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 |
|
hdmap1cl.ne |
|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) ) |
15 |
|
hdmap1cl.x |
|- ( ph -> X e. ( V \ { .0. } ) ) |
16 |
|
hdmap1cl.y |
|- ( ph -> Y e. V ) |
17 |
|
eqid |
|- ( -g ` U ) = ( -g ` U ) |
18 |
|
eqid |
|- ( -g ` C ) = ( -g ` C ) |
19 |
|
eqid |
|- ( 0g ` C ) = ( 0g ` C ) |
20 |
|
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 ) } ) ) ) ) ) |
21 |
1 2 3 17 4 5 6 7 18 19 8 9 10 11 15 12 16 20
|
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 >. ) ) |
22 |
19 20 1 9 2 3 17 4 5 6 7 18 8 11 12 13 15 16 14
|
mapdhcl |
|- ( 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 >. ) e. D ) |
23 |
21 22
|
eqeltrd |
|- ( ph -> ( I ` <. X , F , Y >. ) e. D ) |