Step | Hyp | Ref
| Expression |
1 | | dochsat.h |
. . . . . . . . 9
β’ π» = (LHypβπΎ) |
2 | | dochsat.u |
. . . . . . . . 9
β’ π = ((DVecHβπΎ)βπ) |
3 | | dochsat.k |
. . . . . . . . 9
β’ (π β (πΎ β HL β§ π β π»)) |
4 | 1, 2, 3 | dvhlmod 39976 |
. . . . . . . 8
β’ (π β π β LMod) |
5 | 4 | adantr 481 |
. . . . . . 7
β’ ((π β§ ( β₯ β( β₯
βπ)) β π΄) β π β LMod) |
6 | | dochsat.q |
. . . . . . . 8
β’ (π β π β π) |
7 | 6 | adantr 481 |
. . . . . . 7
β’ ((π β§ ( β₯ β( β₯
βπ)) β π΄) β π β π) |
8 | | eqid 2732 |
. . . . . . . 8
β’
(0gβπ) = (0gβπ) |
9 | | dochsat.s |
. . . . . . . 8
β’ π = (LSubSpβπ) |
10 | 8, 9 | lss0ss 20558 |
. . . . . . 7
β’ ((π β LMod β§ π β π) β {(0gβπ)} β π) |
11 | 5, 7, 10 | syl2anc 584 |
. . . . . 6
β’ ((π β§ ( β₯ β( β₯
βπ)) β π΄) β
{(0gβπ)}
β π) |
12 | | dochsat.a |
. . . . . . . . 9
β’ π΄ = (LSAtomsβπ) |
13 | | simpr 485 |
. . . . . . . . 9
β’ ((π β§ ( β₯ β( β₯
βπ)) β π΄) β ( β₯ β( β₯
βπ)) β π΄) |
14 | 8, 12, 5, 13 | lsatn0 37864 |
. . . . . . . 8
β’ ((π β§ ( β₯ β( β₯
βπ)) β π΄) β ( β₯ β( β₯
βπ)) β
{(0gβπ)}) |
15 | | simpr 485 |
. . . . . . . . . . . . 13
β’ (((π β§ ( β₯ β( β₯
βπ)) β π΄) β§ π = {(0gβπ)}) β π = {(0gβπ)}) |
16 | 15 | fveq2d 6895 |
. . . . . . . . . . . 12
β’ (((π β§ ( β₯ β( β₯
βπ)) β π΄) β§ π = {(0gβπ)}) β ( β₯ βπ) = ( β₯
β{(0gβπ)})) |
17 | 16 | fveq2d 6895 |
. . . . . . . . . . 11
β’ (((π β§ ( β₯ β( β₯
βπ)) β π΄) β§ π = {(0gβπ)}) β ( β₯ β( β₯
βπ)) = ( β₯
β( β₯
β{(0gβπ)}))) |
18 | | dochsat.o |
. . . . . . . . . . . . . 14
β’ β₯ =
((ocHβπΎ)βπ) |
19 | 1, 2, 18, 8, 3 | dochoc0 40226 |
. . . . . . . . . . . . 13
β’ (π β ( β₯ β( β₯
β{(0gβπ)})) = {(0gβπ)}) |
20 | 19 | adantr 481 |
. . . . . . . . . . . 12
β’ ((π β§ ( β₯ β( β₯
βπ)) β π΄) β ( β₯ β( β₯
β{(0gβπ)})) = {(0gβπ)}) |
21 | 20 | adantr 481 |
. . . . . . . . . . 11
β’ (((π β§ ( β₯ β( β₯
βπ)) β π΄) β§ π = {(0gβπ)}) β ( β₯ β( β₯
β{(0gβπ)})) = {(0gβπ)}) |
22 | 17, 21 | eqtrd 2772 |
. . . . . . . . . 10
β’ (((π β§ ( β₯ β( β₯
βπ)) β π΄) β§ π = {(0gβπ)}) β ( β₯ β( β₯
βπ)) =
{(0gβπ)}) |
23 | 22 | ex 413 |
. . . . . . . . 9
β’ ((π β§ ( β₯ β( β₯
βπ)) β π΄) β (π = {(0gβπ)} β ( β₯ β( β₯
βπ)) =
{(0gβπ)})) |
24 | 23 | necon3d 2961 |
. . . . . . . 8
β’ ((π β§ ( β₯ β( β₯
βπ)) β π΄) β (( β₯ β( β₯
βπ)) β
{(0gβπ)}
β π β
{(0gβπ)})) |
25 | 14, 24 | mpd 15 |
. . . . . . 7
β’ ((π β§ ( β₯ β( β₯
βπ)) β π΄) β π β {(0gβπ)}) |
26 | 25 | necomd 2996 |
. . . . . 6
β’ ((π β§ ( β₯ β( β₯
βπ)) β π΄) β
{(0gβπ)}
β π) |
27 | | df-pss 3967 |
. . . . . 6
β’
({(0gβπ)} β π β ({(0gβπ)} β π β§ {(0gβπ)} β π)) |
28 | 11, 26, 27 | sylanbrc 583 |
. . . . 5
β’ ((π β§ ( β₯ β( β₯
βπ)) β π΄) β
{(0gβπ)}
β π) |
29 | 3 | adantr 481 |
. . . . . . 7
β’ ((π β§ ( β₯ β( β₯
βπ)) β π΄) β (πΎ β HL β§ π β π»)) |
30 | | eqid 2732 |
. . . . . . . . . 10
β’
(Baseβπ) =
