Metamath Proof Explorer

Theorem lclkrlem2l

Description: Lemma for lclkr . Eliminate the X =/= .0. , Y =/= .0. hypotheses. (Contributed by NM, 18-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 } ) ) )
lclkrlem2l.x 
|- ( ph -> X e. V )
lclkrlem2l.y 
|- ( ph -> Y e. V )
Assertion lclkrlem2l 
|- ( 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 lclkrlem2l.x 
 |-  ( ph -> X e. V )
24 lclkrlem2l.y 
 |-  ( ph -> Y e. V )
25 15 adantr 
 |-  ( ( ph /\ X = .0. ) -> ( K e. HL /\ W e. H ) )
26 16 adantr 
 |-  ( ( ph /\ X = .0. ) -> B e. ( V \ { .0. } ) )
27 17 adantr 
 |-  ( ( ph /\ X = .0. ) -> E e. F )
28 18 adantr 
 |-  ( ( ph /\ X = .0. ) -> G e. F )
29 19 adantr 
 |-  ( ( ph /\ X = .0. ) -> ( L ` E ) = ( ._|_ ` { X } ) )
30 20 adantr 
 |-  ( ( ph /\ X = .0. ) -> ( L ` G ) = ( ._|_ ` { Y } ) )
31 21 adantr 
 |-  ( ( ph /\ X = .0. ) -> ( ( E .+ G ) ` B ) = Q )
32 22 adantr 
 |-  ( ( ph /\ X = .0. ) -> ( -. X e. ( ._|_ ` { B } ) \/ -. Y e. ( ._|_ ` { B } ) ) )
33 simpr 
 |-  ( ( ph /\ X = .0. ) -> X = .0. )
34 24 adantr 
 |-  ( ( ph /\ X = .0. ) -> Y e. V )
35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 25 26 27 28 29 30 31 32 33 34 lclkrlem2k 
 |-  ( ( ph /\ X = .0. ) -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) )
36 15 adantr 
 |-  ( ( ph /\ Y = .0. ) -> ( K e. HL /\ W e. H ) )
37 16 adantr 
 |-  ( ( ph /\ Y = .0. ) -> B e. ( V \ { .0. } ) )
38 17 adantr 
 |-  ( ( ph /\ Y = .0. ) -> E e. F )
39 18 adantr 
 |-  ( ( ph /\ Y = .0. ) -> G e. F )
40 19 adantr 
 |-  ( ( ph /\ Y = .0. ) -> ( L ` E ) = ( ._|_ ` { X } ) )
41 20 adantr 
 |-  ( ( ph /\ Y = .0. ) -> ( L ` G ) = ( ._|_ ` { Y } ) )
42 21 adantr 
 |-  ( ( ph /\ Y = .0. ) -> ( ( E .+ G ) ` B ) = Q )
43 22 adantr 
 |-  ( ( ph /\ Y = .0. ) -> ( -. X e. ( ._|_ ` { B } ) \/ -. Y e. ( ._|_ ` { B } ) ) )
44 23 adantr 
 |-  ( ( ph /\ Y = .0. ) -> X e. V )
45 simpr 
 |-  ( ( ph /\ Y = .0. ) -> Y = .0. )
46 1 2 3 4 5 6 7 8 9 10 11 12 13 14 36 37 38 39 40 41 42 43 44 45 lclkrlem2j 
 |-  ( ( ph /\ Y = .0. ) -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) )
47 15 adantr 
 |-  ( ( ph /\ ( X =/= .0. /\ Y =/= .0. ) ) -> ( K e. HL /\ W e. H ) )
48 16 adantr 
 |-  ( ( ph /\ ( X =/= .0. /\ Y =/= .0. ) ) -> B e. ( V \ { .0. } ) )
49 17 adantr 
 |-  ( ( ph /\ ( X =/= .0. /\ Y =/= .0. ) ) -> E e. F )
50 18 adantr 
 |-  ( ( ph /\ ( X =/= .0. /\ Y =/= .0. ) ) -> G e. F )
51 19 adantr 
 |-  ( ( ph /\ ( X =/= .0. /\ Y =/= .0. ) ) -> ( L ` E ) = ( ._|_ ` { X } ) )
52 20 adantr 
 |-  ( ( ph /\ ( X =/= .0. /\ Y =/= .0. ) ) -> ( L ` G ) = ( ._|_ ` { Y } ) )
53 21 adantr 
 |-  ( ( ph /\ ( X =/= .0. /\ Y =/= .0. ) ) -> ( ( E .+ G ) ` B ) = Q )
54 22 adantr 
 |-  ( ( ph /\ ( X =/= .0. /\ Y =/= .0. ) ) -> ( -. X e. ( ._|_ ` { B } ) \/ -. Y e. ( ._|_ ` { B } ) ) )
55 23 adantr 
 |-  ( ( ph /\ ( X =/= .0. /\ Y =/= .0. ) ) -> X e. V )
56 simprl 
 |-  ( ( ph /\ ( X =/= .0. /\ Y =/= .0. ) ) -> X =/= .0. )
57 eldifsn 
 |-  ( X e. ( V \ { .0. } ) <-> ( X e. V /\ X =/= .0. ) )
58 55 56 57 sylanbrc 
 |-  ( ( ph /\ ( X =/= .0. /\ Y =/= .0. ) ) -> X e. ( V \ { .0. } ) )
59 24 adantr 
 |-  ( ( ph /\ ( X =/= .0. /\ Y =/= .0. ) ) -> Y e. V )
60 simprr 
 |-  ( ( ph /\ ( X =/= .0. /\ Y =/= .0. ) ) -> Y =/= .0. )
61 eldifsn 
 |-  ( Y e. ( V \ { .0. } ) <-> ( Y e. V /\ Y =/= .0. ) )
62 59 60 61 sylanbrc 
 |-  ( ( ph /\ ( X =/= .0. /\ Y =/= .0. ) ) -> Y e. ( V \ { .0. } ) )
63 1 2 3 4 5 6 7 8 9 10 11 12 13 14 47 48 49 50 51 52 53 54 58 62 lclkrlem2i 
 |-  ( ( ph /\ ( X =/= .0. /\ Y =/= .0. ) ) -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) )
64 35 46 63 pm2.61da2ne 
 |-  ( ph -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) )