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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 2726 | . . 3 β’ (πβ{π}) = (πβ{π}) | |
3 | sneq 4633 | . . . . 5 β’ (π£ = π β {π£} = {π}) | |
4 | 3 | fveq2d 6889 | . . . 4 β’ (π£ = π β (πβ{π£}) = (πβ{π})) |
5 | 4 | rspceeqv 3628 | . . 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 585 | . 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 38374 | . . 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 257 | 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 3064 β cdif 3940 {csn 4623 βcfv 6537 Basecbs 17153 0gc0g 17394 LModclmod 20706 LSpanclspn 20818 LSAtomsclsa 38357 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-lsatoms 38359 |
This theorem is referenced by: lsatspn0 38383 dvh4dimlem 40827 dochsnshp 40837 lclkrlem2a 40891 lclkrlem2c 40893 lclkrlem2e 40895 lcfrlem20 40946 mapdrvallem2 41029 mapdpglem20 41075 mapdpglem30a 41079 mapdpglem30b 41080 hdmaprnlem3eN 41242 hdmaprnlem16N 41246 |
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