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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 2728 | . . 3 โข (Baseโ๐พ) = (Baseโ๐พ) | |
2 | eqid 2728 | . . 3 โข (leโ๐พ) = (leโ๐พ) | |
3 | eqid 2728 | . . 3 โข (joinโ๐พ) = (joinโ๐พ) | |
4 | eqid 2728 | . . 3 โข (Atomsโ๐พ) = (Atomsโ๐พ) | |
5 | 1, 2, 3, 4 | iscvlat 38795 | . 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 497 | 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 395 โ wcel 2099 โwral 3058 class class class wbr 5148 โcfv 6548 (class class class)co 7420 Basecbs 17180 lecple 17240 joincjn 18303 Atomscatm 38735 AtLatcal 38736 CvLatclc 38737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-iota 6500 df-fv 6556 df-ov 7423 df-cvlat 38794 |
This theorem is referenced by: cvllat 38798 cvlexch3 38804 cvlexch4N 38805 cvlatexchb1 38806 cvlcvr1 38811 cvlcvrp 38812 cvlatcvr1 38813 cvlsupr2 38815 hlatl 38832 |
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