Metamath Proof Explorer

Theorem lclkrlem2j

Description: Lemma for lclkr . Kernel closure when Y is zero. (Contributed by NM, 18-Jan-2015)

Ref Expression
Hypotheses lclkrlem2f.h 
|- H = ( LHyp ` K )
lclkrlem2f.o 
|- ._|_ = ( ( ocH ` K ) ` W )
lclkrlem2f.u 
|- U = ( ( DVecH ` K ) ` W )
lclkrlem2f.v 
|- V = ( Base ` U )
lclkrlem2f.s 
|- S = ( Scalar ` U )
lclkrlem2f.q 
|- Q = ( 0g ` S )
lclkrlem2f.z 
|- .0. = ( 0g ` U )
lclkrlem2f.a 
|- .(+) = ( LSSum ` U )
lclkrlem2f.n 
|- N = ( LSpan ` U )
lclkrlem2f.f 
|- F = ( LFnl ` U )
lclkrlem2f.j 
|- J = ( LSHyp ` U )
lclkrlem2f.l 
|- L = ( LKer ` U )
lclkrlem2f.d 
|- D = ( LDual ` U )
lclkrlem2f.p 
|- .+ = ( +g ` D )
lclkrlem2f.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
lclkrlem2f.b 
|- ( ph -> B e. ( V \ { .0. } ) )
lclkrlem2f.e 
|- ( ph -> E e. F )
lclkrlem2f.g 
|- ( ph -> G e. F )
lclkrlem2f.le 
|- ( ph -> ( L ` E ) = ( ._|_ ` { X } ) )
lclkrlem2f.lg 
|- ( ph -> ( L ` G ) = ( ._|_ ` { Y } ) )
lclkrlem2f.kb 
|- ( ph -> ( ( E .+ G ) ` B ) = Q )
lclkrlem2f.nx 
|- ( ph -> ( -. X e. ( ._|_ ` { B } ) \/ -. Y e. ( ._|_ ` { B } ) ) )
lclkrlem2j.x 
|- ( ph -> X e. V )
lclkrlem2j.y 
|- ( ph -> Y = .0. )
Assertion lclkrlem2j 
|- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) )

