Metamath Proof Explorer

Theorem lclkrlem2f

Description: Lemma for lclkr . Construct a closed hyperplane under the kernel of the sum. (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 lclkrlem2f 
|- ( ph -> ( ( ( L ` E ) i^i ( L ` G ) ) .(+) ( N ` { B } ) ) C_ ( 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 3 15 dvhlmod 
 |-  ( ph -> U e. LMod )
28 10 12 13 14 27 17 18 lkrin 
 |-  ( ph -> ( ( L ` E ) i^i ( L ` G ) ) C_ ( L ` ( E .+ G ) ) )
29 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
30 29 11 27 26 lshplss 
 |-  ( ph -> ( L ` ( E .+ G ) ) e. ( LSubSp ` U ) )
31 10 13 14 27 17 18 ldualvaddcl 
 |-  ( ph -> ( E .+ G ) e. F )
32 16 eldifad 
 |-  ( ph -> B e. V )
33 4 5 6 10 12 27 31 32 ellkr2 
 |-  ( ph -> ( B e. ( L ` ( E .+ G ) ) <-> ( ( E .+ G ) ` B ) = Q ) )
34 21 33 mpbird 
 |-  ( ph -> B e. ( L ` ( E .+ G ) ) )
35 29 9 27 30 34 lspsnel5a 
 |-  ( ph -> ( N ` { B } ) C_ ( L ` ( E .+ G ) ) )
36 29 lsssssubg 
 |-  ( U e. LMod -> ( LSubSp ` U ) C_ ( SubGrp ` U ) )
37 27 36 syl 
 |-  ( ph -> ( LSubSp ` U ) C_ ( SubGrp ` U ) )
38 10 12 29 lkrlss 
 |-  ( ( U e. LMod /\ E e. F ) -> ( L ` E ) e. ( LSubSp ` U ) )
39 27 17 38 syl2anc 
 |-  ( ph -> ( L ` E ) e. ( LSubSp ` U ) )
40 10 12 29 lkrlss 
 |-  ( ( U e. LMod /\ G e. F ) -> ( L ` G ) e. ( LSubSp ` U ) )
41 27 18 40 syl2anc 
 |-  ( ph -> ( L ` G ) e. ( LSubSp ` U ) )
42 29 lssincl 
 |-  ( ( U e. LMod /\ ( L ` E ) e. ( LSubSp ` U ) /\ ( L ` G ) e. ( LSubSp ` U ) ) -> ( ( L ` E ) i^i ( L ` G ) ) e. ( LSubSp ` U ) )
43 27 39 41 42 syl3anc 
 |-  ( ph -> ( ( L ` E ) i^i ( L ` G ) ) e. ( LSubSp ` U ) )
44 37 43 sseldd 
 |-  ( ph -> ( ( L ` E ) i^i ( L ` G ) ) e. ( SubGrp ` U ) )
45 4 29 9 lspsncl 
 |-  ( ( U e. LMod /\ B e. V ) -> ( N ` { B } ) e. ( LSubSp ` U ) )
46 27 32 45 syl2anc 
 |-  ( ph -> ( N ` { B } ) e. ( LSubSp ` U ) )
47 37 46 sseldd 
 |-  ( ph -> ( N ` { B } ) e. ( SubGrp ` U ) )
48 37 30 sseldd 
 |-  ( ph -> ( L ` ( E .+ G ) ) e. ( SubGrp ` U ) )
49 8 lsmlub 
 |-  ( ( ( ( L ` E ) i^i ( L ` G ) ) e. ( SubGrp ` U ) /\ ( N ` { B } ) e. ( SubGrp ` U ) /\ ( L ` ( E .+ G ) ) e. ( SubGrp ` U ) ) -> ( ( ( ( L ` E ) i^i ( L ` G ) ) C_ ( L ` ( E .+ G ) ) /\ ( N ` { B } ) C_ ( L ` ( E .+ G ) ) ) <-> ( ( ( L ` E ) i^i ( L ` G ) ) .(+) ( N ` { B } ) ) C_ ( L ` ( E .+ G ) ) ) )
50 44 47 48 49 syl3anc 
 |-  ( ph -> ( ( ( ( L ` E ) i^i ( L ` G ) ) C_ ( L ` ( E .+ G ) ) /\ ( N ` { B } ) C_ ( L ` ( E .+ G ) ) ) <-> ( ( ( L ` E ) i^i ( L ` G ) ) .(+) ( N ` { B } ) ) C_ ( L ` ( E .+ G ) ) ) )
51 28 35 50 mpbi2and 
 |-  ( ph -> ( ( ( L ` E ) i^i ( L ` G ) ) .(+) ( N ` { B } ) ) C_ ( L ` ( E .+ G ) ) )