Step | Hyp | Ref
| Expression |
1 | | dia2dimlem7.k |
. . . . 5
β’ (π β (πΎ β HL β§ π β π»)) |
2 | | dia2dimlem7.f |
. . . . 5
β’ (π β πΉ β π) |
3 | | dia2dimlem7.p |
. . . . 5
β’ (π β (π β π΄ β§ Β¬ π β€ π)) |
4 | | eqid 2737 |
. . . . . 6
β’
(BaseβπΎ) =
(BaseβπΎ) |
5 | | dia2dimlem7.l |
. . . . . 6
β’ β€ =
(leβπΎ) |
6 | | dia2dimlem7.a |
. . . . . 6
β’ π΄ = (AtomsβπΎ) |
7 | | dia2dimlem7.h |
. . . . . 6
β’ π» = (LHypβπΎ) |
8 | | dia2dimlem7.t |
. . . . . 6
β’ π = ((LTrnβπΎ)βπ) |
9 | 4, 5, 6, 7, 8 | ltrnideq 38641 |
. . . . 5
β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ (π β π΄ β§ Β¬ π β€ π)) β (πΉ = ( I βΎ (BaseβπΎ)) β (πΉβπ) = π)) |
10 | 1, 2, 3, 9 | syl3anc 1372 |
. . . 4
β’ (π β (πΉ = ( I βΎ (BaseβπΎ)) β (πΉβπ) = π)) |
11 | | dia2dimlem7.y |
. . . . . . . 8
β’ π = ((DVecAβπΎ)βπ) |
12 | | eqid 2737 |
. . . . . . . 8
β’
(0gβπ) = (0gβπ) |
13 | 4, 7, 8, 11, 12 | dva0g 39493 |
. . . . . . 7
β’ ((πΎ β HL β§ π β π») β (0gβπ) = ( I βΎ
(BaseβπΎ))) |
14 | 1, 13 | syl 17 |
. . . . . 6
β’ (π β (0gβπ) = ( I βΎ
(BaseβπΎ))) |
15 | 7, 11 | dvalvec 39492 |
. . . . . . . 8
β’ ((πΎ β HL β§ π β π») β π β LVec) |
16 | | lveclmod 20570 |
. . . . . . . 8
β’ (π β LVec β π β LMod) |
17 | 1, 15, 16 | 3syl 18 |
. . . . . . 7
β’ (π β π β LMod) |
18 | | dia2dimlem7.u |
. . . . . . . . . . 11
β’ (π β (π β π΄ β§ π β€ π)) |
19 | 18 | simpld 496 |
. . . . . . . . . 10
β’ (π β π β π΄) |
20 | 4, 6 | atbase 37754 |
. . . . . . . . . 10
β’ (π β π΄ β π β (BaseβπΎ)) |
21 | 19, 20 | syl 17 |
. . . . . . . . 9
β’ (π β π β (BaseβπΎ)) |
22 | 18 | simprd 497 |
. . . . . . . . 9
β’ (π β π β€ π) |
23 | | dia2dimlem7.i |
. . . . . . . . . 10
β’ πΌ = ((DIsoAβπΎ)βπ) |
24 | | dia2dimlem7.s |
. . . . . . . . . 10
β’ π = (LSubSpβπ) |
25 | 4, 5, 7, 11, 23, 24 | dialss 39512 |
. . . . . . . . 9
β’ (((πΎ β HL β§ π β π») β§ (π β (BaseβπΎ) β§ π β€ π)) β (πΌβπ) β π) |
26 | 1, 21, 22, 25 | syl12anc 836 |
. . . . . . . 8
β’ (π β (πΌβπ) β π) |
27 | | dia2dimlem7.v |
. . . . . . . . . . 11
β’ (π β (π β π΄ β§ π β€ π)) |
28 | 27 | simpld 496 |
. . . . . . . . . 10
β’ (π β π β π΄) |
29 | 4, 6 | atbase 37754 |
. . . . . . . . . 10
β’ (π β π΄ β π β (BaseβπΎ)) |
30 | 28, 29 | syl 17 |
. . . . . . . . 9
β’ (π β π β (BaseβπΎ)) |
31 | 27 | simprd 497 |
. . . . . . . . 9
β’ (π β π β€ π) |
32 | 4, 5, 7, 11, 23, 24 | dialss 39512 |
. . . . . . . . 9
β’ (((πΎ β HL β§ π β π») β§ (π β (BaseβπΎ) β§ π β€ π)) β (πΌβπ) β π) |
