Step | Hyp | Ref
| Expression |
1 | | cdleme0.u |
. . 3
β’ π = ((π β¨ π) β§ π) |
2 | 1 | oveq2i 7369 |
. 2
β’ (π β¨ π) = (π β¨ ((π β¨ π) β§ π)) |
3 | | simpll 766 |
. . . 4
β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β π΄)) β πΎ β HL) |
4 | | simprll 778 |
. . . 4
β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β π΄)) β π β π΄) |
5 | | hllat 37828 |
. . . . . 6
β’ (πΎ β HL β πΎ β Lat) |
6 | 5 | ad2antrr 725 |
. . . . 5
β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β π΄)) β πΎ β Lat) |
7 | | eqid 2737 |
. . . . . . 7
β’
(BaseβπΎ) =
(BaseβπΎ) |
8 | | cdleme0.a |
. . . . . . 7
β’ π΄ = (AtomsβπΎ) |
9 | 7, 8 | atbase 37754 |
. . . . . 6
β’ (π β π΄ β π β (BaseβπΎ)) |
10 | 4, 9 | syl 17 |
. . . . 5
β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β π΄)) β π β (BaseβπΎ)) |
11 | | simprr 772 |
. . . . . 6
β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β π΄)) β π β π΄) |
12 | 7, 8 | atbase 37754 |
. . . . . 6
β’ (π β π΄ β π β (BaseβπΎ)) |
13 | 11, 12 | syl 17 |
. . . . 5
β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β π΄)) β π β (BaseβπΎ)) |
14 | | cdleme0.j |
. . . . . 6
β’ β¨ =
(joinβπΎ) |
15 | 7, 14 | latjcl 18329 |
. . . . 5
β’ ((πΎ β Lat β§ π β (BaseβπΎ) β§ π β (BaseβπΎ)) β (π β¨ π) β (BaseβπΎ)) |
16 | 6, 10, 13, 15 | syl3anc 1372 |
. . . 4
β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β π΄)) β (π β¨ π) β (BaseβπΎ)) |
17 | | cdleme0.h |
. . . . . 6
β’ π» = (LHypβπΎ) |
18 | 7, 17 | lhpbase 38464 |
. . . . 5
β’ (π β π» β π β (BaseβπΎ)) |
19 | 18 | ad2antlr 726 |
. . . 4
β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β π΄)) β π β (BaseβπΎ)) |
20 | | cdleme0.l |
. . . . . 6
β’ β€ =
(leβπΎ) |
21 | 20, 14, 8 | hlatlej1 37840 |
. . . . 5
β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β π β€ (π β¨ π)) |
22 | 3, 4, 11, 21 | syl3anc 1372 |
. . . 4
β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β π΄)) β π β€ (π β¨ π)) |
23 | | cdleme0.m |
. . . . 5
β’ β§ =
(meetβπΎ) |
24 | 7, 20, 14, 23, 8 | atmod3i1 38330 |
. . . 4
β’ ((πΎ β HL β§ (π β π΄ β§ (π β¨ π) β (BaseβπΎ) β§ π β (BaseβπΎ)) β§ π β€ (π β¨ π)) β (π β¨ ((π β¨ π) β§ π)) = ((π β¨ π) β§ (π β¨ π))) |
25 | 3, 4, 16, 19, 22, 24 | syl131anc 1384 |
. . 3
β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β π΄)) β (π β¨ ((π β¨ π) β§ π)) = ((π β¨ π) β§ (π β¨ π))) |
26 | | eqid 2737 |
. . . . . 6
β’
(1.βπΎ) =
(1.βπΎ) |
27 | 20, 14, 26, 8, 17 | lhpjat2 38487 |
. . . . 5
β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β (π β¨ π) = (1.βπΎ)) |
28 | 27 | adantrr 716 |
. . . 4
β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β π΄)) β (π β¨ π) = (1.βπΎ)) |
29 | 28 | oveq2d 7374 |
. . 3
β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β π΄)) β ((π β¨ π) β§ (π β¨ π)) = ((π β¨ π) β§ (1.βπΎ))) |
30 | | hlol 37826 |
. . . . 5
β’ (πΎ β HL β πΎ β OL) |
31 | 30 | ad2antrr 725 |
. . . 4
β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β π΄)) β πΎ β OL) |
32 | 7, 23, 26 | olm11 37692 |
. . . 4
β’ ((πΎ β OL β§ (π β¨ π) β (BaseβπΎ)) β ((π β¨ π) β§ (1.βπΎ)) = (π β¨ π)) |
33 | 31, 16, 32 | syl2anc 585 |
. . 3
β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β π΄)) β ((π β¨ π) β§ (1.βπΎ)) = (π β¨ π)) |
34 | 25, 29, 33 | 3eqtrd 2781 |
. 2
β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β π΄)) β (π β¨ ((π β¨ π) β§ π)) = (π β¨ π)) |
35 | 2, 34 | eqtrid 2789 |
1
β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β π΄)) β (π β¨ π) = (π β¨ π)) |