Step | Hyp | Ref
| Expression |
1 | | cvllat 37834 |
. . . . . 6
β’ (πΎ β CvLat β πΎ β Lat) |
2 | 1 | 3ad2ant1 1134 |
. . . . 5
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 | | simp21 1207 |
. . . . . 6
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 |
. . . . . . 7
β’
(BaseβπΎ) =
(BaseβπΎ) |
5 | | cvlsupr5.a |
. . . . . . 7
β’ π΄ = (AtomsβπΎ) |
6 | 4, 5 | atbase 37797 |
. . . . . 6
β’ (π β π΄ β π β (BaseβπΎ)) |
7 | 3, 6 | syl 17 |
. . . . 5
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 |
. . . . . 6
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 |
. . . . . 6
β’ (π
β π΄ β π
β (BaseβπΎ)) |
10 | 8, 9 | syl 17 |
. . . . 5
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 | | eqid 2733 |
. . . . . 6
β’
(leβπΎ) =
(leβπΎ) |
12 | | cvlsupr5.j |
. . . . . 6
β’ β¨ =
(joinβπΎ) |
13 | 4, 11, 12 | latlej1 18342 |
. . . . 5
β’ ((πΎ β Lat β§ π β (BaseβπΎ) β§ π
β (BaseβπΎ)) β π(leβπΎ)(π β¨ π
)) |
14 | 2, 7, 10, 13 | syl3anc 1372 |
. . . 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(leβπΎ)(π β¨ π
)) |
15 | | simp3r 1203 |
. . . 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
) = (π β¨ π
)) |
16 | 14, 15 | breqtrd 5132 |
. . 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(leβπΎ)(π β¨ π
)) |
17 | | simp22 1208 |
. . . . 5
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΄) |
18 | 4, 5 | atbase 37797 |
. . . . 5
β’ (π β π΄ β π β (BaseβπΎ)) |
19 | 17, 18 | syl 17 |
. . . 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 (BaseβπΎ)) |
20 | 4, 12 | latjcom 18341 |
. . . 4
β’ ((πΎ β Lat β§ π β (BaseβπΎ) β§ π
β (BaseβπΎ)) β (π β¨ π
) = (π
β¨ π)) |
21 | 2, 19, 10, 20 | syl3anc 1372 |
. . 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¨ π
) = (π
β¨ π)) |
22 | 16, 21 | breqtrd 5132 |
. 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(leβπΎ)(π
β¨ π)) |
23 | | simp1 1137 |
. . 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 CvLat) |
24 | | simp3l 1202 |
. . 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 π) |
25 | 11, 12, 5 | cvlatexchb2 37843 |
. . 3
β’ ((πΎ β CvLat β§ (π β π΄ β§ π
β π΄ β§ π β π΄) β§ π β π) β (π(leβπΎ)(π
β¨ π) β (π β¨ π) = (π
β¨ π))) |
26 | 23, 3, 8, 17, 24, 25 | syl131anc 1384 |
. 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(leβπΎ)(π
β¨ π) β (π β¨ π) = (π
β¨ π))) |
27 | 22, 26 | mpbid 231 |
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)) |