| Step |
Hyp |
Ref |
Expression |
| 1 |
|
dia2dimlem5.l |
|- .<_ = ( le ` K ) |
| 2 |
|
dia2dimlem5.j |
|- .\/ = ( join ` K ) |
| 3 |
|
dia2dimlem5.m |
|- ./\ = ( meet ` K ) |
| 4 |
|
dia2dimlem5.a |
|- A = ( Atoms ` K ) |
| 5 |
|
dia2dimlem5.h |
|- H = ( LHyp ` K ) |
| 6 |
|
dia2dimlem5.t |
|- T = ( ( LTrn ` K ) ` W ) |
| 7 |
|
dia2dimlem5.r |
|- R = ( ( trL ` K ) ` W ) |
| 8 |
|
dia2dimlem5.y |
|- Y = ( ( DVecA ` K ) ` W ) |
| 9 |
|
dia2dimlem5.s |
|- S = ( LSubSp ` Y ) |
| 10 |
|
dia2dimlem5.pl |
|- .(+) = ( LSSum ` Y ) |
| 11 |
|
dia2dimlem5.n |
|- N = ( LSpan ` Y ) |
| 12 |
|
dia2dimlem5.i |
|- I = ( ( DIsoA ` K ) ` W ) |
| 13 |
|
dia2dimlem5.q |
|- Q = ( ( P .\/ U ) ./\ ( ( F ` P ) .\/ V ) ) |
| 14 |
|
dia2dimlem5.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
| 15 |
|
dia2dimlem5.u |
|- ( ph -> ( U e. A /\ U .<_ W ) ) |
| 16 |
|
dia2dimlem5.v |
|- ( ph -> ( V e. A /\ V .<_ W ) ) |
| 17 |
|
dia2dimlem5.p |
|- ( ph -> ( P e. A /\ -. P .<_ W ) ) |
| 18 |
|
dia2dimlem5.f |
|- ( ph -> ( F e. T /\ ( F ` P ) =/= P ) ) |
| 19 |
|
dia2dimlem5.rf |
|- ( ph -> ( R ` F ) .<_ ( U .\/ V ) ) |
| 20 |
|
dia2dimlem5.uv |
|- ( ph -> U =/= V ) |
| 21 |
|
dia2dimlem5.ru |
|- ( ph -> ( R ` F ) =/= U ) |
| 22 |
|
dia2dimlem5.rv |
|- ( ph -> ( R ` F ) =/= V ) |
| 23 |
|
dia2dimlem5.g |
|- ( ph -> G e. T ) |
| 24 |
|
dia2dimlem5.gv |
|- ( ph -> ( G ` P ) = Q ) |
| 25 |
|
dia2dimlem5.d |
|- ( ph -> D e. T ) |
| 26 |
|
dia2dimlem5.dv |
|- ( ph -> ( D ` Q ) = ( F ` P ) ) |
| 27 |
|
eqid |
|- ( +g ` Y ) = ( +g ` Y ) |
| 28 |
5 6 8 27
|
dvavadd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( D e. T /\ G e. T ) ) -> ( D ( +g ` Y ) G ) = ( D o. G ) ) |
| 29 |
14 25 23 28
|
syl12anc |
|- ( ph -> ( D ( +g ` Y ) G ) = ( D o. G ) ) |
| 30 |
18
|
simpld |
|- ( ph -> F e. T ) |
| 31 |
1 4 5 6 14 17 30 23 24 25 26
|
dia2dimlem4 |
|- ( ph -> ( D o. G ) = F ) |
| 32 |
29 31
|
eqtr2d |
|- ( ph -> F = ( D ( +g ` Y ) G ) ) |
| 33 |
5 8
|
dvalvec |
|- ( ( K e. HL /\ W e. H ) -> Y e. LVec ) |
| 34 |
|
lveclmod |
