Step | Hyp | Ref
| Expression |
1 | | dia2dimlem5.k |
. . . . 5
β’ (π β (πΎ β HL β§ π β π»)) |
2 | | dia2dimlem5.d |
. . . . 5
β’ (π β π· β π) |
3 | | dia2dimlem5.g |
. . . . 5
β’ (π β πΊ β π) |
4 | | dia2dimlem5.h |
. . . . . 6
β’ π» = (LHypβπΎ) |
5 | | dia2dimlem5.t |
. . . . . 6
β’ π = ((LTrnβπΎ)βπ) |
6 | | dia2dimlem5.y |
. . . . . 6
β’ π = ((DVecAβπΎ)βπ) |
7 | | eqid 2737 |
. . . . . 6
β’
(+gβπ) = (+gβπ) |
8 | 4, 5, 6, 7 | dvavadd 39481 |
. . . . 5
β’ (((πΎ β HL β§ π β π») β§ (π· β π β§ πΊ β π)) β (π·(+gβπ)πΊ) = (π· β πΊ)) |
9 | 1, 2, 3, 8 | syl12anc 836 |
. . . 4
β’ (π β (π·(+gβπ)πΊ) = (π· β πΊ)) |
10 | | dia2dimlem5.l |
. . . . 5
β’ β€ =
(leβπΎ) |
11 | | dia2dimlem5.a |
. . . . 5
β’ π΄ = (AtomsβπΎ) |
12 | | dia2dimlem5.p |
. . . . 5
β’ (π β (π β π΄ β§ Β¬ π β€ π)) |
13 | | dia2dimlem5.f |
. . . . . 6
β’ (π β (πΉ β π β§ (πΉβπ) β π)) |
14 | 13 | simpld 496 |
. . . . 5
β’ (π β πΉ β π) |
15 | | dia2dimlem5.gv |
. . . . 5
β’ (π β (πΊβπ) = π) |
16 | | dia2dimlem5.dv |
. . . . 5
β’ (π β (π·βπ) = (πΉβπ)) |
17 | 10, 11, 4, 5, 1, 12,
14, 3, 15, 2, 16 | dia2dimlem4 39533 |
. . . 4
β’ (π β (π· β πΊ) = πΉ) |
18 | 9, 17 | eqtr2d 2778 |
. . 3
β’ (π β πΉ = (π·(+gβπ)πΊ)) |
19 | 4, 6 | dvalvec 39492 |
. . . . . . 7
β’ ((πΎ β HL β§ π β π») β π β LVec) |
20 | | lveclmod 20570 |
. . . . . . 7
β’ (π β LVec β π β LMod) |
21 | 1, 19, 20 | 3syl 18 |
. . . . . 6
β’ (π β π β LMod) |
22 | | dia2dimlem5.s |
. . . . . . 7
β’ π = (LSubSpβπ) |
23 | 22 | lsssssubg 20422 |
. . . . . 6
β’ (π β LMod β π β (SubGrpβπ)) |
24 | 21, 23 | syl 17 |
. . . . 5
β’ (π β π β (SubGrpβπ)) |
25 | | dia2dimlem5.v |
. . . . . . . 8
β’ (π β (π β π΄ β§ π β€ π)) |
26 | 25 | simpld 496 |
. . . . . . 7
β’ (π β π β π΄) |
27 | | eqid 2737 |
. . . . . . . 8
β’
(BaseβπΎ) =
(BaseβπΎ) |
28 | 27, 11 | atbase 37754 |
. . . . . . 7
β’ (π β π΄ β π β (BaseβπΎ)) |
29 | 26, 28 | syl 17 |
. . . . . 6
β’ (π β π β (BaseβπΎ)) |
30 | 25 | simprd 497 |
. . . . . 6
β’ (π β π β€ π) |
31 | | dia2dimlem5.i |
. . . . . . 7
β’ πΌ = ((DIsoAβπΎ)βπ) |
32 | 27, 10, 4, 6, 31, 22 | dialss 39512 |
. . . . . 6
β’ (((πΎ β HL β§ π β π») β§ (π β (BaseβπΎ) β§ π β€ π)) β (πΌβπ) β π) |
