Step | Hyp | Ref
| Expression |
1 | | 4thatlem0.c |
. . 3
β’ πΆ = ((π β¨ π) β§ (π β¨ π)) |
2 | | 4thatlem.ph |
. . . . 5
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π
β π΄ β§ Β¬ π
β€ π β§ (π β¨ π
) = (π β¨ π
)) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) |
3 | 2 | 4atexlemkl 38916 |
. . . 4
β’ (π β πΎ β Lat) |
4 | | 4thatlem0.j |
. . . . 5
β’ β¨ =
(joinβπΎ) |
5 | | 4thatlem0.a |
. . . . 5
β’ π΄ = (AtomsβπΎ) |
6 | 2, 4, 5 | 4atexlemqtb 38920 |
. . . 4
β’ (π β (π β¨ π) β (BaseβπΎ)) |
7 | 2, 4, 5 | 4atexlempsb 38919 |
. . . 4
β’ (π β (π β¨ π) β (BaseβπΎ)) |
8 | | eqid 2732 |
. . . . 5
β’
(BaseβπΎ) =
(BaseβπΎ) |
9 | | 4thatlem0.m |
. . . . 5
β’ β§ =
(meetβπΎ) |
10 | 8, 9 | latmcom 18412 |
. . . 4
β’ ((πΎ β Lat β§ (π β¨ π) β (BaseβπΎ) β§ (π β¨ π) β (BaseβπΎ)) β ((π β¨ π) β§ (π β¨ π)) = ((π β¨ π) β§ (π β¨ π))) |
11 | 3, 6, 7, 10 | syl3anc 1371 |
. . 3
β’ (π β ((π β¨ π) β§ (π β¨ π)) = ((π β¨ π) β§ (π β¨ π))) |
12 | 1, 11 | eqtrid 2784 |
. 2
β’ (π β πΆ = ((π β¨ π) β§ (π β¨ π))) |
13 | 2 | 4atexlemk 38906 |
. . 3
β’ (π β πΎ β HL) |
14 | 2 | 4atexlemp 38909 |
. . 3
β’ (π β π β π΄) |
15 | 2 | 4atexlems 38911 |
. . 3
β’ (π β π β π΄) |
16 | 2 | 4atexlemq 38910 |
. . 3
β’ (π β π β π΄) |
17 | 2 | 4atexlemt 38912 |
. . 3
β’ (π β π β π΄) |
18 | | 4thatlem0.l |
. . . 4
β’ β€ =
(leβπΎ) |
19 | 2, 18, 4, 5 | 4atexlempns 38921 |
. . 3
β’ (π β π β π) |
20 | | 4thatlem0.h |
. . . . 5
β’ π» = (LHypβπΎ) |
21 | | 4thatlem0.u |
. . . . 5
β’ π = ((π β¨ π) β§ π) |
22 | | 4thatlem0.v |
. . . . 5
β’ π = ((π β¨ π) β§ π) |
23 | 2, 18, 4, 9, 5, 20,
21, 22 | 4atexlemntlpq 38927 |
. . . 4
β’ (π β Β¬ π β€ (π β¨ π)) |
24 | 18, 4, 5 | atnlej2 38239 |
. . . . 5
β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ Β¬ π β€ (π β¨ π)) β π β π) |
25 | 24 | necomd 2996 |
. . . 4
β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ Β¬ π β€ (π β¨ π)) β π β π) |
26 | 13, 17, 14, 16, 23, 25 | syl131anc 1383 |
. . 3
β’ (π β π β π) |
27 | 2 | 4atexlempnq 38914 |
. . . 4
β’ (π β π β π) |
28 | 2 | 4atexlemnslpq 38915 |
. . . 4
β’ (π β Β¬ π β€ (π β¨ π)) |
29 | 18, 4, 5 | 4atlem0ae 38453 |
. . . 4
β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ (π β π β§ Β¬ π β€ (π β¨ π))) β Β¬ π β€ (π β¨ π)) |
