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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > atlex | Structured version Visualization version GIF version |
Description: Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 31344 analog.) (Contributed by NM, 21-Oct-2011.) |
Ref | Expression |
---|---|
atlex.b | β’ π΅ = (BaseβπΎ) |
atlex.l | β’ β€ = (leβπΎ) |
atlex.z | β’ 0 = (0.βπΎ) |
atlex.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
atlex | β’ ((πΎ β AtLat β§ π β π΅ β§ π β 0 ) β βπ¦ β π΄ π¦ β€ π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atlex.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
2 | eqid 2733 | . . . . 5 β’ (glbβπΎ) = (glbβπΎ) | |
3 | atlex.l | . . . . 5 β’ β€ = (leβπΎ) | |
4 | atlex.z | . . . . 5 β’ 0 = (0.βπΎ) | |
5 | atlex.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
6 | 1, 2, 3, 4, 5 | isatl 37807 | . . . 4 β’ (πΎ β AtLat β (πΎ β Lat β§ π΅ β dom (glbβπΎ) β§ βπ₯ β π΅ (π₯ β 0 β βπ¦ β π΄ π¦ β€ π₯))) |
7 | 6 | simp3bi 1148 | . . 3 β’ (πΎ β AtLat β βπ₯ β π΅ (π₯ β 0 β βπ¦ β π΄ π¦ β€ π₯)) |
8 | neeq1 3003 | . . . . 5 β’ (π₯ = π β (π₯ β 0 β π β 0 )) | |
9 | breq2 5110 | . . . . . 6 β’ (π₯ = π β (π¦ β€ π₯ β π¦ β€ π)) | |
10 | 9 | rexbidv 3172 | . . . . 5 β’ (π₯ = π β (βπ¦ β π΄ π¦ β€ π₯ β βπ¦ β π΄ π¦ β€ π)) |
11 | 8, 10 | imbi12d 345 | . . . 4 β’ (π₯ = π β ((π₯ β 0 β βπ¦ β π΄ π¦ β€ π₯) β (π β 0 β βπ¦ β π΄ π¦ β€ π))) |
12 | 11 | rspccv 3577 | . . 3 β’ (βπ₯ β π΅ (π₯ β 0 β βπ¦ β π΄ π¦ β€ π₯) β (π β π΅ β (π β 0 β βπ¦ β π΄ π¦ β€ π))) |
13 | 7, 12 | syl 17 | . 2 β’ (πΎ β AtLat β (π β π΅ β (π β 0 β βπ¦ β π΄ π¦ β€ π))) |
14 | 13 | 3imp 1112 | 1 β’ ((πΎ β AtLat β§ π β π΅ β§ π β 0 ) β βπ¦ β π΄ π¦ β€ π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2940 βwral 3061 βwrex 3070 class class class wbr 5106 dom cdm 5634 βcfv 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: atnle 37825 atlatmstc 37827 cvratlem 37930 cvrat4 37952 2llnmat 38033 2lnat 38293 |
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