Metamath Proof Explorer

Theorem hdmap1eq4N

Description: Convert mapdheq4 to use HDMap1 function. (Contributed by NM, 17-May-2015) (New usage is discouraged.)

Ref Expression
Hypotheses hdmap1eq2.h 
|- H = ( LHyp ` K )
hdmap1eq2.u 
|- U = ( ( DVecH ` K ) ` W )
hdmap1eq2.v 
|- V = ( Base ` U )
hdmap1eq2.o 
|- .0. = ( 0g ` U )
hdmap1eq2.n 
|- N = ( LSpan ` U )
hdmap1eq2.c 
|- C = ( ( LCDual ` K ) ` W )
hdmap1eq2.d 
|- D = ( Base ` C )
hdmap1eq2.l 
|- L = ( LSpan ` C )
hdmap1eq2.m 
|- M = ( ( mapd ` K ) ` W )
hdmap1eq2.i 
|- I = ( ( HDMap1 ` K ) ` W )
hdmap1eq2.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
hdmap1eq2.f 
|- ( ph -> F e. D )
hdmap1eq2.mn 
|- ( ph -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) )
hdmap1eq4.x 
|- ( ph -> X e. ( V \ { .0. } ) )
hdmap1eq4.y 
|- ( ph -> Y e. ( V \ { .0. } ) )
hdmap1eq4.z 
|- ( ph -> Z e. ( V \ { .0. } ) )
hdmap1eq4.ne 
|- ( ph -> ( N ` { Y } ) =/= ( N ` { Z } ) )
hdmap1eq4.xn 
|- ( ph -> -. X e. ( N ` { Y , Z } ) )
hdmap1eq4.eg 
|- ( ph -> ( I ` <. X , F , Y >. ) = G )
hdmap1eq4.ee 
|- ( ph -> ( I ` <. X , F , Z >. ) = B )
Assertion hdmap1eq4N 
|- ( ph -> ( I ` <. Y , G , Z >. ) = B )

