Metamath Proof Explorer

Theorem mapdh6kN

Description: Lemmma for mapdh6N . Eliminate nonzero vector requirement. (Contributed by NM, 1-May-2015) (New usage is discouraged.)

Ref Expression
Hypotheses mapdh.q 
|- Q = ( 0g ` C )
mapdh.i 
|- I = ( x e. _V |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) )
mapdh.h 
|- H = ( LHyp ` K )
mapdh.m 
|- M = ( ( mapd ` K ) ` W )
mapdh.u 
|- U = ( ( DVecH ` K ) ` W )
mapdh.v 
|- V = ( Base ` U )
mapdh.s 
|- .- = ( -g ` U )
mapdhc.o 
|- .0. = ( 0g ` U )
mapdh.n 
|- N = ( LSpan ` U )
mapdh.c 
|- C = ( ( LCDual ` K ) ` W )
mapdh.d 
|- D = ( Base ` C )
mapdh.r 
|- R = ( -g ` C )
mapdh.j 
|- J = ( LSpan ` C )
mapdh.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
mapdhc.f 
|- ( ph -> F e. D )
mapdh.mn 
|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
mapdhcl.x 
|- ( ph -> X e. ( V \ { .0. } ) )
mapdh.p 
|- .+ = ( +g ` U )
mapdh.a 
|- .+b = ( +g ` C )
mapdh6k.y 
|- ( ph -> Y e. V )
mapdh6k.z 
|- ( ph -> Z e. V )
mapdh6k.xn 
|- ( ph -> -. X e. ( N ` { Y , Z } ) )
Assertion mapdh6kN 
|- ( ph -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) )

Proof

Step Hyp Ref Expression
1 mapdh.q 
 |-  Q = ( 0g ` C )
2 mapdh.i 
 |-  I = ( x e. _V |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) )
3 mapdh.h 
 |-  H = ( LHyp ` K )
4 mapdh.m 
 |-  M = ( ( mapd ` K ) ` W )
5 mapdh.u 
 |-  U = ( ( DVecH ` K ) ` W )
6 mapdh.v 
 |-  V = ( Base ` U )
7 mapdh.s 
 |-  .- = ( -g ` U )
8 mapdhc.o 
 |-  .0. = ( 0g ` U )
9 mapdh.n 
 |-  N = ( LSpan ` U )
10 mapdh.c 
 |-  C = ( ( LCDual ` K ) ` W )
11 mapdh.d 
 |-  D = ( Base ` C )
12 mapdh.r 
 |-  R = ( -g ` C )
13 mapdh.j 
 |-  J = ( LSpan ` C )
14 mapdh.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
15 mapdhc.f 
 |-  ( ph -> F e. D )
16 mapdh.mn 
 |-  ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
17 mapdhcl.x 
 |-  ( ph -> X e. ( V \ { .0. } ) )
18 mapdh.p 
 |-  .+ = ( +g ` U )
19 mapdh.a 
 |-  .+b = ( +g ` C )
20 mapdh6k.y 
 |-  ( ph -> Y e. V )
21 mapdh6k.z 
 |-  ( ph -> Z e. V )
22 mapdh6k.xn 
 |-  ( ph -> -. X e. ( N ` { Y , Z } ) )
23 14 adantr 
 |-  ( ( ph /\ Y = .0. ) -> ( K e. HL /\ W e. H ) )
24 15 adantr 
 |-  ( ( ph /\ Y = .0. ) -> F e. D )
25 16 adantr 
 |-  ( ( ph /\ Y = .0. ) -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
26 17 adantr 
 |-  ( ( ph /\ Y = .0. ) -> X e. ( V \ { .0. } ) )
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 23 24 25 26 18 19 27 28 29 mapdh6bN 
 |-  ( ( ph /\ Y = .0. ) -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) )
31 14 adantr 
 |-  ( ( ph /\ Z = .0. ) -> ( K e. HL /\ W e. H ) )
32 15 adantr 
 |-  ( ( ph /\ Z = .0. ) -> F e. D )
33 16 adantr 
 |-  ( ( ph /\ Z = .0. ) -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
34 17 adantr 
 |-  ( ( ph /\ Z = .0. ) -> X e. ( V \ { .0. } ) )
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 31 32 33 34 18 19 35 36 37 mapdh6cN 
 |-  ( ( ph /\ Z = .0. ) -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) )
39 14 adantr 
 |-  ( ( ph /\ ( Y =/= .0. /\ Z =/= .0. ) ) -> ( K e. HL /\ W e. H ) )
40 15 adantr 
 |-  ( ( ph /\ ( Y =/= .0. /\ Z =/= .0. ) ) -> F e. D )
41 16 adantr 
 |-  ( ( ph /\ ( Y =/= .0. /\ Z =/= .0. ) ) -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
42 17 adantr 
 |-  ( ( ph /\ ( Y =/= .0. /\ Z =/= .0. ) ) -> X e. ( V \ { .0. } ) )
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 39 40 41 42 18 19 43 47 51 mapdh6jN 
 |-  ( ( 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 >. ) ) )