33 | 1, 29, 30, 32 | syl12anc 836 |
. . . . 5
β’ (π β (πΌβπ) β π) |
34 | 24, 33 | sseldd 3946 |
. . . 4
β’ (π β (πΌβπ) β (SubGrpβπ)) |
35 | | dia2dimlem5.u |
. . . . . . . 8
β’ (π β (π β π΄ β§ π β€ π)) |
36 | 35 | simpld 496 |
. . . . . . 7
β’ (π β π β π΄) |
37 | 27, 11 | atbase 37754 |
. . . . . . 7
β’ (π β π΄ β π β (BaseβπΎ)) |
38 | 36, 37 | syl 17 |
. . . . . 6
β’ (π β π β (BaseβπΎ)) |
39 | 35 | simprd 497 |
. . . . . 6
β’ (π β π β€ π) |
40 | 27, 10, 4, 6, 31, 22 | dialss 39512 |
. . . . . 6
β’ (((πΎ β HL β§ π β π») β§ (π β (BaseβπΎ) β§ π β€ π)) β (πΌβπ) β π) |
41 | 1, 38, 39, 40 | syl12anc 836 |
. . . . 5
β’ (π β (πΌβπ) β π) |
42 | 24, 41 | sseldd 3946 |
. . . 4
β’ (π β (πΌβπ) β (SubGrpβπ)) |
43 | | dia2dimlem5.r |
. . . . . . . 8
β’ π
= ((trLβπΎ)βπ) |
44 | | dia2dimlem5.n |
. . . . . . . 8
β’ π = (LSpanβπ) |
45 | 4, 5, 43, 6, 31, 44 | dia1dim2 39528 |
. . . . . . 7
β’ (((πΎ β HL β§ π β π») β§ π· β π) β (πΌβ(π
βπ·)) = (πβ{π·})) |
46 | 1, 2, 45 | syl2anc 585 |
. . . . . 6
β’ (π β (πΌβ(π
βπ·)) = (πβ{π·})) |
47 | | dia2dimlem5.j |
. . . . . . . . . 10
β’ β¨ =
(joinβπΎ) |
48 | | dia2dimlem5.m |
. . . . . . . . . 10
β’ β§ =
(meetβπΎ) |
49 | | dia2dimlem5.q |
. . . . . . . . . 10
β’ π = ((π β¨ π) β§ ((πΉβπ) β¨ π)) |
50 | | dia2dimlem5.rf |
. . . . . . . . . 10
β’ (π β (π
βπΉ) β€ (π β¨ π)) |
51 | | dia2dimlem5.uv |
. . . . . . . . . 10
β’ (π β π β π) |
52 | | dia2dimlem5.ru |
. . . . . . . . . 10
β’ (π β (π
βπΉ) β π) |
53 | | dia2dimlem5.rv |
. . . . . . . . . 10
β’ (π β (π
βπΉ) β π) |
54 | 10, 47, 48, 11, 4, 5, 43, 49, 1, 35, 25, 12, 13, 50, 51, 52, 53, 2, 16 | dia2dimlem3 39532 |
. . . . . . . . 9
β’ (π β (π
βπ·) = π) |
55 | 54 | fveq2d 6847 |
. . . . . . . 8
β’ (π β (πΌβ(π
βπ·)) = (πΌβπ)) |
56 | | eqss 3960 |
. . . . . . . 8
β’ ((πΌβ(π
βπ·)) = (πΌβπ) β ((πΌβ(π
βπ·)) β (πΌβπ) β§ (πΌβπ) β (πΌβ(π
βπ·)))) |
57 | 55, 56 | sylib 217 |
. . . . . . 7
β’ (π β ((πΌβ(π
βπ·)) β (πΌβπ) β§ (πΌβπ) β (πΌβ(π
βπ·)))) |
58 | 57 | simpld 496 |
. . . . . 6
β’ (π β (πΌβ(π
βπ·)) β (πΌβπ)) |
59 | 46, 58 | eqsstrrd 3984 |
. . . . 5
β’ (π β (πβ{π·}) β (πΌβπ)) |
60 | | eqid 2737 |
. . . . . 6
β’
(Baseβπ) =
(Baseβπ) |
