Metamath Proof Explorer

Theorem mapdh6iN

Description: Lemmma for mapdh6N . Eliminate auxiliary vector w . (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 )
mapdh6i.xn 
|- ( ph -> -. X e. ( N ` { Y , Z } ) )
mapdh6i.y 
|- ( ph -> Y e. ( V \ { .0. } ) )
mapdh6i.z 
|- ( ph -> Z e. ( V \ { .0. } ) )
mapdh6i.yz 
|- ( ph -> ( N ` { Y } ) = ( N ` { Z } ) )
Assertion mapdh6iN 
|- ( 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 mapdh6i.xn 
 |-  ( ph -> -. X e. ( N ` { Y , Z } ) )
21 mapdh6i.y 
 |-  ( ph -> Y e. ( V \ { .0. } ) )
22 mapdh6i.z 
 |-  ( ph -> Z e. ( V \ { .0. } ) )
23 mapdh6i.yz 
 |-  ( ph -> ( N ` { Y } ) = ( N ` { Z } ) )
24 17 eldifad 
 |-  ( ph -> X e. V )
25 21 eldifad 
 |-  ( ph -> Y e. V )
26 3 5 6 9 14 24 25 dvh3dim 
 |-  ( ph -> E. w e. V -. w e. ( N ` { X , Y } ) )
27 14 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> ( K e. HL /\ W e. H ) )
28 15 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> F e. D )
29 16 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
30 17 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> X e. ( V \ { .0. } ) )
31 20 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> -. X e. ( N ` { Y , Z } ) )
32 23 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> ( N ` { Y } ) = ( N ` { Z } ) )
33 21 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> Y e. ( V \ { .0. } ) )
34 22 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> Z e. ( V \ { .0. } ) )
35 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
36 3 5 14 dvhlmod 
 |-  ( ph -> U e. LMod )
37 36 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> U e. LMod )
38 6 35 9 36 24 25 lspprcl 
 |-  ( ph -> ( N ` { X , Y } ) e. ( LSubSp ` U ) )
39 38 3ad2ant1 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> ( N ` { X , Y } ) e. ( LSubSp ` U ) )
40 simp2 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> w e. V )
41 simp3 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> -. w e. ( N ` { X , Y } ) )
42 8 35 37 39 40 41 lssneln0 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> w e. ( V \ { .0. } ) )
43 1 2 3 4 5 6 7 8 9 10 11 12 13 27 28 29 30 18 19 31 32 33 34 42 41 mapdh6hN 
 |-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) )
44 43 rexlimdv3a 
 |-  ( ph -> ( E. w e. V -. w e. ( N ` { X , Y } ) -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) ) )
45 26 44 mpd 
 |-  ( ph -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) )