Metamath Proof Explorer

Theorem hdmap1eulemOLDN

Description: Lemma for hdmap1euOLDN . TODO: combine with hdmap1euOLDN or at least share some hypotheses. (Contributed by NM, 15-May-2015) (New usage is discouraged.)

Ref Expression
Hypotheses hdmap1eulem.h 
|- H = ( LHyp ` K )
hdmap1eulem.u 
|- U = ( ( DVecH ` K ) ` W )
hdmap1eulem.v 
|- V = ( Base ` U )
hdmap1eulem.s 
|- .- = ( -g ` U )
hdmap1eulem.o 
|- .0. = ( 0g ` U )
hdmap1eulem.n 
|- N = ( LSpan ` U )
hdmap1eulem.c 
|- C = ( ( LCDual ` K ) ` W )
hdmap1eulem.d 
|- D = ( Base ` C )
hdmap1eulem.r 
|- R = ( -g ` C )
hdmap1eulem.q 
|- Q = ( 0g ` C )
hdmap1eulem.j 
|- J = ( LSpan ` C )
hdmap1eulem.m 
|- M = ( ( mapd ` K ) ` W )
hdmap1eulem.i 
|- I = ( ( HDMap1 ` K ) ` W )
hdmap1eulem.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
hdmap1eulem.mn 
|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
hdmap1eulem.x 
|- ( ph -> X e. ( V \ { .0. } ) )
hdmap1eulem.f 
|- ( ph -> F e. D )
hdmap1eulem.y 
|- ( ph -> T e. V )
hdmap1eulem.l 
|- L = ( 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 ) } ) ) ) ) )
Assertion hdmap1eulemOLDN 
|- ( ph -> E! y e. D A. z e. V ( -. z e. ( N ` { X , T } ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) )

Proof

Step Hyp Ref Expression
1 hdmap1eulem.h 
 |-  H = ( LHyp ` K )
2 hdmap1eulem.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 hdmap1eulem.v 
 |-  V = ( Base ` U )
4 hdmap1eulem.s 
 |-  .- = ( -g ` U )
5 hdmap1eulem.o 
 |-  .0. = ( 0g ` U )
6 hdmap1eulem.n 
 |-  N = ( LSpan ` U )
7 hdmap1eulem.c 
 |-  C = ( ( LCDual ` K ) ` W )
8 hdmap1eulem.d 
 |-  D = ( Base ` C )
9 hdmap1eulem.r 
 |-  R = ( -g ` C )
10 hdmap1eulem.q 
 |-  Q = ( 0g ` C )
11 hdmap1eulem.j 
 |-  J = ( LSpan ` C )
12 hdmap1eulem.m 
 |-  M = ( ( mapd ` K ) ` W )
13 hdmap1eulem.i 
 |-  I = ( ( HDMap1 ` K ) ` W )
14 hdmap1eulem.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
15 hdmap1eulem.mn 
 |-  ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
16 hdmap1eulem.x 
 |-  ( ph -> X e. ( V \ { .0. } ) )
17 hdmap1eulem.f 
 |-  ( ph -> F e. D )
18 hdmap1eulem.y 
 |-  ( ph -> T e. V )
19 hdmap1eulem.l 
 |-  L = ( 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 ) } ) ) ) ) )
20 1 2 3 4 5 6 7 8 9 10 11 12 19 14 17 15 16 18 mapdh9aOLDN 
 |-  ( ph -> E! y e. D A. z e. V ( -. z e. ( N ` { X , T } ) -> y = ( L ` <. z , ( L ` <. X , F , z >. ) , T >. ) ) )
21 14 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> ( K e. HL /\ W e. H ) )
22 16 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> X e. ( V \ { .0. } ) )
23 17 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> F e. D )
24 simplr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> z e. V )
25 1 2 3 4 5 6 7 8 9 10 11 12 13 21 22 23 24 19 hdmap1valc 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> ( I ` <. X , F , z >. ) = ( L ` <. X , F , z >. ) )
26 25 oteq2d 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> <. z , ( I ` <. X , F , z >. ) , T >. = <. z , ( L ` <. X , F , z >. ) , T >. )
27 26 fveq2d 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) = ( I ` <. z , ( L ` <. X , F , z >. ) , T >. ) )
28 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
29 1 2 14 dvhlmod 
 |-  ( ph -> U e. LMod )
30 29 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> U e. LMod )
31 16 eldifad 
 |-  ( ph -> X e. V )
32 3 28 6 29 31 18 lspprcl 
 |-  ( ph -> ( N ` { X , T } ) e. ( LSubSp ` U ) )
33 32 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> ( N ` { X , T } ) e. ( LSubSp ` U ) )
34 simpr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> -. z e. ( N ` { X , T } ) )
35 5 28 30 33 24 34 lssneln0 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> z e. ( V \ { .0. } ) )
36 15 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
37 1 2 14 dvhlvec 
 |-  ( ph -> U e. LVec )
38 37 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> U e. LVec )
39 31 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> X e. V )
40 18 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> T e. V )
41 3 6 38 24 39 40 34 lspindpi 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) )
42 41 simpld 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> ( N ` { z } ) =/= ( N ` { X } ) )
43 42 necomd 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> ( N ` { X } ) =/= ( N ` { z } ) )
44 10 19 1 12 2 3 4 5 6 7 8 9 11 21 23 36 22 24 43 mapdhcl 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> ( L ` <. X , F , z >. ) e. D )
45 1 2 3 4 5 6 7 8 9 10 11 12 13 21 35 44 40 19 hdmap1valc 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> ( I ` <. z , ( L ` <. X , F , z >. ) , T >. ) = ( L ` <. z , ( L ` <. X , F , z >. ) , T >. ) )
46 27 45 eqtrd 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) = ( L ` <. z , ( L ` <. X , F , z >. ) , T >. ) )
47 46 eqeq2d 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> ( y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) <-> y = ( L ` <. z , ( L ` <. X , F , z >. ) , T >. ) ) )
48 47 pm5.74da 
 |-  ( ( ph /\ z e. V ) -> ( ( -. z e. ( N ` { X , T } ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) <-> ( -. z e. ( N ` { X , T } ) -> y = ( L ` <. z , ( L ` <. X , F , z >. ) , T >. ) ) ) )
49 48 ralbidva 
 |-  ( ph -> ( A. z e. V ( -. z e. ( N ` { X , T } ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) <-> A. z e. V ( -. z e. ( N ` { X , T } ) -> y = ( L ` <. z , ( L ` <. X , F , z >. ) , T >. ) ) ) )
50 49 reubidv 
 |-  ( ph -> ( E! y e. D A. z e. V ( -. z e. ( N ` { X , T } ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) <-> E! y e. D A. z e. V ( -. z e. ( N ` { X , T } ) -> y = ( L ` <. z , ( L ` <. X , F , z >. ) , T >. ) ) ) )
51 20 50 mpbird 
 |-  ( ph -> E! y e. D A. z e. V ( -. z e. ( N ` { X , T } ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) )