Metamath Proof Explorer

Theorem mapdh7eN

Description: Part (7) of Baer p. 48 line 10 (5 of 6 cases). (Note: 1 of 6 and 2 of 6 are hypotheses a and b.) (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 } ) )
mapdh7b 
|- ( ph -> ( I ` <. u , F , w >. ) = E )
Assertion mapdh7eN 
|- ( ph -> ( I ` <. w , E , u >. ) = F )

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 mapdh7b 
 |-  ( ph -> ( I ` <. u , F , w >. ) = E )
23 19 eldifad 
 |-  ( ph -> w e. V )
24 1 2 14 dvhlvec 
 |-  ( ph -> U e. LVec )
25 17 eldifad 
 |-  ( ph -> u e. V )
26 18 eldifad 
 |-  ( ph -> v e. V )
27 3 6 24 23 25 26 21 lspindpi 
 |-  ( ph -> ( ( N ` { w } ) =/= ( N ` { u } ) /\ ( N ` { w } ) =/= ( N ` { v } ) ) )
28 27 simpld 
 |-  ( ph -> ( N ` { w } ) =/= ( N ` { u } ) )
29 28 necomd 
 |-  ( ph -> ( N ` { u } ) =/= ( N ` { w } ) )
30 10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 17 23 29 mapdhcl 
 |-  ( ph -> ( I ` <. u , F , w >. ) e. D )
31 22 30 eqeltrrd 
 |-  ( ph -> E e. D )
32 10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 17 19 31 29 mapdheq2 
 |-  ( ph -> ( ( I ` <. u , F , w >. ) = E -> ( I ` <. w , E , u >. ) = F ) )
33 22 32 mpd 
 |-  ( ph -> ( I ` <. w , E , u >. ) = F )