Metamath Proof Explorer

Theorem mapdpglem9

Description: Lemma for mapdpg . Baer p. 45, line 4: "...so that x would consequently belong to Fy." (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 )
mapdpglem4.g0 
|- ( ph -> g = .0. )
Assertion mapdpglem9 
|- ( ph -> X e. ( N ` { Y } ) )

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 mapdpglem4.g0 
 |-  ( ph -> g = .0. )
30 1 3 8 dvhlmod 
 |-  ( ph -> U e. LMod )
31 eqid 
 |-  ( +g ` U ) = ( +g ` U )
32 4 31 5 lmodvnpcan 
 |-  ( ( U e. LMod /\ X e. V /\ Y e. V ) -> ( ( X .- Y ) ( +g ` U ) Y ) = X )
33 30 9 10 32 syl3anc 
 |-  ( ph -> ( ( X .- Y ) ( +g ` U ) Y ) = X )
34 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
35 4 34 6 lspsncl 
 |-  ( ( U e. LMod /\ Y e. V ) -> ( N ` { Y } ) e. ( LSubSp ` U ) )
36 30 10 35 syl2anc 
 |-  ( ph -> ( N ` { Y } ) e. ( LSubSp ` U ) )
37 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 mapdpglem8 
 |-  ( ph -> ( N ` { ( X .- Y ) } ) C_ ( N ` { Y } ) )
38 4 5 lmodvsubcl 
 |-  ( ( U e. LMod /\ X e. V /\ Y e. V ) -> ( X .- Y ) e. V )
39 30 9 10 38 syl3anc 
 |-  ( ph -> ( X .- Y ) e. V )
40 4 6 lspsnid 
 |-  ( ( U e. LMod /\ ( X .- Y ) e. V ) -> ( X .- Y ) e. ( N ` { ( X .- Y ) } ) )
41 30 39 40 syl2anc 
 |-  ( ph -> ( X .- Y ) e. ( N ` { ( X .- Y ) } ) )
42 37 41 sseldd 
 |-  ( ph -> ( X .- Y ) e. ( N ` { Y } ) )
43 4 6 lspsnid 
 |-  ( ( U e. LMod /\ Y e. V ) -> Y e. ( N ` { Y } ) )
44 30 10 43 syl2anc 
 |-  ( ph -> Y e. ( N ` { Y } ) )
45 31 34 lssvacl 
 |-  ( ( ( U e. LMod /\ ( N ` { Y } ) e. ( LSubSp ` U ) ) /\ ( ( X .- Y ) e. ( N ` { Y } ) /\ Y e. ( N ` { Y } ) ) ) -> ( ( X .- Y ) ( +g ` U ) Y ) e. ( N ` { Y } ) )
46 30 36 42 44 45 syl22anc 
 |-  ( ph -> ( ( X .- Y ) ( +g ` U ) Y ) e. ( N ` { Y } ) )
47 33 46 eqeltrrd 
 |-  ( ph -> X e. ( N ` { Y } ) )