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 ellspsn 
 |-  ( ( 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 bitrid 
 |-  ( 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 bitrid 
 |-  ( 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 ) )