33 | 1, 30, 31, 32 | syl12anc 836 |
. . . . . . . 8
β’ (π β (πΌβπ) β π) |
34 | | dia2dimlem7.pl |
. . . . . . . . 9
β’ β =
(LSSumβπ) |
35 | 24, 34 | lsmcl 20547 |
. . . . . . . 8
β’ ((π β LMod β§ (πΌβπ) β π β§ (πΌβπ) β π) β ((πΌβπ) β (πΌβπ)) β π) |
36 | 17, 26, 33, 35 | syl3anc 1372 |
. . . . . . 7
β’ (π β ((πΌβπ) β (πΌβπ)) β π) |
37 | 12, 24 | lss0cl 20410 |
. . . . . . 7
β’ ((π β LMod β§ ((πΌβπ) β (πΌβπ)) β π) β (0gβπ) β ((πΌβπ) β (πΌβπ))) |
38 | 17, 36, 37 | syl2anc 585 |
. . . . . 6
β’ (π β (0gβπ) β ((πΌβπ) β (πΌβπ))) |
39 | 14, 38 | eqeltrrd 2839 |
. . . . 5
β’ (π β ( I βΎ
(BaseβπΎ)) β
((πΌβπ) β (πΌβπ))) |
40 | | eleq1a 2833 |
. . . . 5
β’ (( I
βΎ (BaseβπΎ))
β ((πΌβπ) β (πΌβπ)) β (πΉ = ( I βΎ (BaseβπΎ)) β πΉ β ((πΌβπ) β (πΌβπ)))) |
41 | 39, 40 | syl 17 |
. . . 4
β’ (π β (πΉ = ( I βΎ (BaseβπΎ)) β πΉ β ((πΌβπ) β (πΌβπ)))) |
42 | 10, 41 | sylbird 260 |
. . 3
β’ (π β ((πΉβπ) = π β πΉ β ((πΌβπ) β (πΌβπ)))) |
43 | 42 | imp 408 |
. 2
β’ ((π β§ (πΉβπ) = π) β πΉ β ((πΌβπ) β (πΌβπ))) |
44 | | dia2dimlem7.j |
. . 3
β’ β¨ =
(joinβπΎ) |
45 | | dia2dimlem7.m |
. . 3
β’ β§ =
(meetβπΎ) |
46 | | dia2dimlem7.r |
. . 3
β’ π
= ((trLβπΎ)βπ) |
47 | | dia2dimlem7.n |
. . 3
β’ π = (LSpanβπ) |
48 | | dia2dimlem7.q |
. . 3
β’ π = ((π β¨ π) β§ ((πΉβπ) β¨ π)) |
49 | 1 | adantr 482 |
. . 3
β’ ((π β§ (πΉβπ) β π) β (πΎ β HL β§ π β π»)) |
50 | 18 | adantr 482 |
. . 3
β’ ((π β§ (πΉβπ) β π) β (π β π΄ β§ π β€ π)) |
51 | 27 | adantr 482 |
. . 3
β’ ((π β§ (πΉβπ) β π) β (π β π΄ β§ π β€ π)) |
52 | 3 | adantr 482 |
. . 3
β’ ((π β§ (πΉβπ) β π) β (π β π΄ β§ Β¬ π β€ π)) |
53 | 2 | anim1i 616 |
. . 3
β’ ((π β§ (πΉβπ) β π) β (πΉ β π β§ (πΉβπ) β π)) |
54 | | dia2dimlem7.rf |
. . . 4
β’ (π β (π
βπΉ) β€ (π β¨ π)) |
55 | 54 | adantr 482 |
. . 3
β’ ((π β§ (πΉβπ) β π) β (π
βπΉ) β€ (π β¨ π)) |
56 | | dia2dimlem7.uv |
. . . 4
β’ (π β π β π) |
57 | 56 | adantr 482 |
. . 3
β’ ((π β§ (πΉβπ) β π) β π β π) |
58 | | dia2dimlem7.ru |
. . . 4
β’ (π β (π
βπΉ) β π) |
59 | 58 | adantr 482 |
. . 3
β’ ((π β§ (πΉβπ) β π) β (π
βπΉ) β π) |
60 | | dia2dimlem7.rv |
. . . 4
β’ (π β (π
βπΉ) β π) |
61 | 60 | adantr 482 |
. . 3
β’ ((π β§ (πΉβπ) β π) β (π
βπΉ) β π) |
62 | 5, 44, 45, 6, 7, 8,
46, 11, 24, 34, 47, 23, 48, 49, 50, 51, 52, 53, 55, 57, 59, 61 | dia2dimlem6 39535 |
. 2
β’ ((π β§ (πΉβπ) β π) β πΉ β ((πΌβπ) β (πΌβπ))) |
63 | 43, 62 | pm2.61dane 3033 |
1
β’ (π β πΉ β ((πΌβπ) β (πΌβπ))) |