Metamath Proof Explorer

Theorem mapdheq4

Description: Lemma for ~? mapdh . Part (4) in Baer p. 46. (Contributed by NM, 12-Apr-2015)

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. } ) )
mapdhe4.y 
|- ( ph -> Y e. ( V \ { .0. } ) )
mapdhe.z 
|- ( ph -> Z e. ( V \ { .0. } ) )
mapdh.xn 
|- ( ph -> -. X e. ( N ` { Y , Z } ) )
mapdh.yz 
|- ( ph -> ( N ` { Y } ) =/= ( N ` { Z } ) )
mapdh.eg 
|- ( ph -> ( I ` <. X , F , Y >. ) = G )
mapdh.ee 
|- ( ph -> ( I ` <. X , F , Z >. ) = E )
Assertion mapdheq4 
|- ( ph -> ( I ` <. Y , G , Z >. ) = E )

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 mapdhe4.y 
 |-  ( ph -> Y e. ( V \ { .0. } ) )
19 mapdhe.z 
 |-  ( ph -> Z e. ( V \ { .0. } ) )
20 mapdh.xn 
 |-  ( ph -> -. X e. ( N ` { Y , Z } ) )
21 mapdh.yz 
 |-  ( ph -> ( N ` { Y } ) =/= ( N ` { Z } ) )
22 mapdh.eg 
 |-  ( ph -> ( I ` <. X , F , Y >. ) = G )
23 mapdh.ee 
 |-  ( ph -> ( I ` <. X , F , Z >. ) = E )
24 19 eldifad 
 |-  ( ph -> Z e. V )
25 3 5 14 dvhlvec 
 |-  ( ph -> U e. LVec )
26 17 eldifad 
 |-  ( ph -> X e. V )
27 6 8 9 25 18 24 26 21 20 lspindp1 
 |-  ( ph -> ( ( N ` { X } ) =/= ( N ` { Z } ) /\ -. Y e. ( N ` { X , Z } ) ) )
28 27 simpld 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { Z } ) )
29 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 24 28 mapdhcl 
 |-  ( ph -> ( I ` <. X , F , Z >. ) e. D )
30 23 29 eqeltrrd 
 |-  ( ph -> E e. D )
31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 19 30 28 mapdheq 
 |-  ( ph -> ( ( I ` <. X , F , Z >. ) = E <-> ( ( M ` ( N ` { Z } ) ) = ( J ` { E } ) /\ ( M ` ( N ` { ( X .- Z ) } ) ) = ( J ` { ( F R E ) } ) ) ) )
32 23 31 mpbid 
 |-  ( ph -> ( ( M ` ( N ` { Z } ) ) = ( J ` { E } ) /\ ( M ` ( N ` { ( X .- Z ) } ) ) = ( J ` { ( F R E ) } ) ) )
33 32 simpld 
 |-  ( ph -> ( M ` ( N ` { Z } ) ) = ( J ` { E } ) )
34 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 mapdheq4lem 
 |-  ( ph -> ( M ` ( N ` { ( Y .- Z ) } ) ) = ( J ` { ( G R E ) } ) )
35 18 eldifad 
 |-  ( ph -> Y e. V )
36 6 8 9 25 35 19 26 21 20 lspindp2 
 |-  ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) /\ -. Z e. ( N ` { X , Y } ) ) )
37 36 simpld 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 35 37 mapdhcl 
 |-  ( ph -> ( I ` <. X , F , Y >. ) e. D )
39 22 38 eqeltrrd 
 |-  ( ph -> G e. D )
40 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 39 37 mapdheq 
 |-  ( ph -> ( ( I ` <. X , F , Y >. ) = G <-> ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) ) )
41 22 40 mpbid 
 |-  ( ph -> ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) )
42 41 simpld 
 |-  ( ph -> ( M ` ( N ` { Y } ) ) = ( J ` { G } ) )
43 1 2 3 4 5 6 7 8 9 10 11 12 13 14 39 42 18 19 30 21 mapdheq 
 |-  ( ph -> ( ( I ` <. Y , G , Z >. ) = E <-> ( ( M ` ( N ` { Z } ) ) = ( J ` { E } ) /\ ( M ` ( N ` { ( Y .- Z ) } ) ) = ( J ` { ( G R E ) } ) ) ) )
44 33 34 43 mpbir2and 
 |-  ( ph -> ( I ` <. Y , G , Z >. ) = E )