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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvlatl | Structured version Visualization version GIF version |
Description: An atomic lattice with the covering property is an atomic lattice. (Contributed by NM, 5-Nov-2012.) |
Ref | Expression |
---|---|
cvlatl | โข (๐พ โ CvLat โ ๐พ โ AtLat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 โข (Baseโ๐พ) = (Baseโ๐พ) | |
2 | eqid 2733 | . . 3 โข (leโ๐พ) = (leโ๐พ) | |
3 | eqid 2733 | . . 3 โข (joinโ๐พ) = (joinโ๐พ) | |
4 | eqid 2733 | . . 3 โข (Atomsโ๐พ) = (Atomsโ๐พ) | |
5 | 1, 2, 3, 4 | iscvlat 37831 | . 2 โข (๐พ โ CvLat โ (๐พ โ AtLat โง โ๐ โ (Atomsโ๐พ)โ๐ โ (Atomsโ๐พ)โ๐ฅ โ (Baseโ๐พ)((ยฌ ๐(leโ๐พ)๐ฅ โง ๐(leโ๐พ)(๐ฅ(joinโ๐พ)๐)) โ ๐(leโ๐พ)(๐ฅ(joinโ๐พ)๐)))) |
6 | 5 | simplbi 499 | 1 โข (๐พ โ CvLat โ ๐พ โ AtLat) |
Colors of variables: wff setvar class |
Syntax hints: ยฌ wn 3 โ wi 4 โง wa 397 โ wcel 2107 โwral 3061 class class class wbr 5106 โcfv 6497 (class class class)co 7358 Basecbs 17088 lecple 17145 joincjn 18205 Atomscatm 37771 AtLatcal 37772 CvLatclc 37773 |
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-ext 2704 |
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-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-ov 7361 df-cvlat 37830 |
This theorem is referenced by: cvllat 37834 cvlexch3 37840 cvlexch4N 37841 cvlatexchb1 37842 cvlcvr1 37847 cvlcvrp 37848 cvlatcvr1 37849 cvlsupr2 37851 hlatl 37868 |
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