# 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 )`
` |-  ( 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 ) ) )`