Metamath Proof Explorer


Theorem hdmapglem7a

Description: Lemma for hdmapg . (Contributed by NM, 14-Jun-2015)

Ref Expression
Hypotheses hdmapglem7.h
|- H = ( LHyp ` K )
hdmapglem7.e
|- E = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >.
hdmapglem7.o
|- O = ( ( ocH ` K ) ` W )
hdmapglem7.u
|- U = ( ( DVecH ` K ) ` W )
hdmapglem7.v
|- V = ( Base ` U )
hdmapglem7.p
|- .+ = ( +g ` U )
hdmapglem7.q
|- .x. = ( .s ` U )
hdmapglem7.r
|- R = ( Scalar ` U )
hdmapglem7.b
|- B = ( Base ` R )
hdmapglem7.a
|- .(+) = ( LSSum ` U )
hdmapglem7.n
|- N = ( LSpan ` U )
hdmapglem7.k
|- ( ph -> ( K e. HL /\ W e. H ) )
hdmapglem7.x
|- ( ph -> X e. V )
Assertion hdmapglem7a
|- ( ph -> E. u e. ( O ` { E } ) E. k e. B X = ( ( k .x. E ) .+ u ) )

Proof

Step Hyp Ref Expression
1 hdmapglem7.h
 |-  H = ( LHyp ` K )
2 hdmapglem7.e
 |-  E = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >.
3 hdmapglem7.o
 |-  O = ( ( ocH ` K ) ` W )
4 hdmapglem7.u
 |-  U = ( ( DVecH ` K ) ` W )
5 hdmapglem7.v
 |-  V = ( Base ` U )
6 hdmapglem7.p
 |-  .+ = ( +g ` U )
7 hdmapglem7.q
 |-  .x. = ( .s ` U )
8 hdmapglem7.r
 |-  R = ( Scalar ` U )
9 hdmapglem7.b
 |-  B = ( Base ` R )
10 hdmapglem7.a
 |-  .(+) = ( LSSum ` U )
11 hdmapglem7.n
 |-  N = ( LSpan ` U )
12 hdmapglem7.k
 |-  ( ph -> ( K e. HL /\ W e. H ) )
13 hdmapglem7.x
 |-  ( ph -> X e. V )
14 eqid
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
15 1 4 12 dvhlmod
 |-  ( ph -> U e. LMod )
16 eqid
 |-  ( Base ` K ) = ( Base ` K )
17 eqid
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
18 eqid
 |-  ( 0g ` U ) = ( 0g ` U )
19 1 16 17 4 5 18 2 12 dvheveccl
 |-  ( ph -> E e. ( V \ { ( 0g ` U ) } ) )
20 19 eldifad
 |-  ( ph -> E e. V )
21 5 14 11 lspsncl
 |-  ( ( U e. LMod /\ E e. V ) -> ( N ` { E } ) e. ( LSubSp ` U ) )
22 15 20 21 syl2anc
 |-  ( ph -> ( N ` { E } ) e. ( LSubSp ` U ) )
23 20 snssd
 |-  ( ph -> { E } C_ V )
24 1 4 3 5 11 12 23 dochocsp
 |-  ( ph -> ( O ` ( N ` { E } ) ) = ( O ` { E } ) )
25 24 fveq2d
 |-  ( ph -> ( O ` ( O ` ( N ` { E } ) ) ) = ( O ` ( O ` { E } ) ) )
26 1 4 3 5 11 12 20 dochocsn
 |-  ( ph -> ( O ` ( O ` { E } ) ) = ( N ` { E } ) )
27 25 26 eqtrd
 |-  ( ph -> ( O ` ( O ` ( N ` { E } ) ) ) = ( N ` { E } ) )
28 1 3 4 5 14 10 12 22 27 dochexmid
 |-  ( ph -> ( ( N ` { E } ) .(+) ( O ` ( N ` { E } ) ) ) = V )
29 24 oveq2d
 |-  ( ph -> ( ( N ` { E } ) .(+) ( O ` ( N ` { E } ) ) ) = ( ( N ` { E } ) .(+) ( O ` { E } ) ) )
30 28 29 eqtr3d
 |-  ( ph -> V = ( ( N ` { E } ) .(+) ( O ` { E } ) ) )
31 13 30 eleqtrd
 |-  ( ph -> X e. ( ( N ` { E } ) .(+) ( O ` { E } ) ) )
32 14 lsssssubg
 |-  ( U e. LMod -> ( LSubSp ` U ) C_ ( SubGrp ` U ) )
33 15 32 syl
 |-  ( ph -> ( LSubSp ` U ) C_ ( SubGrp ` U ) )
34 33 22 sseldd
 |-  ( ph -> ( N ` { E } ) e. ( SubGrp ` U ) )
35 1 4 5 14 3 dochlss
 |-  ( ( ( K e. HL /\ W e. H ) /\ { E } C_ V ) -> ( O ` { E } ) e. ( LSubSp ` U ) )
