Step | Hyp | Ref
| Expression |
1 | | cvllat 37834 |
. . . 4
β’ (πΎ β CvLat β πΎ β Lat) |
2 | 1 | 3ad2ant1 1134 |
. . 3
β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π
β π΄) β§ (π β π β§ (π β¨ π
) = (π β¨ π
))) β πΎ β Lat) |
3 | | simp22 1208 |
. . . 4
β’ ((πΎ β CvLat 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΄) |
4 | | eqid 2733 |
. . . . 5
β’
(BaseβπΎ) =
(BaseβπΎ) |
5 | | cvlsupr5.a |
. . . . 5
β’ π΄ = (AtomsβπΎ) |
6 | 4, 5 | atbase 37797 |
. . . 4
β’ (π β π΄ β π β (BaseβπΎ)) |
7 | 3, 6 | syl 17 |
. . 3
β’ ((πΎ β CvLat 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 (BaseβπΎ)) |
8 | | simp23 1209 |
. . . 4
β’ ((πΎ β CvLat 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΄) |
9 | 4, 5 | atbase 37797 |
. . . 4
β’ (π
β π΄ β π
β (BaseβπΎ)) |
10 | 8, 9 | syl 17 |
. . 3
β’ ((πΎ β CvLat 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 (BaseβπΎ)) |
11 | | cvlsupr5.j |
. . . 4
β’ β¨ =
(joinβπΎ) |
12 | 4, 11 | latjcom 18341 |
. . 3
β’ ((πΎ β Lat β§ π β (BaseβπΎ) β§ π
β (BaseβπΎ)) β (π β¨ π
) = (π
β¨ π)) |
13 | 2, 7, 10, 12 | syl3anc 1372 |
. 2
β’ ((πΎ β CvLat 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
) = (π
β¨ π)) |
14 | | simp3r 1203 |
. 2
β’ ((πΎ β CvLat 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
) = (π β¨ π
)) |
15 | 5, 11 | cvlsupr7 37856 |
. 2
β’ ((πΎ β CvLat 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) = (π
β¨ π)) |
16 | 13, 14, 15 | 3eqtr4rd 2784 |
1
β’ ((πΎ β CvLat 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) = (π β¨ π
)) |