Metamath Proof Explorer


Theorem hdmap1eu

Description: Convert mapdh9a to use the HDMap1 notation. (Contributed by NM, 15-May-2015)

Ref Expression
Hypotheses hdmap1eu.h
|- H = ( LHyp ` K )
hdmap1eu.u
|- U = ( ( DVecH ` K ) ` W )
hdmap1eu.v
|- V = ( Base ` U )
hdmap1eu.o
|- .0. = ( 0g ` U )
hdmap1eu.n
|- N = ( LSpan ` U )
hdmap1eu.c
|- C = ( ( LCDual ` K ) ` W )
hdmap1eu.d
|- D = ( Base ` C )
hdmap1eu.l
|- L = ( LSpan ` C )
hdmap1eu.m
|- M = ( ( mapd ` K ) ` W )
hdmap1eu.i
|- I = ( ( HDMap1 ` K ) ` W )
hdmap1eu.k
|- ( ph -> ( K e. HL /\ W e. H ) )
hdmap1eu.mn
|- ( ph -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) )
hdmap1eu.x
|- ( ph -> X e. ( V \ { .0. } ) )
hdmap1eu.f
|- ( ph -> F e. D )
hdmap1eu.t
|- ( ph -> T e. V )
Assertion hdmap1eu
|- ( ph -> E! y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) )

Proof

Step Hyp Ref Expression
1 hdmap1eu.h
 |-  H = ( LHyp ` K )
2 hdmap1eu.u
 |-  U = ( ( DVecH ` K ) ` W )
3 hdmap1eu.v
 |-  V = ( Base ` U )
4 hdmap1eu.o
 |-  .0. = ( 0g ` U )
5 hdmap1eu.n
 |-  N = ( LSpan ` U )
6 hdmap1eu.c
 |-  C = ( ( LCDual ` K ) ` W )
7 hdmap1eu.d
 |-  D = ( Base ` C )
8 hdmap1eu.l
 |-  L = ( LSpan ` C )
9 hdmap1eu.m
 |-  M = ( ( mapd ` K ) ` W )
10 hdmap1eu.i
 |-  I = ( ( HDMap1 ` K ) ` W )
11 hdmap1eu.k
 |-  ( ph -> ( K e. HL /\ W e. H ) )
12 hdmap1eu.mn
 |-  ( ph -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) )
13 hdmap1eu.x
 |-  ( ph -> X e. ( V \ { .0. } ) )
14 hdmap1eu.f
 |-  ( ph -> F e. D )
15 hdmap1eu.t
 |-  ( ph -> T e. V )
16 eqid
 |-  ( -g ` U ) = ( -g ` U )
17 eqid
 |-  ( -g ` C ) = ( -g ` C )
18 eqid
 |-  ( 0g ` C ) = ( 0g ` C )
19 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 ) } ) ) ) ) )
20 19 hdmap1cbv
 |-  ( 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 ) } ) ) ) ) ) = ( w e. _V |-> if ( ( 2nd ` w ) = .0. , ( 0g ` C ) , ( iota_ g e. D ( ( M ` ( N ` { ( 2nd ` w ) } ) ) = ( L ` { g } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` w ) ) ( -g ` U ) ( 2nd ` w ) ) } ) ) = ( L ` { ( ( 2nd ` ( 1st ` w ) ) ( -g ` C ) g ) } ) ) ) ) )
21 1 2 3 16 4 5 6 7 17 18 8 9 10 11 12 13 14 15 20 hdmap1eulem
 |-  ( ph -> E! y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) )