Step |
Hyp |
Ref |
Expression |
1 |
|
mapdpglem.h |
|- H = ( LHyp ` K ) |
2 |
|
mapdpglem.m |
|- M = ( ( mapd ` K ) ` W ) |
3 |
|
mapdpglem.u |
|- U = ( ( DVecH ` K ) ` W ) |
4 |
|
mapdpglem.v |
|- V = ( Base ` U ) |
5 |
|
mapdpglem.s |
|- .- = ( -g ` U ) |
6 |
|
mapdpglem.n |
|- N = ( LSpan ` U ) |
7 |
|
mapdpglem.c |
|- C = ( ( LCDual ` K ) ` W ) |
8 |
|
mapdpglem.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
9 |
|
mapdpglem.x |
|- ( ph -> X e. V ) |
10 |
|
mapdpglem.y |
|- ( ph -> Y e. V ) |
11 |
|
mapdpglem1.p |
|- .(+) = ( LSSum ` C ) |
12 |
|
mapdpglem2.j |
|- J = ( LSpan ` C ) |
13 |
|
mapdpglem3.f |
|- F = ( Base ` C ) |
14 |
|
mapdpglem3.te |
|- ( ph -> t e. ( ( M ` ( N ` { X } ) ) .(+) ( M ` ( N ` { Y } ) ) ) ) |
15 |
|
mapdpglem3.a |
|- A = ( Scalar ` U ) |
16 |
|
mapdpglem3.b |
|- B = ( Base ` A ) |
17 |
|
mapdpglem3.t |
|- .x. = ( .s ` C ) |
18 |
|
mapdpglem3.r |
|- R = ( -g ` C ) |
19 |
|
mapdpglem3.g |
|- ( ph -> G e. F ) |
20 |
|
mapdpglem3.e |
|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { G } ) ) |
21 |
|
mapdpglem4.q |
|- Q = ( 0g ` U ) |
22 |
|
mapdpglem.ne |
|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) ) |
23 |
|
mapdpglem4.jt |
|- ( ph -> ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { t } ) ) |
24 |
|
mapdpglem4.z |
|- .0. = ( 0g ` A ) |
25 |
|
mapdpglem4.g4 |
|- ( ph -> g e. B ) |
26 |
|
mapdpglem4.z4 |
|- ( ph -> z e. ( M ` ( N ` { Y } ) ) ) |
27 |
|
mapdpglem4.t4 |
|- ( ph -> t = ( ( g .x. G ) R z ) ) |
28 |
|
mapdpglem4.xn |
|- ( ph -> X =/= Q ) |
29 |
|
mapdpglem12.yn |
|- ( ph -> Y =/= Q ) |
30 |
|
mapdpglem12.g0 |
|- ( ph -> z = ( 0g ` C ) ) |
31 |
1 7 8
|
lcdlmod |
|- ( ph -> C e. LMod ) |
32 |
|
eqid |
|- ( LSubSp ` U ) = ( LSubSp ` U ) |
33 |
|
eqid |
|- ( LSubSp ` C ) = ( LSubSp ` C ) |
34 |
1 3 8
|
dvhlmod |
|- ( ph -> U e. LMod ) |
35 |
4 32 6
|
lspsncl |
|- ( ( U e. LMod /\ X e. V ) -> ( N ` { X } ) e. ( LSubSp ` U ) ) |
36 |
34 9 35
|
syl2anc |
|- ( ph -> ( N ` { X } ) e. ( LSubSp ` U ) ) |
37 |
1 2 3 32 7 33 8 36
|
mapdcl2 |
|- ( ph -> ( M ` ( N ` { X } ) ) e. ( LSubSp ` C ) ) |
38 |
13 12
|
lspsnid |
|- ( ( C e. LMod /\ G e. F ) -> G e. ( J ` { G } ) ) |
39 |
31 19 38
|
syl2anc |
|- ( ph -> G e. ( J ` { G } ) ) |
40 |
39 20
|
eleqtrrd |
|- ( ph -> G e. ( M ` ( N ` { X } ) ) ) |
41 |
1 3 15 16 7 13 17 33 8 37 25 40
|
lcdlssvscl |
|- ( ph -> ( g .x. G ) e. ( M ` ( N ` { X } ) ) ) |
42 |
|
eqid |
|- ( 0g ` C ) = ( 0g ` C ) |
43 |
42 33
|
lss0cl |
|- ( ( C e. LMod /\ ( M ` ( N ` { X } ) ) e. ( LSubSp ` C ) ) -> ( 0g ` C ) e. ( M ` ( N ` { X } ) ) ) |
44 |
31 37 43
|
syl2anc |
|- ( ph -> ( 0g ` C ) e. ( M ` ( N ` { X } ) ) ) |
45 |
30 44
|
eqeltrd |
|- ( ph -> z e. ( M ` ( N ` { X } ) ) ) |
46 |
18 33
|
lssvsubcl |
|- ( ( ( C e. LMod /\ ( M ` ( N ` { X } ) ) e. ( LSubSp ` C ) ) /\ ( ( g .x. G ) e. ( M ` ( N ` { X } ) ) /\ z e. ( M ` ( N ` { X } ) ) ) ) -> ( ( g .x. G ) R z ) e. ( M ` ( N ` { X } ) ) ) |
47 |
31 37 41 45 46
|
syl22anc |
|- ( ph -> ( ( g .x. G ) R z ) e. ( M ` ( N ` { X } ) ) ) |
48 |
27 47
|
eqeltrd |
|- ( ph -> t e. ( M ` ( N ` { X } ) ) ) |