(Baseβπ) |
31 | 30, 9 | lssss 20546 |
. . . . . . . . 9
β’ (π β π β π β (Baseβπ)) |
32 | 6, 31 | syl 17 |
. . . . . . . 8
β’ (π β π β (Baseβπ)) |
33 | 32 | adantr 481 |
. . . . . . 7
β’ ((π β§ ( β₯ β( β₯
βπ)) β π΄) β π β (Baseβπ)) |
34 | 1, 2, 30, 18 | dochocss 40232 |
. . . . . . 7
β’ (((πΎ β HL β§ π β π») β§ π β (Baseβπ)) β π β ( β₯ β( β₯
βπ))) |
35 | 29, 33, 34 | syl2anc 584 |
. . . . . 6
β’ ((π β§ ( β₯ β( β₯
βπ)) β π΄) β π β ( β₯ β( β₯
βπ))) |
36 | 9, 12, 5, 13 | lsatlssel 37862 |
. . . . . . . 8
β’ ((π β§ ( β₯ β( β₯
βπ)) β π΄) β ( β₯ β( β₯
βπ)) β π) |
37 | 9 | lsssubg 20567 |
. . . . . . . 8
β’ ((π β LMod β§ ( β₯
β( β₯ βπ)) β π) β ( β₯ β( β₯
βπ)) β
(SubGrpβπ)) |
38 | 5, 36, 37 | syl2anc 584 |
. . . . . . 7
β’ ((π β§ ( β₯ β( β₯
βπ)) β π΄) β ( β₯ β( β₯
βπ)) β
(SubGrpβπ)) |
39 | | eqid 2732 |
. . . . . . . 8
β’
(LSSumβπ) =
(LSSumβπ) |
40 | 8, 39 | lsm02 19539 |
. . . . . . 7
β’ (( β₯
β( β₯ βπ)) β (SubGrpβπ) β
({(0gβπ)}
(LSSumβπ)( β₯
β( β₯ βπ))) = ( β₯ β( β₯
βπ))) |
41 | 38, 40 | syl 17 |
. . . . . 6
β’ ((π β§ ( β₯ β( β₯
βπ)) β π΄) β
({(0gβπ)}
(LSSumβπ)( β₯
β( β₯ βπ))) = ( β₯ β( β₯
βπ))) |
42 | 35, 41 | sseqtrrd 4023 |
. . . . 5
β’ ((π β§ ( β₯ β( β₯
βπ)) β π΄) β π β ({(0gβπ)} (LSSumβπ)( β₯ β( β₯
βπ)))) |
43 | 1, 2, 3 | dvhlvec 39975 |
. . . . . . 7
β’ (π β π β LVec) |
44 | 43 | adantr 481 |
. . . . . 6
β’ ((π β§ ( β₯ β( β₯
βπ)) β π΄) β π β LVec) |
45 | 8, 9 | lsssn0 20557 |
. . . . . . 7
β’ (π β LMod β
{(0gβπ)}
β π) |
46 | 5, 45 | syl 17 |
. . . . . 6
β’ ((π β§ ( β₯ β( β₯
βπ)) β π΄) β
{(0gβπ)}
β π) |
47 | 9, 39, 12, 44, 46, 7, 13 | lsmsatcv 37875 |
. . . . 5
β’ (((π β§ ( β₯ β( β₯
βπ)) β π΄) β§
{(0gβπ)}
β π β§ π β
({(0gβπ)}
(LSSumβπ)( β₯
β( β₯ βπ)))) β π = ({(0gβπ)} (LSSumβπ)( β₯ β( β₯
βπ)))) |
48 | 28, 42, 47 | mpd3an23 1463 |
. . . 4
β’ ((π β§ ( β₯ β( β₯
βπ)) β π΄) β π = ({(0gβπ)} (LSSumβπ)( β₯ β( β₯
βπ)))) |
49 | 48, 41 | eqtr2d 2773 |
. . 3
β’ ((π β§ ( β₯ β( β₯
βπ)) β π΄) β ( β₯ β( β₯
βπ)) = π) |
50 | 49, 13 | eqeltrrd 2834 |
. 2
β’ ((π β§ ( β₯ β( β₯
βπ)) β π΄) β π β π΄) |
51 | 3 | adantr 481 |
. . . 4
β’ ((π β§ π β π΄) β (πΎ β HL β§ π β π»)) |
52 | | eqid 2732 |
. . . . . 6
β’
((DIsoHβπΎ)βπ) = ((DIsoHβπΎ)βπ) |
53 | 1, 2, 52, 12 | dih1dimat 40196 |
. . . . 5
β’ (((πΎ β HL β§ π β π») β§ π β π΄) β π β ran ((DIsoHβπΎ)βπ)) |
54 | 3, 53 | sylan 580 |
. . . 4
β’ ((π β§ π β π΄) β π β ran ((DIsoHβπΎ)βπ)) |
55 | 1, 52, 18 | dochoc 40233 |
. . . 4
β’ (((πΎ β HL β§ π β π») β§ π β ran ((DIsoHβπΎ)βπ)) β ( β₯ β( β₯
βπ)) = π) |
56 | 51, 54, 55 | syl2anc 584 |
. . 3
β’ ((π β§ π β π΄) β ( β₯ β( β₯
βπ)) = π) |
57 | | simpr 485 |
. . 3
β’ ((π β§ π β π΄) β π β π΄) |
58 | 56, 57 | eqeltrd 2833 |
. 2
β’ ((π β§ π β π΄) β ( β₯ β( β₯
βπ)) β π΄) |
59 | 50, 58 | impbida 799 |
1
β’ (π β (( β₯ β( β₯
βπ)) β π΄ β π β π΄)) |