Proof

Step Hyp Ref Expression
1 hdmap1eq2.h 
 |-  H = ( LHyp ` K )
2 hdmap1eq2.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 hdmap1eq2.v 
 |-  V = ( Base ` U )
4 hdmap1eq2.o 
 |-  .0. = ( 0g ` U )
5 hdmap1eq2.n 
 |-  N = ( LSpan ` U )
6 hdmap1eq2.c 
 |-  C = ( ( LCDual ` K ) ` W )
7 hdmap1eq2.d 
 |-  D = ( Base ` C )
8 hdmap1eq2.l 
 |-  L = ( LSpan ` C )
9 hdmap1eq2.m 
 |-  M = ( ( mapd ` K ) ` W )
10 hdmap1eq2.i 
 |-  I = ( ( HDMap1 ` K ) ` W )
11 hdmap1eq2.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
12 hdmap1eq2.f 
 |-  ( ph -> F e. D )
13 hdmap1eq2.mn 
 |-  ( ph -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) )
14 hdmap1eq4.x 
 |-  ( ph -> X e. ( V \ { .0. } ) )
15 hdmap1eq4.y 
 |-  ( ph -> Y e. ( V \ { .0. } ) )
16 hdmap1eq4.z 
 |-  ( ph -> Z e. ( V \ { .0. } ) )
17 hdmap1eq4.ne 
 |-  ( ph -> ( N ` { Y } ) =/= ( N ` { Z } ) )
18 hdmap1eq4.xn 
 |-  ( ph -> -. X e. ( N ` { Y , Z } ) )
19 hdmap1eq4.eg 
 |-  ( ph -> ( I ` <. X , F , Y >. ) = G )
20 hdmap1eq4.ee 
 |-  ( ph -> ( I ` <. X , F , Z >. ) = B )
21 eqid 
 |-  ( -g ` U ) = ( -g ` U )
22 eqid 
 |-  ( -g ` C ) = ( -g ` C )
23 eqid 
 |-  ( 0g ` C ) = ( 0g ` C )
24 1 2 11 dvhlvec 
 |-  ( ph -> U e. LVec )
25 14 eldifad 
 |-  ( ph -> X e. V )
26 15 eldifad 
 |-  ( ph -> Y e. V )
27 16 eldifad 
 |-  ( ph -> Z e. V )
28 3 5 24 25 26 27 18 lspindpi 
 |-  ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) /\ ( N ` { X } ) =/= ( N ` { Z } ) ) )
29 28 simpld 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
30 1 2 3 4 5 6 7 8 9 10 11 12 13 29 14 26 hdmap1cl 
 |-  ( ph -> ( I ` <. X , F , Y >. ) e. D )
31 19 30 eqeltrrd 
 |-  ( ph -> G e. D )
32 eqid 
 |-  ( x e. _V |-> if ( ( 2nd ` x ) = .0. , ( 0g ` C ) , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( L ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` U ) ( 2nd ` x ) ) } ) ) = ( L ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` C ) h ) } ) ) ) ) ) = ( x e. _V |-> if ( ( 2nd ` x ) = .0. , ( 0g ` C ) , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( L ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` U ) ( 2nd ` x ) ) } ) ) = ( L ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` C ) h ) } ) ) ) ) )
33 1 2 3 21 4 5 6 7 22 23 8 9 10 11 15 31 27 32 hdmap1valc 
 |-  ( ph -> ( I ` <. Y , G , Z >. ) = ( ( x e. _V |-> if ( ( 2nd ` x ) = .0. , ( 0g ` C ) , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( L ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` U ) ( 2nd ` x ) ) } ) ) = ( L ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` C ) h ) } ) ) ) ) ) ` <. Y , G , Z >. ) )
34 1 2 3 21 4 5 6 7 22 23 8 9 10 11 14 12 26 32 hdmap1valc 
 |-  ( ph -> ( I ` <. X , F , Y >. ) = ( ( x e. _V |-> if ( ( 2nd ` x ) = .0. , ( 0g ` C ) , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( L ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` U ) ( 2nd ` x ) ) } ) ) = ( L ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` C ) h ) } ) ) ) ) ) ` <. X , F , Y >. ) )
35 34 19 eqtr3d 
 |-  ( ph -> ( ( x e. _V |-> if ( ( 2nd ` x ) = .0. , ( 0g ` C ) , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( L ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` U ) ( 2nd ` x ) ) } ) ) = ( L ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` C ) h ) } ) ) ) ) ) ` <. X , F , Y >. ) = G )
36 1 2 3 21 4 5 6 7 22 23 8 9 10 11 14 12 27 32 hdmap1valc 
 |-  ( ph -> ( I ` <. X , F , Z >. ) = ( ( x e. _V |-> if ( ( 2nd ` x ) = .0. , ( 0g ` C ) , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( L ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` U ) ( 2nd ` x ) ) } ) ) = ( L ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` C ) h ) } ) ) ) ) ) ` <. X , F , Z >. ) )
37 36 20 eqtr3d 
 |-  ( ph -> ( ( x e. _V |-> if ( ( 2nd ` x ) = .0. , ( 0g ` C ) , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( L ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` U ) ( 2nd ` x ) ) } ) ) = ( L ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` C ) h ) } ) ) ) ) ) ` <. X , F , Z >. ) = B )
38 23 32 1 9 2 3 21 4 5 6 7 22 8 11 12 13 14 15 16 18 17 35 37 mapdheq4 
 |-  ( ph -> ( ( x e. _V |-> if ( ( 2nd ` x ) = .0. , ( 0g ` C ) , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( L ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` U ) ( 2nd ` x ) ) } ) ) = ( L ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` C ) h ) } ) ) ) ) ) ` <. Y , G , Z >. ) = B )
39 33 38 eqtrd 
 |-  ( ph -> ( I ` <. Y , G , Z >. ) = B )