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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 2733 | . . 3 โข (Baseโ๐พ) = (Baseโ๐พ) | |
2 | eqid 2733 | . . 3 โข (glbโ๐พ) = (glbโ๐พ) | |
3 | eqid 2733 | . . 3 โข (leโ๐พ) = (leโ๐พ) | |
4 | eqid 2733 | . . 3 โข (0.โ๐พ) = (0.โ๐พ) | |
5 | eqid 2733 | . . 3 โข (Atomsโ๐พ) = (Atomsโ๐พ) | |
6 | 1, 2, 3, 4, 5 | isatl 37807 | . 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 1146 | 1 โข (๐พ โ AtLat โ ๐พ โ Lat) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โ wcel 2107 โ wne 2940 โwral 3061 โwrex 3070 class class class wbr 5106 dom cdm 5634 โcfv 6497 Basecbs 17088 lecple 17145 glbcglb 18204 0.cp0 18317 Latclat 18325 Atomscatm 37771 AtLatcal 37772 |
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-ne 2941 df-ral 3062 df-rex 3071 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-dm 5644 df-iota 6449 df-fv 6505 df-atl 37806 |
This theorem is referenced by: atlpos 37809 atnle 37825 atlatmstc 37827 cvllat 37834 hllat 37871 snatpsubN 38259 |
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