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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlatjcl | Structured version Visualization version GIF version |
Description: Closure of join operation. Frequently-used special case of latjcl 18333 for atoms. (Contributed by NM, 15-Jun-2012.) |
Ref | Expression |
---|---|
hlatjcl.b | β’ π΅ = (BaseβπΎ) |
hlatjcl.j | β’ β¨ = (joinβπΎ) |
hlatjcl.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
hlatjcl | β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π β¨ π) β π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hllat 37871 | . 2 β’ (πΎ β HL β πΎ β Lat) | |
2 | hlatjcl.b | . . 3 β’ π΅ = (BaseβπΎ) | |
3 | hlatjcl.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
4 | 2, 3 | atbase 37797 | . 2 β’ (π β π΄ β π β π΅) |
5 | 2, 3 | atbase 37797 | . 2 β’ (π β π΄ β π β π΅) |
6 | hlatjcl.j | . . 3 β’ β¨ = (joinβπΎ) | |
7 | 2, 6 | latjcl 18333 | . 2 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β¨ π) β π΅) |
8 | 1, 4, 5, 7 | syl3an 1161 | 1 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π β¨ π) β π΅) |
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