| 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 |
|
mapdpglem17.ep |
|- E = ( ( ( invr ` A ) ` g ) .x. z ) |
| 31 |
|
eqid |
|- ( LSubSp ` C ) = ( LSubSp ` C ) |
| 32 |
|
eqid |
|- ( LSubSp ` U ) = ( LSubSp ` U ) |
| 33 |
1 3 8
|
dvhlmod |
|- ( ph -> U e. LMod ) |
| 34 |
4 32 6
|
lspsncl |
|- ( ( U e. LMod /\ Y e. V ) -> ( N ` { Y } ) e. ( LSubSp ` U ) ) |
| 35 |
33 10 34
|
syl2anc |
|- ( ph -> ( N ` { Y } ) e. ( LSubSp ` U ) ) |
| 36 |
1 2 3 32 7 31 8 35
|
mapdcl2 |
|- ( ph -> ( M ` ( N ` { Y } ) ) e. ( LSubSp ` C ) ) |
| 37 |
1 3 8
|
dvhlvec |
|- ( ph -> U e. LVec ) |
| 38 |
15
|
lvecdrng |
|- ( U e. LVec -> A e. DivRing ) |
| 39 |
37 38
|
syl |
|- ( ph -> A e. DivRing ) |
| 40 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
|
mapdpglem11 |
|- ( ph -> g =/= .0. ) |
| 41 |
|
eqid |
|- ( invr ` A ) = ( invr ` A ) |
| 42 |
16 24 41
|
drnginvrcl |
|- ( ( A e. DivRing /\ g e. B /\ g =/= .0. ) -> ( ( invr ` A ) ` g ) e. B ) |
| 43 |
39 25 40 42
|
syl3anc |
|- ( ph -> ( ( invr ` A ) ` g ) e. B ) |
| 44 |
1 3 15 16 7 13 17 31 8 36 43 26
|
lcdlssvscl |
|- ( ph -> ( ( ( invr ` A ) ` g ) .x. z ) e. ( M ` ( N ` { Y } ) ) ) |
| 45 |
30 44
|
eqeltrid |
|- ( ph -> E e. ( M ` ( N ` { Y } ) ) ) |