Metamath Proof Explorer

Theorem mapdh7dN

Description: Part (7) of Baer p. 48 line 10 (4 of 6 cases). (Contributed by NM, 2-May-2015) (New usage is discouraged.)

Ref Expression
Hypotheses mapdh7.h 
|- H = ( LHyp ` K )
mapdh7.u 
|- U = ( ( DVecH ` K ) ` W )
mapdh7.v 
|- V = ( Base ` U )
mapdh7.s 
|- .- = ( -g ` U )
mapdh7.o 
|- .0. = ( 0g ` U )
mapdh7.n 
|- N = ( LSpan ` U )
mapdh7.c 
|- C = ( ( LCDual ` K ) ` W )
mapdh7.d 
|- D = ( Base ` C )
mapdh7.r 
|- R = ( -g ` C )
mapdh7.q 
|- Q = ( 0g ` C )
mapdh7.j 
|- J = ( LSpan ` C )
mapdh7.m 
|- M = ( ( mapd ` K ) ` W )
mapdh7.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 ) } ) ) ) ) )
mapdh7.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
mapdh7.f 
|- ( ph -> F e. D )
mapdh7.mn 
|- ( ph -> ( M ` ( N ` { u } ) ) = ( J ` { F } ) )
mapdh7.x 
|- ( ph -> u e. ( V \ { .0. } ) )
mapdh7.y 
|- ( ph -> v e. ( V \ { .0. } ) )
mapdh7.z 
|- ( ph -> w e. ( V \ { .0. } ) )
mapdh7.ne 
|- ( ph -> ( N ` { u } ) =/= ( N ` { v } ) )
mapdh7.wn 
|- ( ph -> -. w e. ( N ` { u , v } ) )
mapdh7a 
|- ( ph -> ( I ` <. u , F , v >. ) = G )
mapdh7.b 
|- ( ph -> ( I ` <. u , F , w >. ) = E )
Assertion mapdh7dN 
|- ( ph -> ( I ` <. v , G , w >. ) = E )

Proof

Step Hyp Ref Expression
1 mapdh7.h 
 |-  H = ( LHyp ` K )
2 mapdh7.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 mapdh7.v 
 |-  V = ( Base ` U )
4 mapdh7.s 
 |-  .- = ( -g ` U )
5 mapdh7.o 
 |-  .0. = ( 0g ` U )
6 mapdh7.n 
 |-  N = ( LSpan ` U )
7 mapdh7.c 
 |-  C = ( ( LCDual ` K ) ` W )
8 mapdh7.d 
 |-  D = ( Base ` C )
9 mapdh7.r 
 |-  R = ( -g ` C )
10 mapdh7.q 
 |-  Q = ( 0g ` C )
11 mapdh7.j 
 |-  J = ( LSpan ` C )
12 mapdh7.m 
 |-  M = ( ( mapd ` K ) ` W )
13 mapdh7.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 ) } ) ) ) ) )
14 mapdh7.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
15 mapdh7.f 
 |-  ( ph -> F e. D )
16 mapdh7.mn 
 |-  ( ph -> ( M ` ( N ` { u } ) ) = ( J ` { F } ) )
17 mapdh7.x 
 |-  ( ph -> u e. ( V \ { .0. } ) )
18 mapdh7.y 
 |-  ( ph -> v e. ( V \ { .0. } ) )
19 mapdh7.z 
 |-  ( ph -> w e. ( V \ { .0. } ) )
20 mapdh7.ne 
 |-  ( ph -> ( N ` { u } ) =/= ( N ` { v } ) )
21 mapdh7.wn 
 |-  ( ph -> -. w e. ( N ` { u , v } ) )
22 mapdh7a 
 |-  ( ph -> ( I ` <. u , F , v >. ) = G )
23 mapdh7.b 
 |-  ( ph -> ( I ` <. u , F , w >. ) = E )
24 1 2 14 dvhlvec 
 |-  ( ph -> U e. LVec )
25 18 eldifad 
 |-  ( ph -> v e. V )
26 19 eldifad 
 |-  ( ph -> w e. V )
27 3 5 6 24 17 25 26 20 21 lspindp1 
 |-  ( ph -> ( ( N ` { w } ) =/= ( N ` { v } ) /\ -. u e. ( N ` { w , v } ) ) )
28 27 simprd 
 |-  ( ph -> -. u e. ( N ` { w , v } ) )
29 prcom 
 |-  { v , w } = { w , v }
30 29 fveq2i 
 |-  ( N ` { v , w } ) = ( N ` { w , v } )
31 30 eleq2i 
 |-  ( u e. ( N ` { v , w } ) <-> u e. ( N ` { w , v } ) )
32 28 31 sylnibr 
 |-  ( ph -> -. u e. ( N ` { v , w } ) )
33 17 eldifad 
 |-  ( ph -> u e. V )
34 3 6 24 26 33 25 21 lspindpi 
 |-  ( ph -> ( ( N ` { w } ) =/= ( N ` { u } ) /\ ( N ` { w } ) =/= ( N ` { v } ) ) )
35 34 simprd 
 |-  ( ph -> ( N ` { w } ) =/= ( N ` { v } ) )
36 35 necomd 
 |-  ( ph -> ( N ` { v } ) =/= ( N ` { w } ) )
37 10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 17 18 19 32 36 22 23 mapdheq4 
 |-  ( ph -> ( I ` <. v , G , w >. ) = E )