30 | 13, 14, 16, 15, 27, 28, 29 | syl132anc 1388 |
. . 3
β’ (π β Β¬ π β€ (π β¨ π)) |
31 | 8, 5 | atbase 38147 |
. . . . 5
β’ (π β π΄ β π β (BaseβπΎ)) |
32 | 17, 31 | syl 17 |
. . . 4
β’ (π β π β (BaseβπΎ)) |
33 | 2, 18, 4, 9, 5, 20,
21 | 4atexlemu 38923 |
. . . . 5
β’ (π β π β π΄) |
34 | 2, 18, 4, 9, 5, 20,
21, 22 | 4atexlemv 38924 |
. . . . 5
β’ (π β π β π΄) |
35 | 8, 4, 5 | hlatjcl 38225 |
. . . . 5
β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π β¨ π) β (BaseβπΎ)) |
36 | 13, 33, 34, 35 | syl3anc 1371 |
. . . 4
β’ (π β (π β¨ π) β (BaseβπΎ)) |
37 | 8, 5 | atbase 38147 |
. . . . . 6
β’ (π β π΄ β π β (BaseβπΎ)) |
38 | 16, 37 | syl 17 |
. . . . 5
β’ (π β π β (BaseβπΎ)) |
39 | 8, 4 | latjcl 18388 |
. . . . 5
β’ ((πΎ β Lat β§ (π β¨ π) β (BaseβπΎ) β§ π β (BaseβπΎ)) β ((π β¨ π) β¨ π) β (BaseβπΎ)) |
40 | 3, 7, 38, 39 | syl3anc 1371 |
. . . 4
β’ (π β ((π β¨ π) β¨ π) β (BaseβπΎ)) |
41 | 2 | 4atexlemkc 38917 |
. . . . 5
β’ (π β πΎ β CvLat) |
42 | 2, 18, 4, 9, 5, 20,
21, 22 | 4atexlemunv 38925 |
. . . . 5
β’ (π β π β π) |
43 | 2 | 4atexlemutvt 38913 |
. . . . 5
β’ (π β (π β¨ π) = (π β¨ π)) |
44 | 5, 18, 4 | cvlsupr4 38203 |
. . . . 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 β¨ π)) |
45 | 41, 33, 34, 17, 42, 43, 44 | syl132anc 1388 |
. . . 4
β’ (π β π β€ (π β¨ π)) |
46 | 8, 4, 5 | hlatjcl 38225 |
. . . . . . . . 9
β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π β¨ π) β (BaseβπΎ)) |
47 | 13, 14, 16, 46 | syl3anc 1371 |
. . . . . . . 8
β’ (π β (π β¨ π) β (BaseβπΎ)) |
48 | 2, 20 | 4atexlemwb 38918 |
. . . . . . . 8
β’ (π β π β (BaseβπΎ)) |
49 | 8, 18, 9 | latmle1 18413 |
. . . . . . . 8
β’ ((πΎ β Lat β§ (π β¨ π) β (BaseβπΎ) β§ π β (BaseβπΎ)) β ((π β¨ π) β§ π) β€ (π β¨ π)) |
50 | 3, 47, 48, 49 | syl3anc 1371 |
. . . . . . 7
β’ (π β ((π β¨ π) β§ π) β€ (π β¨ π)) |
51 | 21, 50 | eqbrtrid 5182 |
. . . . . 6
β’ (π β π β€ (π β¨ π)) |
52 | 8, 18, 9 | latmle1 18413 |
. . . . . . . 8
β’ ((πΎ β Lat β§ (π β¨ π) β (BaseβπΎ) β§ π β (BaseβπΎ)) β ((π β¨ π) β§ π) β€ (π β¨ π)) |
53 | 3, 7, 48, 52 | syl3anc 1371 |
. . . . . . 7
β’ (π β ((π β¨ π) β§ π) β€ (π β¨ π)) |
54 | 22, 53 | eqbrtrid 5182 |
. . . . . 6
β’ (π β π β€ (π β¨ π)) |
55 | 8, 5 | atbase 38147 |
. . . . . . . 8
β’ (π β π΄ β π β (BaseβπΎ)) |
56 | 33, 55 | syl 17 |