|- ( Y e. LVec -> Y e. LMod ) |
| 35 |
14 33 34
|
3syl |
|- ( ph -> Y e. LMod ) |
| 36 |
9
|
lsssssubg |
|- ( Y e. LMod -> S C_ ( SubGrp ` Y ) ) |
| 37 |
35 36
|
syl |
|- ( ph -> S C_ ( SubGrp ` Y ) ) |
| 38 |
16
|
simpld |
|- ( ph -> V e. A ) |
| 39 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
| 40 |
39 4
|
atbase |
|- ( V e. A -> V e. ( Base ` K ) ) |
| 41 |
38 40
|
syl |
|- ( ph -> V e. ( Base ` K ) ) |
| 42 |
16
|
simprd |
|- ( ph -> V .<_ W ) |
| 43 |
39 1 5 8 12 9
|
dialss |
|- ( ( ( K e. HL /\ W e. H ) /\ ( V e. ( Base ` K ) /\ V .<_ W ) ) -> ( I ` V ) e. S ) |
| 44 |
14 41 42 43
|
syl12anc |
|- ( ph -> ( I ` V ) e. S ) |
| 45 |
37 44
|
sseldd |
|- ( ph -> ( I ` V ) e. ( SubGrp ` Y ) ) |
| 46 |
15
|
simpld |
|- ( ph -> U e. A ) |
| 47 |
39 4
|
atbase |
|- ( U e. A -> U e. ( Base ` K ) ) |
| 48 |
46 47
|
syl |
|- ( ph -> U e. ( Base ` K ) ) |
| 49 |
15
|
simprd |
|- ( ph -> U .<_ W ) |
| 50 |
39 1 5 8 12 9
|
dialss |
|- ( ( ( K e. HL /\ W e. H ) /\ ( U e. ( Base ` K ) /\ U .<_ W ) ) -> ( I ` U ) e. S ) |
| 51 |
14 48 49 50
|
syl12anc |
|- ( ph -> ( I ` U ) e. S ) |
| 52 |
37 51
|
sseldd |
|- ( ph -> ( I ` U ) e. ( SubGrp ` Y ) ) |
| 53 |
5 6 7 8 12 11
|
dia1dim2 |
|- ( ( ( K e. HL /\ W e. H ) /\ D e. T ) -> ( I ` ( R ` D ) ) = ( N ` { D } ) ) |
| 54 |
14 25 53
|
syl2anc |
|- ( ph -> ( I ` ( R ` D ) ) = ( N ` { D } ) ) |
| 55 |
1 2 3 4 5 6 7 13 14 15 16 17 18 19 20 21 22 25 26
|
dia2dimlem3 |
|- ( ph -> ( R ` D ) = V ) |
| 56 |
55
|
fveq2d |
|- ( ph -> ( I ` ( R ` D ) ) = ( I ` V ) ) |
| 57 |
|
eqss |
|- ( ( I ` ( R ` D ) ) = ( I ` V ) <-> ( ( I ` ( R ` D ) ) C_ ( I ` V ) /\ ( I ` V ) C_ ( I ` ( R ` D ) ) ) ) |
| 58 |
56 57
|
sylib |
|- ( ph -> ( ( I ` ( R ` D ) ) C_ ( I ` V ) /\ ( I ` V ) C_ ( I ` ( R ` D ) ) ) ) |
| 59 |
58
|
simpld |
|- ( ph -> ( I ` ( R ` D ) ) C_ ( I ` V ) ) |
| 60 |
54 59
|
eqsstrrd |
|- ( ph -> ( N ` { D } ) C_ ( I ` V ) ) |
| 61 |
|
eqid |
|- ( Base ` Y ) = ( Base ` Y ) |
| 62 |
5 6 8 61
|
dvavbase |
|- ( ( K e. HL /\ W e. H ) -> ( Base ` Y ) = T ) |
| 63 |
