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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 2724 | . . 3 β’ (leβπΎ) = (leβπΎ) | |
4 | eqid 2724 | . . 3 β’ (0.βπΎ) = (0.βπΎ) | |
5 | eqid 2724 | . . 3 β’ (AtomsβπΎ) = (AtomsβπΎ) | |
6 | 1, 2, 3, 4, 5 | isatl 38673 | . 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 1143 | 1 β’ (πΎ β AtLat β π΅ β dom πΊ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wne 2932 βwral 3053 βwrex 3062 class class class wbr 5139 dom cdm 5667 βcfv 6534 Basecbs 17149 lecple 17209 lubclub 18270 glbcglb 18271 0.cp0 18384 Latclat 18392 Atomscatm 38637 AtLatcal 38638 |
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 2695 |
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 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-dm 5677 df-iota 6486 df-fv 6542 df-atl 38672 |
This theorem is referenced by: atl0cl 38677 atl0le 38678 |
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