Proof

Step Hyp Ref Expression
1 lclkrlem2f.h 
 |-  H = ( LHyp ` K )
2 lclkrlem2f.o 
 |-  ._|_ = ( ( ocH ` K ) ` W )
3 lclkrlem2f.u 
 |-  U = ( ( DVecH ` K ) ` W )
4 lclkrlem2f.v 
 |-  V = ( Base ` U )
5 lclkrlem2f.s 
 |-  S = ( Scalar ` U )
6 lclkrlem2f.q 
 |-  Q = ( 0g ` S )
7 lclkrlem2f.z 
 |-  .0. = ( 0g ` U )
8 lclkrlem2f.a 
 |-  .(+) = ( LSSum ` U )
9 lclkrlem2f.n 
 |-  N = ( LSpan ` U )
10 lclkrlem2f.f 
 |-  F = ( LFnl ` U )
11 lclkrlem2f.j 
 |-  J = ( LSHyp ` U )
12 lclkrlem2f.l 
 |-  L = ( LKer ` U )
13 lclkrlem2f.d 
 |-  D = ( LDual ` U )
14 lclkrlem2f.p 
 |-  .+ = ( +g ` D )
15 lclkrlem2f.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
16 lclkrlem2f.b 
 |-  ( ph -> B e. ( V \ { .0. } ) )
17 lclkrlem2f.e 
 |-  ( ph -> E e. F )
18 lclkrlem2f.g 
 |-  ( ph -> G e. F )
19 lclkrlem2f.le 
 |-  ( ph -> ( L ` E ) = ( ._|_ ` { X } ) )
20 lclkrlem2f.lg 
 |-  ( ph -> ( L ` G ) = ( ._|_ ` { Y } ) )
21 lclkrlem2f.kb 
 |-  ( ph -> ( ( E .+ G ) ` B ) = Q )
22 lclkrlem2f.nx 
 |-  ( ph -> ( -. X e. ( ._|_ ` { B } ) \/ -. Y e. ( ._|_ ` { B } ) ) )
23 lclkrlem2j.x 
 |-  ( ph -> X e. V )
24 lclkrlem2j.y 
 |-  ( ph -> Y = .0. )
25 23 snssd 
 |-  ( ph -> { X } C_ V )
26 eqid 
 |-  ( ( DIsoH ` K ) ` W ) = ( ( DIsoH ` K ) ` W )
27 1 26 3 4 2 dochcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ { X } C_ V ) -> ( ._|_ ` { X } ) e. ran ( ( DIsoH ` K ) ` W ) )
28 15 25 27 syl2anc 
 |-  ( ph -> ( ._|_ ` { X } ) e. ran ( ( DIsoH ` K ) ` W ) )
29 1 26 2 dochoc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ._|_ ` { X } ) e. ran ( ( DIsoH ` K ) ` W ) ) -> ( ._|_ ` ( ._|_ ` ( ._|_ ` { X } ) ) ) = ( ._|_ ` { X } ) )
30 15 28 29 syl2anc 
 |-  ( ph -> ( ._|_ ` ( ._|_ ` ( ._|_ ` { X } ) ) ) = ( ._|_ ` { X } ) )
31 24 sneqd 
 |-  ( ph -> { Y } = { .0. } )
32 31 fveq2d 
 |-  ( ph -> ( ._|_ ` { Y } ) = ( ._|_ ` { .0. } ) )
33 eqid 
 |-  ( Base ` U ) = ( Base ` U )
34 1 3 2 33 7 doch0 
 |-  ( ( K e. HL /\ W e. H ) -> ( ._|_ ` { .0. } ) = ( Base ` U ) )
35 15 34 syl 
 |-  ( ph -> ( ._|_ ` { .0. } ) = ( Base ` U ) )
36 20 32 35 3eqtrd 
 |-  ( ph -> ( L ` G ) = ( Base ` U ) )
37 1 3 15 dvhlmod 
 |-  ( ph -> U e. LMod )
38 5 6 33 10 12 lkr0f 
 |-  ( ( U e. LMod /\ G e. F ) -> ( ( L ` G ) = ( Base ` U ) <-> G = ( ( Base ` U ) X. { Q } ) ) )
39 37 18 38 syl2anc 
 |-  ( ph -> ( ( L ` G ) = ( Base ` U ) <-> G = ( ( Base ` U ) X. { Q } ) ) )
40 36 39 mpbid 
 |-  ( ph -> G = ( ( Base ` U ) X. { Q } ) )
41 eqid 
 |-  ( 0g ` D ) = ( 0g ` D )
42 33 5 6 13 41 37 ldual0v 
 |-  ( ph -> ( 0g ` D ) = ( ( Base ` U ) X. { Q } ) )
43 40 42 eqtr4d 
 |-  ( ph -> G = ( 0g ` D ) )
44 43 oveq2d 
 |-  ( ph -> ( E .+ G ) = ( E .+ ( 0g ` D ) ) )
45 13 37 lduallmod 
 |-  ( ph -> D e. LMod )
46 eqid 
 |-  ( Base ` D ) = ( Base ` D )
47 10 13 46 37 17 ldualelvbase 
 |-  ( ph -> E e. ( Base ` D ) )
48 46 14 41 lmod0vrid 
 |-  ( ( D e. LMod /\ E e. ( Base ` D ) ) -> ( E .+ ( 0g ` D ) ) = E )
49 45 47 48 syl2anc 
 |-  ( ph -> ( E .+ ( 0g ` D ) ) = E )
50 44 49 eqtrd 
 |-  ( ph -> ( E .+ G ) = E )
51 50 fveq2d 
 |-  ( ph -> ( L ` ( E .+ G ) ) = ( L ` E ) )
52 51 19 eqtr2d 
 |-  ( ph -> ( ._|_ ` { X } ) = ( L ` ( E .+ G ) ) )
53 52 fveq2d 
 |-  ( ph -> ( ._|_ ` ( ._|_ ` { X } ) ) = ( ._|_ ` ( L ` ( E .+ G ) ) ) )
54 53 fveq2d 
 |-  ( ph -> ( ._|_ ` ( ._|_ ` ( ._|_ ` { X } ) ) ) = ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) )
55 30 54 52 3eqtr3d 
 |-  ( ph -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) )