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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > atllat | Structured version Visualization version GIF version |
Description: An atomic lattice is a lattice. (Contributed by NM, 21-Oct-2011.) |
Ref | Expression |
---|---|
atllat | โข (๐พ โ AtLat โ ๐พ โ Lat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . 3 โข (Baseโ๐พ) = (Baseโ๐พ) | |
2 | eqid 2728 | . . 3 โข (glbโ๐พ) = (glbโ๐พ) | |
3 | eqid 2728 | . . 3 โข (leโ๐พ) = (leโ๐พ) | |
4 | eqid 2728 | . . 3 โข (0.โ๐พ) = (0.โ๐พ) | |
5 | eqid 2728 | . . 3 โข (Atomsโ๐พ) = (Atomsโ๐พ) | |
6 | 1, 2, 3, 4, 5 | isatl 38771 | . 2 โข (๐พ โ AtLat โ (๐พ โ Lat โง (Baseโ๐พ) โ dom (glbโ๐พ) โง โ๐ฅ โ (Baseโ๐พ)(๐ฅ โ (0.โ๐พ) โ โ๐ โ (Atomsโ๐พ)๐(leโ๐พ)๐ฅ))) |
7 | 6 | simp1bi 1143 | 1 โข (๐พ โ AtLat โ ๐พ โ Lat) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โ wcel 2099 โ wne 2937 โwral 3058 โwrex 3067 class class class wbr 5148 dom cdm 5678 โcfv 6548 Basecbs 17180 lecple 17240 glbcglb 18302 0.cp0 18415 Latclat 18423 Atomscatm 38735 AtLatcal 38736 |
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-ne 2938 df-ral 3059 df-rex 3068 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-dm 5688 df-iota 6500 df-fv 6556 df-atl 38770 |
This theorem is referenced by: atlpos 38773 atnle 38789 atlatmstc 38791 cvllat 38798 hllat 38835 snatpsubN 39223 |
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