Metamath Proof Explorer

Theorem hdmap1l6k

Description: Lemmma for hdmap1l6 . Eliminate nonzero vector requirement. (Contributed by NM, 1-May-2015)

Ref Expression
Hypotheses hdmap1l6.h 
|- H = ( LHyp ` K )
hdmap1l6.u 
|- U = ( ( DVecH ` K ) ` W )
hdmap1l6.v 
|- V = ( Base ` U )
hdmap1l6.p 
|- .+ = ( +g ` U )
hdmap1l6.s 
|- .- = ( -g ` U )
hdmap1l6c.o 
|- .0. = ( 0g ` U )
hdmap1l6.n 
|- N = ( LSpan ` U )
hdmap1l6.c 
|- C = ( ( LCDual ` K ) ` W )
hdmap1l6.d 
|- D = ( Base ` C )
hdmap1l6.a 
|- .+b = ( +g ` C )
hdmap1l6.r 
|- R = ( -g ` C )
hdmap1l6.q 
|- Q = ( 0g ` C )
hdmap1l6.l 
|- L = ( LSpan ` C )
hdmap1l6.m 
|- M = ( ( mapd ` K ) ` W )
hdmap1l6.i 
|- I = ( ( HDMap1 ` K ) ` W )
hdmap1l6.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
hdmap1l6.f 
|- ( ph -> F e. D )
hdmap1l6cl.x 
|- ( ph -> X e. ( V \ { .0. } ) )
hdmap1l6.mn 
|- ( ph -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) )
hdmap1l6k.y 
|- ( ph -> Y e. V )
hdmap1l6k.z 
|- ( ph -> Z e. V )
hdmap1l6k.xn 
|- ( ph -> -. X e. ( N ` { Y , Z } ) )
Assertion hdmap1l6k 
|- ( ph -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) )

Proof

Step Hyp Ref Expression
1 hdmap1l6.h 
 |-  H = ( LHyp ` K )
2 hdmap1l6.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 hdmap1l6.v 
 |-  V = ( Base ` U )
4 hdmap1l6.p 
 |-  .+ = ( +g ` U )
5 hdmap1l6.s 
 |-  .- = ( -g ` U )
6 hdmap1l6c.o 
 |-  .0. = ( 0g ` U )
7 hdmap1l6.n 
 |-  N = ( LSpan ` U )
8 hdmap1l6.c 
 |-  C = ( ( LCDual ` K ) ` W )
9 hdmap1l6.d 
 |-  D = ( Base ` C )
10 hdmap1l6.a 
 |-  .+b = ( +g ` C )
11 hdmap1l6.r 
 |-  R = ( -g ` C )
12 hdmap1l6.q 
 |-  Q = ( 0g ` C )
13 hdmap1l6.l 
 |-  L = ( LSpan ` C )
14 hdmap1l6.m 
 |-  M = ( ( mapd ` K ) ` W )
15 hdmap1l6.i 
 |-  I = ( ( HDMap1 ` K ) ` W )
16 hdmap1l6.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
17 hdmap1l6.f 
 |-  ( ph -> F e. D )
18 hdmap1l6cl.x 
 |-  ( ph -> X e. ( V \ { .0. } ) )
19 hdmap1l6.mn 
 |-  ( ph -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) )
20 hdmap1l6k.y 
 |-  ( ph -> Y e. V )
21 hdmap1l6k.z 
 |-  ( ph -> Z e. V )
22 hdmap1l6k.xn 
 |-  ( ph -> -. X e. ( N ` { Y , Z } ) )
23 16 adantr 
 |-  ( ( ph /\ Y = .0. ) -> ( K e. HL /\ W e. H ) )
24 17 adantr 
 |-  ( ( ph /\ Y = .0. ) -> F e. D )
25 18 adantr 
 |-  ( ( ph /\ Y = .0. ) -> X e. ( V \ { .0. } ) )
26 19 adantr 
 |-  ( ( ph /\ Y = .0. ) -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) )
27 simpr 
 |-  ( ( ph /\ Y = .0. ) -> Y = .0. )
28 21 adantr 
 |-  ( ( ph /\ Y = .0. ) -> Z e. V )
29 22 adantr 
 |-  ( ( ph /\ Y = .0. ) -> -. X e. ( N ` { Y , Z } ) )
30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 23 24 25 26 27 28 29 hdmap1l6b 
 |-  ( ( ph /\ Y = .0. ) -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) )
31 16 adantr 
 |-  ( ( ph /\ Z = .0. ) -> ( K e. HL /\ W e. H ) )
32 17 adantr 
 |-  ( ( ph /\ Z = .0. ) -> F e. D )
33 18 adantr 
 |-  ( ( ph /\ Z = .0. ) -> X e. ( V \ { .0. } ) )
34 19 adantr 
 |-  ( ( ph /\ Z = .0. ) -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) )
35 20 adantr 
 |-  ( ( ph /\ Z = .0. ) -> Y e. V )
36 simpr 
 |-  ( ( ph /\ Z = .0. ) -> Z = .0. )
37 22 adantr 
 |-  ( ( ph /\ Z = .0. ) -> -. X e. ( N ` { Y , Z } ) )
38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 31 32 33 34 35 36 37 hdmap1l6c 
 |-  ( ( ph /\ Z = .0. ) -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) )
39 16 adantr 
 |-  ( ( ph /\ ( Y =/= .0. /\ Z =/= .0. ) ) -> ( K e. HL /\ W e. H ) )
40 17 adantr 
 |-  ( ( ph /\ ( Y =/= .0. /\ Z =/= .0. ) ) -> F e. D )
41 18 adantr 
 |-  ( ( ph /\ ( Y =/= .0. /\ Z =/= .0. ) ) -> X e. ( V \ { .0. } ) )
42 19 adantr 
 |-  ( ( ph /\ ( Y =/= .0. /\ Z =/= .0. ) ) -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) )
43 22 adantr 
 |-  ( ( ph /\ ( Y =/= .0. /\ Z =/= .0. ) ) -> -. X e. ( N ` { Y , Z } ) )
44 20 adantr 
 |-  ( ( ph /\ ( Y =/= .0. /\ Z =/= .0. ) ) -> Y e. V )
45 simprl 
 |-  ( ( ph /\ ( Y =/= .0. /\ Z =/= .0. ) ) -> Y =/= .0. )
46 eldifsn 
 |-  ( Y e. ( V \ { .0. } ) <-> ( Y e. V /\ Y =/= .0. ) )
47 44 45 46 sylanbrc 
 |-  ( ( ph /\ ( Y =/= .0. /\ Z =/= .0. ) ) -> Y e. ( V \ { .0. } ) )
48 21 adantr 
 |-  ( ( ph /\ ( Y =/= .0. /\ Z =/= .0. ) ) -> Z e. V )
49 simprr 
 |-  ( ( ph /\ ( Y =/= .0. /\ Z =/= .0. ) ) -> Z =/= .0. )
50 eldifsn 
 |-  ( Z e. ( V \ { .0. } ) <-> ( Z e. V /\ Z =/= .0. ) )
51 48 49 50 sylanbrc 
 |-  ( ( ph /\ ( Y =/= .0. /\ Z =/= .0. ) ) -> Z e. ( V \ { .0. } ) )
52 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 39 40 41 42 43 47 51 hdmap1l6j 
 |-  ( ( ph /\ ( Y =/= .0. /\ Z =/= .0. ) ) -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) )
53 30 38 52 pm2.61da2ne 
 |-  ( ph -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) )