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Mirrors > Home > MPE Home > Th. List > Mathboxes > atl0dm | Structured version Visualization version GIF version |
Description: Condition necessary for zero element to exist. (Contributed by NM, 14-Sep-2018.) |
Ref | Expression |
---|---|
atl01dm.b | β’ π΅ = (BaseβπΎ) |
atl01dm.u | β’ π = (lubβπΎ) |
atl01dm.g | β’ πΊ = (glbβπΎ) |
Ref | Expression |
---|---|
atl0dm | β’ (πΎ β AtLat β π΅ β dom πΊ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atl01dm.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | atl01dm.g | . . 3 β’ πΊ = (glbβπΎ) | |
3 | eqid 2728 | . . 3 β’ (leβπΎ) = (leβπΎ) | |
4 | eqid 2728 | . . 3 β’ (0.βπΎ) = (0.βπΎ) | |
5 | eqid 2728 | . . 3 β’ (AtomsβπΎ) = (AtomsβπΎ) | |
6 | 1, 2, 3, 4, 5 | isatl 38771 | . 2 β’ (πΎ β AtLat β (πΎ β Lat β§ π΅ β dom πΊ β§ βπ₯ β π΅ (π₯ β (0.βπΎ) β βπ¦ β (AtomsβπΎ)π¦(leβπΎ)π₯))) |
7 | 6 | simp2bi 1144 | 1 β’ (πΎ β AtLat β π΅ β dom πΊ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 β wne 2937 βwral 3058 βwrex 3067 class class class wbr 5148 dom cdm 5678 βcfv 6548 Basecbs 17180 lecple 17240 lubclub 18301 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: atl0cl 38775 atl0le 38776 |
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