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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 2726 | . . 3 โข (Baseโ๐พ) = (Baseโ๐พ) | |
2 | eqid 2726 | . . 3 โข (leโ๐พ) = (leโ๐พ) | |
3 | eqid 2726 | . . 3 โข (joinโ๐พ) = (joinโ๐พ) | |
4 | eqid 2726 | . . 3 โข (Atomsโ๐พ) = (Atomsโ๐พ) | |
5 | 1, 2, 3, 4 | iscvlat 38704 | . 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 2098 โwral 3055 class class class wbr 5141 โcfv 6536 (class class class)co 7404 Basecbs 17151 lecple 17211 joincjn 18274 Atomscatm 38644 AtLatcal 38645 CvLatclc 38646 |
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-ext 2697 |
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-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 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-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-iota 6488 df-fv 6544 df-ov 7407 df-cvlat 38703 |
This theorem is referenced by: cvllat 38707 cvlexch3 38713 cvlexch4N 38714 cvlatexchb1 38715 cvlcvr1 38720 cvlcvrp 38721 cvlatcvr1 38722 cvlsupr2 38724 hlatl 38741 |
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