mapdrvallem2 41174 mapdpglem20 41220 mapdpglem30a 41224 mapdpglem30b 41225 hdmaprnlem3eN 41387 hdmaprnlem16N 41391 |
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