Step | Hyp | Ref
| Expression |
1 | | dia2dimlem9.k |
. . . . . . 7
β’ (π β (πΎ β HL β§ π β π»)) |
2 | | dia2dimlem9.h |
. . . . . . . 8
β’ π» = (LHypβπΎ) |
3 | | dia2dimlem9.y |
. . . . . . . 8
β’ π = ((DVecAβπΎ)βπ) |
4 | 2, 3 | dvalvec 39492 |
. . . . . . 7
β’ ((πΎ β HL β§ π β π») β π β LVec) |
5 | | lveclmod 20570 |
. . . . . . 7
β’ (π β LVec β π β LMod) |
6 | | dia2dimlem9.s |
. . . . . . . 8
β’ π = (LSubSpβπ) |
7 | 6 | lsssssubg 20422 |
. . . . . . 7
β’ (π β LMod β π β (SubGrpβπ)) |
8 | 1, 4, 5, 7 | 4syl 19 |
. . . . . 6
β’ (π β π β (SubGrpβπ)) |
9 | | dia2dimlem9.u |
. . . . . . . . 9
β’ (π β (π β π΄ β§ π β€ π)) |
10 | 9 | simpld 496 |
. . . . . . . 8
β’ (π β π β π΄) |
11 | | eqid 2737 |
. . . . . . . . 9
β’
(BaseβπΎ) =
(BaseβπΎ) |
12 | | dia2dimlem9.a |
. . . . . . . . 9
β’ π΄ = (AtomsβπΎ) |
13 | 11, 12 | atbase 37754 |
. . . . . . . 8
β’ (π β π΄ β π β (BaseβπΎ)) |
14 | 10, 13 | syl 17 |
. . . . . . 7
β’ (π β π β (BaseβπΎ)) |
15 | 9 | simprd 497 |
. . . . . . 7
β’ (π β π β€ π) |
16 | | dia2dimlem9.l |
. . . . . . . 8
β’ β€ =
(leβπΎ) |
17 | | dia2dimlem9.i |
. . . . . . . 8
β’ πΌ = ((DIsoAβπΎ)βπ) |
18 | 11, 16, 2, 3, 17, 6 | dialss 39512 |
. . . . . . 7
β’ (((πΎ β HL β§ π β π») β§ (π β (BaseβπΎ) β§ π β€ π)) β (πΌβπ) β π) |
19 | 1, 14, 15, 18 | syl12anc 836 |
. . . . . 6
β’ (π β (πΌβπ) β π) |
20 | 8, 19 | sseldd 3946 |
. . . . 5
β’ (π β (πΌβπ) β (SubGrpβπ)) |
21 | | dia2dimlem9.v |
. . . . . . . . 9
β’ (π β (π β π΄ β§ π β€ π)) |
22 | 21 | simpld 496 |
. . . . . . . 8
β’ (π β π β π΄) |
23 | 11, 12 | atbase 37754 |
. . . . . . . 8
β’ (π β π΄ β π β (BaseβπΎ)) |
24 | 22, 23 | syl 17 |
. . . . . . 7
β’ (π β π β (BaseβπΎ)) |
25 | 21 | simprd 497 |
. . . . . . 7
β’ (π β π β€ π) |
26 | 11, 16, 2, 3, 17, 6 | dialss 39512 |
. . . . . . 7
β’ (((πΎ β HL β§ π β π») β§ (π β (BaseβπΎ) β§ π β€ π)) β (πΌβπ) β π) |
27 | 1, 24, 25, 26 | syl12anc 836 |
. . . . . 6
β’ (π β (πΌβπ) β π) |
28 | 8, 27 | sseldd 3946 |
. . . . 5
β’ (π β (πΌβπ) β (SubGrpβπ)) |
29 | | dia2dimlem9.pl |
. . . . . 6
β’ β =
(LSSumβπ) |
30 | 29 | lsmub1 19440 |
. . . . 5
β’ (((πΌβπ) β (SubGrpβπ) β§ (πΌβπ) β (SubGrpβπ)) β (πΌβπ) β ((πΌβπ) β (πΌβπ))) |
31 | 20, 28, 30 | syl2anc 585 |
. . . 4
β’ (π β (πΌβπ) β ((πΌβπ) β (πΌβπ))) |
32 | 31 | adantr 482 |
. . 3
β’ ((π β§ (π
βπΉ) = π) β (πΌβπ) β ((πΌβπ) β (πΌβπ))) |
33 | | dia2dimlem9.f |
. . . . . 6
β’ (π β πΉ β π) |
34 | | dia2dimlem9.t |
. . . . . . 7
β’ π = ((LTrnβπΎ)βπ) |
35 | | dia2dimlem9.r |
. . . . . . 7
β’ π
= ((trLβπΎ)βπ) |
36 | 2, 34, 35, 17 | dia1dimid 39529 |
