Metamath Proof Explorer


Theorem mapdh6b0N

Description: Lemmma for mapdh6N . (Contributed by NM, 23-Apr-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 )
mapdh6b0.y
|- ( ph -> Y e. V )
mapdh6b0.z
|- ( ph -> Z e. V )
mapdh6b0.ne
|- ( ph -> ( ( N ` { X } ) i^i ( N ` { Y , Z } ) ) = { .0. } )
Assertion mapdh6b0N
|- ( ph -> -. X e. ( N ` { Y , 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 mapdh6b0.y
 |-  ( ph -> Y e. V )
21 mapdh6b0.z
 |-  ( ph -> Z e. V )
22 mapdh6b0.ne
 |-  ( ph -> ( ( N ` { X } ) i^i ( N ` { Y , Z } ) ) = { .0. } )
23 eqid
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
24 3 5 14 dvhlvec
 |-  ( ph -> U e. LVec )
25 3 5 14 dvhlmod
 |-  ( ph -> U e. LMod )
26 6 23 9 25 20 21 lspprcl
 |-  ( ph -> ( N ` { Y , Z } ) e. ( LSubSp ` U ) )
27 6 8 9 23 24 26 17 lspdisjb
 |-  ( ph -> ( -. X e. ( N ` { Y , Z } ) <-> ( ( N ` { X } ) i^i ( N ` { Y , Z } ) ) = { .0. } ) )
28 22 27 mpbird
 |-  ( ph -> -. X e. ( N ` { Y , Z } ) )