Metamath Proof Explorer


Theorem mapdpglem14

Description: Lemma for mapdpg . (Contributed by NM, 20-Mar-2015)

Ref Expression
Hypotheses mapdpglem.h
|- H = ( LHyp ` K )
mapdpglem.m
|- M = ( ( mapd ` K ) ` W )
mapdpglem.u
|- U = ( ( DVecH ` K ) ` W )
mapdpglem.v
|- V = ( Base ` U )
mapdpglem.s
|- .- = ( -g ` U )
mapdpglem.n
|- N = ( LSpan ` U )
mapdpglem.c
|- C = ( ( LCDual ` K ) ` W )
mapdpglem.k
|- ( ph -> ( K e. HL /\ W e. H ) )
mapdpglem.x
|- ( ph -> X e. V )
mapdpglem.y
|- ( ph -> Y e. V )
mapdpglem1.p
|- .(+) = ( LSSum ` C )
mapdpglem2.j
|- J = ( LSpan ` C )
mapdpglem3.f
|- F = ( Base ` C )
mapdpglem3.te
|- ( ph -> t e. ( ( M ` ( N ` { X } ) ) .(+) ( M ` ( N ` { Y } ) ) ) )
mapdpglem3.a
|- A = ( Scalar ` U )
mapdpglem3.b
|- B = ( Base ` A )
mapdpglem3.t
|- .x. = ( .s ` C )
mapdpglem3.r
|- R = ( -g ` C )
mapdpglem3.g
|- ( ph -> G e. F )
mapdpglem3.e
|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { G } ) )
mapdpglem4.q
|- Q = ( 0g ` U )
mapdpglem.ne
|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
mapdpglem4.jt
|- ( ph -> ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { t } ) )
mapdpglem4.z
|- .0. = ( 0g ` A )
mapdpglem4.g4
|- ( ph -> g e. B )
mapdpglem4.z4
|- ( ph -> z e. ( M ` ( N ` { Y } ) ) )
mapdpglem4.t4
|- ( ph -> t = ( ( g .x. G ) R z ) )
mapdpglem4.xn
|- ( ph -> X =/= Q )
mapdpglem12.yn
|- ( ph -> Y =/= Q )
mapdpglem12.g0
|- ( ph -> z = ( 0g ` C ) )
Assertion mapdpglem14
|- ( ph -> Y e. ( N ` { X } ) )

Proof

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 } ) )