14 62
|
syl |
|- ( ph -> ( Base ` Y ) = T ) |
| 64 |
25 63
|
eleqtrrd |
|- ( ph -> D e. ( Base ` Y ) ) |
| 65 |
61 9 11 35 44 64
|
lspsnel5 |
|- ( ph -> ( D e. ( I ` V ) <-> ( N ` { D } ) C_ ( I ` V ) ) ) |
| 66 |
60 65
|
mpbird |
|- ( ph -> D e. ( I ` V ) ) |
| 67 |
5 6 7 8 12 11
|
dia1dim2 |
|- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( I ` ( R ` G ) ) = ( N ` { G } ) ) |
| 68 |
14 23 67
|
syl2anc |
|- ( ph -> ( I ` ( R ` G ) ) = ( N ` { G } ) ) |
| 69 |
1 2 3 4 5 6 7 13 14 15 16 17 18 19 22 23 24
|
dia2dimlem2 |
|- ( ph -> ( R ` G ) = U ) |
| 70 |
69
|
fveq2d |
|- ( ph -> ( I ` ( R ` G ) ) = ( I ` U ) ) |
| 71 |
|
eqss |
|- ( ( I ` ( R ` G ) ) = ( I ` U ) <-> ( ( I ` ( R ` G ) ) C_ ( I ` U ) /\ ( I ` U ) C_ ( I ` ( R ` G ) ) ) ) |
| 72 |
70 71
|
sylib |
|- ( ph -> ( ( I ` ( R ` G ) ) C_ ( I ` U ) /\ ( I ` U ) C_ ( I ` ( R ` G ) ) ) ) |
| 73 |
72
|
simpld |
|- ( ph -> ( I ` ( R ` G ) ) C_ ( I ` U ) ) |
| 74 |
68 73
|
eqsstrrd |
|- ( ph -> ( N ` { G } ) C_ ( I ` U ) ) |
| 75 |
23 63
|
eleqtrrd |
|- ( ph -> G e. ( Base ` Y ) ) |
| 76 |
61 9 11 35 51 75
|
lspsnel5 |
|- ( ph -> ( G e. ( I ` U ) <-> ( N ` { G } ) C_ ( I ` U ) ) ) |
| 77 |
74 76
|
mpbird |
|- ( ph -> G e. ( I ` U ) ) |
| 78 |
27 10
|
lsmelvali |
|- ( ( ( ( I ` V ) e. ( SubGrp ` Y ) /\ ( I ` U ) e. ( SubGrp ` Y ) ) /\ ( D e. ( I ` V ) /\ G e. ( I ` U ) ) ) -> ( D ( +g ` Y ) G ) e. ( ( I ` V ) .(+) ( I ` U ) ) ) |
| 79 |
45 52 66 77 78
|
syl22anc |
|- ( ph -> ( D ( +g ` Y ) G ) e. ( ( I ` V ) .(+) ( I ` U ) ) ) |
| 80 |
32 79
|
eqeltrd |
|- ( ph -> F e. ( ( I ` V ) .(+) ( I ` U ) ) ) |
| 81 |
|
lmodabl |
|- ( Y e. LMod -> Y e. Abel ) |
| 82 |
35 81
|
syl |
|- ( ph -> Y e. Abel ) |
| 83 |
10
|
lsmcom |
|- ( ( Y e. Abel /\ ( I ` V ) e. ( SubGrp ` Y ) /\ ( I ` U ) e. ( SubGrp ` Y ) ) -> ( ( I ` V ) .(+) ( I ` U ) ) = ( ( I ` U ) .(+) ( I ` V ) ) ) |
| 84 |
82 45 52 83
|
syl3anc |
|- ( ph -> ( ( I ` V ) .(+) ( I ` U ) ) = ( ( I ` U ) .(+) ( I ` V ) ) ) |
| 85 |
80 84
|
eleqtrd |
|- ( ph -> F e. ( ( I ` U ) .(+) ( I ` V ) ) ) |