Step | Hyp | Ref
| Expression |
1 | | cvlatexch.l |
. . 3
β’ β€ =
(leβπΎ) |
2 | | cvlatexch.j |
. . 3
β’ β¨ =
(joinβπΎ) |
3 | | cvlatexch.a |
. . 3
β’ π΄ = (AtomsβπΎ) |
4 | 1, 2, 3 | cvlatexchb1 37799 |
. 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
β¨ π) β (π
β¨ π) = (π
β¨ π))) |
5 | | cvllat 37791 |
. . . . 5
β’ (πΎ β CvLat β πΎ β Lat) |
6 | 5 | 3ad2ant1 1134 |
. . . 4
β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π
β π΄) β§ π β π
) β πΎ β Lat) |
7 | | 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΄) |
8 | | eqid 2737 |
. . . . . 6
β’
(BaseβπΎ) =
(BaseβπΎ) |
9 | 8, 3 | atbase 37754 |
. . . . 5
β’ (π β π΄ β π β (BaseβπΎ)) |
10 | 7, 9 | 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 (BaseβπΎ)) |
11 | | simp23 1209 |
. . . . 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΄) |
12 | 8, 3 | atbase 37754 |
. . . . 5
β’ (π
β π΄ β π
β (BaseβπΎ)) |
13 | 11, 12 | 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 (BaseβπΎ)) |
14 | 8, 2 | latjcom 18337 |
. . . 4
β’ ((πΎ β Lat β§ π β (BaseβπΎ) β§ π
β (BaseβπΎ)) β (π β¨ π
) = (π
β¨ π)) |
15 | 6, 10, 13, 14 | 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)) |
16 | 15 | breq2d 5118 |
. 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 β¨ π
) β π β€ (π
β¨ π))) |
17 | | simp21 1207 |
. . . . 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΄) |
18 | 8, 3 | atbase 37754 |
. . . . 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 (BaseβπΎ)) |
20 | 8, 2 | latjcom 18337 |
. . . 4
β’ ((πΎ β Lat β§ π β (BaseβπΎ) β§ π
β (BaseβπΎ)) β (π β¨ π
) = (π
β¨ π)) |
21 | 6, 19, 13, 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)) |
22 | 21, 15 | eqeq12d 2753 |
. 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))) |
23 | 4, 16, 22 | 3bitr4d 311 |
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 β¨ π
) β (π β¨ π
) = (π β¨ π
))) |