. . . . . 6
β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β πΉ β (πΌβ(π
βπΉ))) |
37 | 1, 33, 36 | syl2anc 585 |
. . . . 5
β’ (π β πΉ β (πΌβ(π
βπΉ))) |
38 | 37 | adantr 482 |
. . . 4
β’ ((π β§ (π
βπΉ) = π) β πΉ β (πΌβ(π
βπΉ))) |
39 | | fveq2 6843 |
. . . . 5
β’ ((π
βπΉ) = π β (πΌβ(π
βπΉ)) = (πΌβπ)) |
40 | 39 | adantl 483 |
. . . 4
β’ ((π β§ (π
βπΉ) = π) β (πΌβ(π
βπΉ)) = (πΌβπ)) |
41 | 38, 40 | eleqtrd 2840 |
. . 3
β’ ((π β§ (π
βπΉ) = π) β πΉ β (πΌβπ)) |
42 | 32, 41 | sseldd 3946 |
. 2
β’ ((π β§ (π
βπΉ) = π) β πΉ β ((πΌβπ) β (πΌβπ))) |
43 | 20 | adantr 482 |
. . . 4
β’ ((π β§ (π
βπΉ) = π) β (πΌβπ) β (SubGrpβπ)) |
44 | 28 | adantr 482 |
. . . 4
β’ ((π β§ (π
βπΉ) = π) β (πΌβπ) β (SubGrpβπ)) |
45 | 29 | lsmub2 19441 |
. . . 4
β’ (((πΌβπ) β (SubGrpβπ) β§ (πΌβπ) β (SubGrpβπ)) β (πΌβπ) β ((πΌβπ) β (πΌβπ))) |
46 | 43, 44, 45 | syl2anc 585 |
. . 3
β’ ((π β§ (π
βπΉ) = π) β (πΌβπ) β ((πΌβπ) β (πΌβπ))) |
47 | 37 | adantr 482 |
. . . 4
β’ ((π β§ (π
βπΉ) = π) β πΉ β (πΌβ(π
βπΉ))) |
48 | | fveq2 6843 |
. . . . 5
β’ ((π
βπΉ) = π β (πΌβ(π
βπΉ)) = (πΌβπ)) |
49 | 48 | adantl 483 |
. . . 4
β’ ((π β§ (π
βπΉ) = π) β (πΌβ(π
βπΉ)) = (πΌβπ)) |
50 | 47, 49 | eleqtrd 2840 |
. . 3
β’ ((π β§ (π
βπΉ) = π) β πΉ β (πΌβπ)) |
51 | 46, 50 | sseldd 3946 |
. 2
β’ ((π β§ (π
βπΉ) = π) β πΉ β ((πΌβπ) β (πΌβπ))) |
52 | | dia2dimlem9.j |
. . 3
β’ β¨ =
(joinβπΎ) |
53 | | dia2dimlem9.m |
. . 3
β’ β§ =
(meetβπΎ) |
54 | | dia2dimlem9.n |
. . 3
β’ π = (LSpanβπ) |
55 | 1 | adantr 482 |
. . 3
β’ ((π β§ ((π
βπΉ) β π β§ (π
βπΉ) β π)) β (πΎ β HL β§ π β π»)) |
56 | 9 | adantr 482 |
. . 3
β’ ((π β§ ((π
βπΉ) β π β§ (π
βπΉ) β π)) β (π β π΄ β§ π β€ π)) |
57 | 21 | adantr 482 |
. . 3
β’ ((π β§ ((π
βπΉ) β π β§ (π
βπΉ) β π)) β (π β π΄ β§ π β€ π)) |
58 | 33 | adantr 482 |
. . 3
β’ ((π β§ ((π
βπΉ) β π β§ (π
βπΉ) β π)) β πΉ β π) |
59 | | dia2dimlem9.rf |
. . . 4
β’ (π β (π
βπΉ) β€ (π β¨ π)) |
60 | 59 | adantr 482 |
. . 3
β’ ((π β§ ((π
βπΉ) β π β§ (π
βπΉ) β π)) β (π
βπΉ) β€ (π β¨ π)) |
61 | | dia2dimlem9.uv |
. . . 4
β’ (π β π β π) |
62 | 61 | adantr 482 |
. . 3
β’ ((π β§ ((π
βπΉ) β π β§ (π
βπΉ) β π)) β π β π) |
63 | | simprl 770 |
. . 3
β’ ((π β§ ((π
βπΉ) β π β§ (π
βπΉ) β π)) β (π
βπΉ) β π) |
64 | | simprr 772 |
. . 3
β’ ((π β§ ((π
βπΉ) β π β§ (π
βπΉ) β π)) β (π
βπΉ) β π) |
65 | 16, 52, 53, 12, 2, 34, 35, 3, 6, 29, 54, 17, 55, 56, 57, 58, 60, 62, 63, 64 | dia2dimlem8 39537 |
. 2
β’ ((π β§ ((π
βπΉ) β π β§ (π
βπΉ) β π)) β πΉ β ((πΌβπ) β (πΌβπ))) |
66 | 42, 51, 65 | pm2.61da2ne 3034 |
1
β’ (π β πΉ β ((πΌβπ) β (πΌβπ))) |