61 | 4, 5, 6, 60 | dvavbase 39479 |
. . . . . . . 8
β’ ((πΎ β HL β§ π β π») β (Baseβπ) = π) |
62 | 1, 61 | syl 17 |
. . . . . . 7
β’ (π β (Baseβπ) = π) |
63 | 2, 62 | eleqtrrd 2841 |
. . . . . 6
β’ (π β π· β (Baseβπ)) |
64 | 60, 22, 44, 21, 33, 63 | lspsnel5 20459 |
. . . . 5
β’ (π β (π· β (πΌβπ) β (πβ{π·}) β (πΌβπ))) |
65 | 59, 64 | mpbird 257 |
. . . 4
β’ (π β π· β (πΌβπ)) |
66 | 4, 5, 43, 6, 31, 44 | dia1dim2 39528 |
. . . . . . 7
β’ (((πΎ β HL β§ π β π») β§ πΊ β π) β (πΌβ(π
βπΊ)) = (πβ{πΊ})) |
67 | 1, 3, 66 | syl2anc 585 |
. . . . . 6
β’ (π β (πΌβ(π
βπΊ)) = (πβ{πΊ})) |
68 | 10, 47, 48, 11, 4, 5, 43, 49, 1, 35, 25, 12, 13, 50, 53, 3, 15 | dia2dimlem2 39531 |
. . . . . . . . 9
β’ (π β (π
βπΊ) = π) |
69 | 68 | fveq2d 6847 |
. . . . . . . 8
β’ (π β (πΌβ(π
βπΊ)) = (πΌβπ)) |
70 | | eqss 3960 |
. . . . . . . 8
β’ ((πΌβ(π
βπΊ)) = (πΌβπ) β ((πΌβ(π
βπΊ)) β (πΌβπ) β§ (πΌβπ) β (πΌβ(π
βπΊ)))) |
71 | 69, 70 | sylib 217 |
. . . . . . 7
β’ (π β ((πΌβ(π
βπΊ)) β (πΌβπ) β§ (πΌβπ) β (πΌβ(π
βπΊ)))) |
72 | 71 | simpld 496 |
. . . . . 6
β’ (π β (πΌβ(π
βπΊ)) β (πΌβπ)) |
73 | 67, 72 | eqsstrrd 3984 |
. . . . 5
β’ (π β (πβ{πΊ}) β (πΌβπ)) |
74 | 3, 62 | eleqtrrd 2841 |
. . . . . 6
β’ (π β πΊ β (Baseβπ)) |
75 | 60, 22, 44, 21, 41, 74 | lspsnel5 20459 |
. . . . 5
β’ (π β (πΊ β (πΌβπ) β (πβ{πΊ}) β (πΌβπ))) |
76 | 73, 75 | mpbird 257 |
. . . 4
β’ (π β πΊ β (πΌβπ)) |
77 | | dia2dimlem5.pl |
. . . . 5
β’ β =
(LSSumβπ) |
78 | 7, 77 | lsmelvali 19433 |
. . . 4
β’ ((((πΌβπ) β (SubGrpβπ) β§ (πΌβπ) β (SubGrpβπ)) β§ (π· β (πΌβπ) β§ πΊ β (πΌβπ))) β (π·(+gβπ)πΊ) β ((πΌβπ) β (πΌβπ))) |
79 | 34, 42, 65, 76, 78 | syl22anc 838 |
. . 3
β’ (π β (π·(+gβπ)πΊ) β ((πΌβπ) β (πΌβπ))) |
80 | 18, 79 | eqeltrd 2838 |
. 2
β’ (π β πΉ β ((πΌβπ) β (πΌβπ))) |
81 | | lmodabl 20372 |
. . . 4
β’ (π β LMod β π β Abel) |
82 | 21, 81 | syl 17 |
. . 3
β’ (π β π β Abel) |
83 | 77 | lsmcom 19637 |
. . 3
β’ ((π β Abel β§ (πΌβπ) β (SubGrpβπ) β§ (πΌβπ) β (SubGrpβπ)) β ((πΌβπ) β (πΌβπ)) = ((πΌβπ) β (πΌβπ))) |
84 | 82, 34, 42, 83 | syl3anc 1372 |
. 2
β’ (π β ((πΌβπ) β (πΌβπ)) = ((πΌβπ) β (πΌβπ))) |
85 | 80, 84 | eleqtrd 2840 |
1
β’ (π β πΉ β ((πΌβπ) β (πΌβπ))) |