36 12 23 35 syl2anc
 |-  ( ph -> ( O ` { E } ) e. ( LSubSp ` U ) )
37 33 36 sseldd
 |-  ( ph -> ( O ` { E } ) e. ( SubGrp ` U ) )
38 6 10 lsmelval
 |-  ( ( ( N ` { E } ) e. ( SubGrp ` U ) /\ ( O ` { E } ) e. ( SubGrp ` U ) ) -> ( X e. ( ( N ` { E } ) .(+) ( O ` { E } ) ) <-> E. a e. ( N ` { E } ) E. u e. ( O ` { E } ) X = ( a .+ u ) ) )
39 34 37 38 syl2anc
 |-  ( ph -> ( X e. ( ( N ` { E } ) .(+) ( O ` { E } ) ) <-> E. a e. ( N ` { E } ) E. u e. ( O ` { E } ) X = ( a .+ u ) ) )
40 31 39 mpbid
 |-  ( ph -> E. a e. ( N ` { E } ) E. u e. ( O ` { E } ) X = ( a .+ u ) )
41 rexcom
 |-  ( E. a e. ( N ` { E } ) E. u e. ( O ` { E } ) X = ( a .+ u ) <-> E. u e. ( O ` { E } ) E. a e. ( N ` { E } ) X = ( a .+ u ) )
42 df-rex
 |-  ( E. a e. ( N ` { E } ) X = ( a .+ u ) <-> E. a ( a e. ( N ` { E } ) /\ X = ( a .+ u ) ) )
43 8 9 5 7 11 lspsnel
 |-  ( ( U e. LMod /\ E e. V ) -> ( a e. ( N ` { E } ) <-> E. k e. B a = ( k .x. E ) ) )
44 15 20 43 syl2anc
 |-  ( ph -> ( a e. ( N ` { E } ) <-> E. k e. B a = ( k .x. E ) ) )
45 44 anbi1d
 |-  ( ph -> ( ( a e. ( N ` { E } ) /\ X = ( a .+ u ) ) <-> ( E. k e. B a = ( k .x. E ) /\ X = ( a .+ u ) ) ) )
46 r19.41v
 |-  ( E. k e. B ( a = ( k .x. E ) /\ X = ( a .+ u ) ) <-> ( E. k e. B a = ( k .x. E ) /\ X = ( a .+ u ) ) )
47 45 46 bitr4di
 |-  ( ph -> ( ( a e. ( N ` { E } ) /\ X = ( a .+ u ) ) <-> E. k e. B ( a = ( k .x. E ) /\ X = ( a .+ u ) ) ) )
48 47 exbidv
 |-  ( ph -> ( E. a ( a e. ( N ` { E } ) /\ X = ( a .+ u ) ) <-> E. a E. k e. B ( a = ( k .x. E ) /\ X = ( a .+ u ) ) ) )
49 rexcom4
 |-  ( E. k e. B E. a ( a = ( k .x. E ) /\ X = ( a .+ u ) ) <-> E. a E. k e. B ( a = ( k .x. E ) /\ X = ( a .+ u ) ) )
50 ovex
 |-  ( k .x. E ) e. _V
51 oveq1
 |-  ( a = ( k .x. E ) -> ( a .+ u ) = ( ( k .x. E ) .+ u ) )
52 51 eqeq2d
 |-  ( a = ( k .x. E ) -> ( X = ( a .+ u ) <-> X = ( ( k .x. E ) .+ u ) ) )
53 50 52 ceqsexv
 |-  ( E. a ( a = ( k .x. E ) /\ X = ( a .+ u ) ) <-> X = ( ( k .x. E ) .+ u ) )
54 53 rexbii
 |-  ( E. k e. B E. a ( a = ( k .x. E ) /\ X = ( a .+ u ) ) <-> E. k e. B X = ( ( k .x. E ) .+ u ) )
55 49 54 bitr3i
 |-  ( E. a E. k e. B ( a = ( k .x. E ) /\ X = ( a .+ u ) ) <-> E. k e. B X = ( ( k .x. E ) .+ u ) )
56 48 55 bitrdi
 |-  ( ph -> ( E. a ( a e. ( N ` { E } ) /\ X = ( a .+ u ) ) <-> E. k e. B X = ( ( k .x. E ) .+ u ) ) )
57 42 56 syl5bb
 |-  ( ph -> ( E. a e. ( N ` { E } ) X = ( a .+ u ) <-> E. k e. B X = ( ( k .x. E ) .+ u ) ) )
58 57 rexbidv
 |-  ( ph -> ( E. u e. ( O ` { E } ) E. a e. ( N ` { E } ) X = ( a .+ u ) <-> E. u e. ( O ` { E } ) E. k e. B X = ( ( k .x. E ) .+ u ) ) )
59 41 58 syl5bb
 |-  ( ph -> ( E. a e. ( N ` { E } ) E. u e. ( O ` { E } ) X = ( a .+ u ) <-> E. u e. ( O ` { E } ) E. k e. B X = ( ( k .x. E ) .+ u ) ) )
60 40 59 mpbid
 |-  ( ph -> E. u e. ( O ` { E } ) E. k e. B X = ( ( k .x. E ) .+ u ) )