. . . . . . 7
β’ (π β π β (BaseβπΎ)) |
57 | 8, 5 | atbase 38147 |
. . . . . . . 8
β’ (π β π΄ β π β (BaseβπΎ)) |
58 | 34, 57 | syl 17 |
. . . . . . 7
β’ (π β π β (BaseβπΎ)) |
59 | 8, 18, 4 | latjlej12 18404 |
. . . . . . 7
β’ ((πΎ β Lat β§ (π β (BaseβπΎ) β§ (π β¨ π) β (BaseβπΎ)) β§ (π β (BaseβπΎ) β§ (π β¨ π) β (BaseβπΎ))) β ((π β€ (π β¨ π) β§ π β€ (π β¨ π)) β (π β¨ π) β€ ((π β¨ π) β¨ (π β¨ π)))) |
60 | 3, 56, 47, 58, 7, 59 | syl122anc 1379 |
. . . . . 6
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)))) |
61 | 51, 54, 60 | mp2and 697 |
. . . . 5
β’ (π β (π β¨ π) β€ ((π β¨ π) β¨ (π β¨ π))) |
62 | 4, 5 | hlatjass 38228 |
. . . . . . 7
β’ ((πΎ β HL β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β ((π β¨ π) β¨ π) = (π β¨ (π β¨ π))) |
63 | 13, 14, 16, 15, 62 | syl13anc 1372 |
. . . . . 6
β’ (π β ((π β¨ π) β¨ π) = (π β¨ (π β¨ π))) |
64 | 8, 5 | atbase 38147 |
. . . . . . . 8
β’ (π β π΄ β π β (BaseβπΎ)) |
65 | 14, 64 | syl 17 |
. . . . . . 7
β’ (π β π β (BaseβπΎ)) |
66 | 8, 5 | atbase 38147 |
. . . . . . . 8
β’ (π β π΄ β π β (BaseβπΎ)) |
67 | 15, 66 | syl 17 |
. . . . . . 7
β’ (π β π β (BaseβπΎ)) |
68 | 8, 4 | latj32 18434 |
. . . . . . 7
β’ ((πΎ β Lat β§ (π β (BaseβπΎ) β§ π β (BaseβπΎ) β§ π β (BaseβπΎ))) β ((π β¨ π) β¨ π) = ((π β¨ π) β¨ π)) |
69 | 3, 65, 38, 67, 68 | syl13anc 1372 |
. . . . . 6
β’ (π β ((π β¨ π) β¨ π) = ((π β¨ π) β¨ π)) |
70 | 8, 4 | latjjdi 18440 |
. . . . . . 7
β’ ((πΎ β Lat β§ (π β (BaseβπΎ) β§ π β (BaseβπΎ) β§ π β (BaseβπΎ))) β (π β¨ (π β¨ π)) = ((π β¨ π) β¨ (π β¨ π))) |
71 | 3, 65, 38, 67, 70 | syl13anc 1372 |
. . . . . 6
β’ (π β (π β¨ (π β¨ π)) = ((π β¨ π) β¨ (π β¨ π))) |
72 | 63, 69, 71 | 3eqtr3rd 2781 |
. . . . 5
β’ (π β ((π β¨ π) β¨ (π β¨ π)) = ((π β¨ π) β¨ π)) |
73 | 61, 72 | breqtrd 5173 |
. . . 4
β’ (π β (π β¨ π) β€ ((π β¨ π) β¨ π)) |
74 | 8, 18, 3, 32, 36, 40, 45, 73 | lattrd 18395 |
. . 3
β’ (π β π β€ ((π β¨ π) β¨ π)) |
75 | 18, 4, 9, 5 | 2atmat 38420 |
. . 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 π β§ Β¬ π β€ (π β¨ π) β§ π β€ ((π β¨ π) β¨ π))) β ((π β¨ π) β§ (π β¨ π)) β π΄) |
76 | 13, 14, 15, 16, 17, 19, 26, 30, 74, 75 | syl333anc 1402 |
. 2
β’ (π β ((π β¨ π) β§ (π β¨ π)) β π΄) |
77 | 12, 76 | eqeltrd 2833 |
1
β’ (π β πΆ β π΄) |