Step | Hyp | Ref
| Expression |
1 | | simpl1 1192 |
. . . 4
β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π
β π΄ β§ π β π΄)) β πΎ β HL) |
2 | 1 | hllatd 37855 |
. . 3
β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π
β π΄ β§ π β π΄)) β πΎ β Lat) |
3 | | eqid 2737 |
. . . . 5
β’
(BaseβπΎ) =
(BaseβπΎ) |
4 | | 4at.j |
. . . . 5
β’ β¨ =
(joinβπΎ) |
5 | | 4at.a |
. . . . 5
β’ π΄ = (AtomsβπΎ) |
6 | 3, 4, 5 | hlatjcl 37858 |
. . . 4
β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π β¨ π) β (BaseβπΎ)) |
7 | 6 | adantr 482 |
. . 3
β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π
β π΄ β§ π β π΄)) β (π β¨ π) β (BaseβπΎ)) |
8 | 3, 5 | atbase 37780 |
. . . 4
β’ (π
β π΄ β π
β (BaseβπΎ)) |
9 | 8 | ad2antrl 727 |
. . 3
β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π
β π΄ β§ π β π΄)) β π
β (BaseβπΎ)) |
10 | 3, 5 | atbase 37780 |
. . . 4
β’ (π β π΄ β π β (BaseβπΎ)) |
11 | 10 | ad2antll 728 |
. . 3
β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π
β π΄ β§ π β π΄)) β π β (BaseβπΎ)) |
12 | 3, 4 | latjass 18379 |
. . 3
β’ ((πΎ β Lat β§ ((π β¨ π) β (BaseβπΎ) β§ π
β (BaseβπΎ) β§ π β (BaseβπΎ))) β (((π β¨ π) β¨ π
) β¨ π) = ((π β¨ π) β¨ (π
β¨ π))) |
13 | 2, 7, 9, 11, 12 | syl13anc 1373 |
. 2
β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π
β π΄ β§ π β π΄)) β (((π β¨ π) β¨ π
) β¨ π) = ((π β¨ π) β¨ (π
β¨ π))) |
14 | 3, 4 | latjcl 18335 |
. . . 4
β’ ((πΎ β Lat β§ (π β¨ π) β (BaseβπΎ) β§ π
β (BaseβπΎ)) β ((π β¨ π) β¨ π
) β (BaseβπΎ)) |
15 | 2, 7, 9, 14 | syl3anc 1372 |
. . 3
β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π
β π΄ β§ π β π΄)) β ((π β¨ π) β¨ π
) β (BaseβπΎ)) |
16 | 3, 4 | latjcom 18343 |
. . 3
β’ ((πΎ β Lat β§ ((π β¨ π) β¨ π
) β (BaseβπΎ) β§ π β (BaseβπΎ)) β (((π β¨ π) β¨ π
) β¨ π) = (π β¨ ((π β¨ π) β¨ π
))) |
17 | 2, 15, 11, 16 | syl3anc 1372 |
. 2
β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π
β π΄ β§ π β π΄)) β (((π β¨ π) β¨ π
) β¨ π) = (π β¨ ((π β¨ π) β¨ π
))) |
18 | 13, 17 | eqtr3d 2779 |
1
β’ (((πΎ β HL β§ π β π΄ β§ π β π΄) β§ (π
β π΄ β§ π β π΄)) β ((π β¨ π) β¨ (π
β¨ π)) = (π β¨ ((π β¨ π) β¨ π
))) |