Metamath Proof Explorer

Theorem mapdh75fN

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

Ref Expression
Hypotheses mapdh75.h 
|- H = ( LHyp ` K )
mapdh75.u 
|- U = ( ( DVecH ` K ) ` W )
mapdh75.v 
|- V = ( Base ` U )
mapdh75.s 
|- .- = ( -g ` U )
mapdh75.o 
|- .0. = ( 0g ` U )
mapdh75.n 
|- N = ( LSpan ` U )
mapdh75.c 
|- C = ( ( LCDual ` K ) ` W )
mapdh75.d 
|- D = ( Base ` C )
mapdh75.r 
|- R = ( -g ` C )
mapdh75.q 
|- Q = ( 0g ` C )
mapdh75.j 
|- J = ( LSpan ` C )
mapdh75.m 
|- M = ( ( mapd ` K ) ` W )
mapdh75.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 ) } ) ) ) ) )
mapdh75.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
mapdh75.f 
|- ( ph -> F e. D )
mapdh75.mn 
|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
mapdh75a 
|- ( ph -> ( I ` <. X , F , Y >. ) = G )
mapdh75d.b 
|- ( ph -> ( I ` <. X , F , Z >. ) = E )
mapdh75d.vw 
|- ( ph -> ( N ` { Y } ) =/= ( N ` { Z } ) )
mapdh75d.un 
|- ( ph -> -. X e. ( N ` { Y , Z } ) )
mapdh75d.x 
|- ( ph -> X e. ( V \ { .0. } ) )
mapdh75d.y 
|- ( ph -> Y e. ( V \ { .0. } ) )
mapdh75d.z 
|- ( ph -> Z e. ( V \ { .0. } ) )
Assertion mapdh75fN 
|- ( ph -> ( I ` <. Z , E , Y >. ) = G )

Proof

Step Hyp Ref Expression
1 mapdh75.h 
 |-  H = ( LHyp ` K )
2 mapdh75.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 mapdh75.v 
 |-  V = ( Base ` U )
4 mapdh75.s 
 |-  .- = ( -g ` U )
5 mapdh75.o 
 |-  .0. = ( 0g ` U )
6 mapdh75.n 
 |-  N = ( LSpan ` U )
7 mapdh75.c 
 |-  C = ( ( LCDual ` K ) ` W )
8 mapdh75.d 
 |-  D = ( Base ` C )
9 mapdh75.r 
 |-  R = ( -g ` C )
10 mapdh75.q 
 |-  Q = ( 0g ` C )
11 mapdh75.j 
 |-  J = ( LSpan ` C )
12 mapdh75.m 
 |-  M = ( ( mapd ` K ) ` W )
13 mapdh75.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 mapdh75.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
15 mapdh75.f 
 |-  ( ph -> F e. D )
16 mapdh75.mn 
 |-  ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
17 mapdh75a 
 |-  ( ph -> ( I ` <. X , F , Y >. ) = G )
18 mapdh75d.b 
 |-  ( ph -> ( I ` <. X , F , Z >. ) = E )
19 mapdh75d.vw 
 |-  ( ph -> ( N ` { Y } ) =/= ( N ` { Z } ) )
20 mapdh75d.un 
 |-  ( ph -> -. X e. ( N ` { Y , Z } ) )
21 mapdh75d.x 
 |-  ( ph -> X e. ( V \ { .0. } ) )
22 mapdh75d.y 
 |-  ( ph -> Y e. ( V \ { .0. } ) )
23 mapdh75d.z 
 |-  ( ph -> Z e. ( V \ { .0. } ) )
24 22 eldifad 
 |-  ( ph -> Y e. V )
25 1 2 14 dvhlvec 
 |-  ( ph -> U e. LVec )
26 21 eldifad 
 |-  ( ph -> X e. V )
27 23 eldifad 
 |-  ( ph -> Z e. V )
28 3 6 25 26 24 27 20 lspindpi 
 |-  ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) /\ ( N ` { X } ) =/= ( N ` { Z } ) ) )
29 28 simpld 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
30 10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 21 24 29 mapdhcl 
 |-  ( ph -> ( I ` <. X , F , Y >. ) e. D )
31 17 30 eqeltrrd 
 |-  ( ph -> G e. D )
32 10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 21 22 31 29 mapdheq 
 |-  ( ph -> ( ( I ` <. X , F , Y >. ) = G <-> ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) ) )
33 17 32 mpbid 
 |-  ( ph -> ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) )
34 33 simpld 
 |-  ( ph -> ( M ` ( N ` { Y } ) ) = ( J ` { G } ) )
35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 mapdh75d 
 |-  ( ph -> ( I ` <. Y , G , Z >. ) = E )
36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 31 34 35 19 22 23 mapdh75e 
 |-  ( ph -> ( I ` <. Z , E , Y >. ) = G )