Metamath Proof Explorer

Theorem lclkrlem2g

Description: Lemma for lclkr . Comparable hyperplanes are equal, so the kernel of the sum is closed. (Contributed by NM, 16-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 } ) ) )
lclkrlem2f.x 
|- ( ph -> X e. ( V \ { .0. } ) )
lclkrlem2f.y 
|- ( ph -> Y e. ( V \ { .0. } ) )
lclkrlem2f.ne 
|- ( ph -> ( L ` E ) =/= ( L ` G ) )
lclkrlem2f.lp 
|- ( ph -> ( L ` ( E .+ G ) ) e. J )
Assertion lclkrlem2g 
|- ( 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 lclkrlem2f.x 
 |-  ( ph -> X e. ( V \ { .0. } ) )
24 lclkrlem2f.y 
 |-  ( ph -> Y e. ( V \ { .0. } ) )
25 lclkrlem2f.ne 
 |-  ( ph -> ( L ` E ) =/= ( L ` G ) )
26 lclkrlem2f.lp 
 |-  ( ph -> ( L ` ( E .+ G ) ) e. J )
27 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 lclkrlem2f 
 |-  ( ph -> ( ( ( L ` E ) i^i ( L ` G ) ) .(+) ( N ` { B } ) ) C_ ( L ` ( E .+ G ) ) )
28 1 3 15 dvhlvec 
 |-  ( ph -> U e. LVec )
29 19 20 ineq12d 
 |-  ( ph -> ( ( L ` E ) i^i ( L ` G ) ) = ( ( ._|_ ` { X } ) i^i ( ._|_ ` { Y } ) ) )
30 29 oveq1d 
 |-  ( ph -> ( ( ( L ` E ) i^i ( L ` G ) ) .(+) ( N ` { B } ) ) = ( ( ( ._|_ ` { X } ) i^i ( ._|_ ` { Y } ) ) .(+) ( N ` { B } ) ) )
31 eqid 
 |-  ( LSAtoms ` U ) = ( LSAtoms ` U )
32 25 19 20 3netr3d 
 |-  ( ph -> ( ._|_ ` { X } ) =/= ( ._|_ ` { Y } ) )
33 1 2 3 4 7 8 9 31 15 16 23 24 32 22 11 lclkrlem2c 
 |-  ( ph -> ( ( ( ._|_ ` { X } ) i^i ( ._|_ ` { Y } ) ) .(+) ( N ` { B } ) ) e. J )
34 30 33 eqeltrd 
 |-  ( ph -> ( ( ( L ` E ) i^i ( L ` G ) ) .(+) ( N ` { B } ) ) e. J )
35 11 28 34 26 lshpcmp 
 |-  ( ph -> ( ( ( ( L ` E ) i^i ( L ` G ) ) .(+) ( N ` { B } ) ) C_ ( L ` ( E .+ G ) ) <-> ( ( ( L ` E ) i^i ( L ` G ) ) .(+) ( N ` { B } ) ) = ( L ` ( E .+ G ) ) ) )
36 27 35 mpbid 
 |-  ( ph -> ( ( ( L ` E ) i^i ( L ` G ) ) .(+) ( N ` { B } ) ) = ( L ` ( E .+ G ) ) )
37 eqid 
 |-  ( ( DIsoH ` K ) ` W ) = ( ( DIsoH ` K ) ` W )
38 1 2 3 4 7 8 9 31 15 16 23 24 32 22 37 lclkrlem2d 
 |-  ( ph -> ( ( ( ._|_ ` { X } ) i^i ( ._|_ ` { Y } ) ) .(+) ( N ` { B } ) ) e. ran ( ( DIsoH ` K ) ` W ) )
39 30 38 eqeltrd 
 |-  ( ph -> ( ( ( L ` E ) i^i ( L ` G ) ) .(+) ( N ` { B } ) ) e. ran ( ( DIsoH ` K ) ` W ) )
40 36 39 eqeltrrd 
 |-  ( ph -> ( L ` ( E .+ G ) ) e. ran ( ( DIsoH ` K ) ` W ) )
41 1 3 37 4 dihrnss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( L ` ( E .+ G ) ) e. ran ( ( DIsoH ` K ) ` W ) ) -> ( L ` ( E .+ G ) ) C_ V )
42 15 40 41 syl2anc 
 |-  ( ph -> ( L ` ( E .+ G ) ) C_ V )
43 1 37 3 4 2 15 42 dochoccl 
 |-  ( ph -> ( ( L ` ( E .+ G ) ) e. ran ( ( DIsoH ` K ) ` W ) <-> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) ) )
44 40 43 mpbid 
 |-  ( ph -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) )