| 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 3 8
|
dvhlmod |
|- ( ph -> U e. LMod ) |
| 32 |
|
eqid |
|- ( +g ` U ) = ( +g ` U ) |
| 33 |
4 32 5
|
lmodvnpcan |
|- ( ( U e. LMod /\ Y e. V /\ X e. V ) -> ( ( Y .- X ) ( +g ` U ) X ) = Y ) |
| 34 |
31 10 9 33
|
syl3anc |
|- ( ph -> ( ( Y .- X ) ( +g ` U ) X ) = Y ) |
| 35 |
|
eqid |
|- ( LSubSp ` U ) = ( LSubSp ` U ) |
| 36 |
4 35 6
|
lspsncl |
|- ( ( U e. LMod /\ X e. V ) -> ( N ` { X } ) e. ( LSubSp ` U ) ) |
| 37 |
31 9 36
|
syl2anc |
|- ( ph -> ( N ` { X } ) e. ( LSubSp ` U ) ) |
| 38 |
|
lmodgrp |
|- ( U e. LMod -> U e. Grp ) |
| 39 |
31 38
|
syl |
|- ( ph -> U e. Grp ) |
| 40 |
|
eqid |
|- ( invg ` U ) = ( invg ` U ) |
| 41 |
4 5 40
|
grpinvsub |
|- ( ( U e. Grp /\ X e. V /\ Y e. V ) -> ( ( invg ` U ) ` ( X .- Y ) ) = ( Y .- X ) ) |
| 42 |
39 9 10 41
|
syl3anc |
|- ( ph -> ( ( invg ` U ) ` ( X .- Y ) ) = ( Y .- X ) ) |
| 43 |
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 29 30
|
mapdpglem13 |
|- ( ph -> ( N ` { ( X .- Y ) } ) C_ ( N ` { X } ) ) |
| 44 |
4 5
|
lmodvsubcl |
|- ( ( U e. LMod /\ X e. V /\ Y e. V ) -> ( X .- Y ) e. V ) |
| 45 |
31 9 10 44
|
syl3anc |
|- ( ph -> ( X .- Y ) e. V ) |
| 46 |
4 6
|
lspsnid |
|- ( ( U e. LMod /\ ( X .- Y ) e. V ) -> ( X .- Y ) e. ( N ` { ( X .- Y ) } ) ) |
| 47 |
31 45 46
|
syl2anc |
|- ( ph -> ( X .- Y ) e. ( N ` { ( X .- Y ) } ) ) |
| 48 |
43 47
|
sseldd |
|- ( ph -> ( X .- Y ) e. ( N ` { X } ) ) |
| 49 |
35 40
|
lssvnegcl |
|- ( ( U e. LMod /\ ( N ` { X } ) e. ( LSubSp ` U ) /\ ( X .- Y ) e. ( N ` { X } ) ) -> ( ( invg ` U ) ` ( X .- Y ) ) e. ( N ` { X } ) ) |
| 50 |
31 37 48 49
|
syl3anc |
|- ( ph -> ( ( invg ` U ) ` ( X .- Y ) ) e. ( N ` { X } ) ) |
| 51 |
42 50
|
eqeltrrd |
|- ( ph -> ( Y .- X ) e. ( N ` { X } ) ) |
| 52 |
4 6
|
lspsnid |
|- ( ( U e. LMod /\ X e. V ) -> X e. ( N ` { X } ) ) |
| 53 |
31 9 52
|
syl2anc |
|- ( ph -> X e. ( N ` { X } ) ) |
| 54 |
32 35
|
lssvacl |
|- ( ( ( U e. LMod /\ ( N ` { X } ) e. ( LSubSp ` U ) ) /\ ( ( Y .- X ) e. ( N ` { X } ) /\ X e. ( N ` { X } ) ) ) -> ( ( Y .- X ) ( +g ` U ) X ) e. ( N ` { X } ) ) |
| 55 |
31 37 51 53 54
|
syl22anc |
|- ( ph -> ( ( Y .- X ) ( +g ` U ) X ) e. ( N ` { X } ) ) |
| 56 |
34 55
|
eqeltrrd |
|- ( ph -> Y e. ( N ` { X } ) ) |