# 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 } ) )
|-  ( ph -> X e. V )
|-  ( ph -> Y e. V )
26 3 5 6 9 14 24 25 dvh3dim
|-  ( ph -> E. w e. V -. w e. ( N ` { X , Y } ) )
|-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> ( K e. HL /\ W e. H ) )
|-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> F e. D )
|-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
|-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> X e. ( V \ { .0. } ) )
|-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> -. X e. ( N ` { Y , Z } ) )
|-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> ( N ` { Y } ) = ( N ` { Z } ) )
|-  ( ( ph /\ w e. V /\ -. w e. ( N ` { X , Y } ) ) -> Y e. ( V \ { .0. } ) )
|-  ( ( 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 )
|-  ( ( 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 ) )