Metamath Proof Explorer

Theorem mapdh9a

Description: Lemma for part (9) in Baer p. 48. TODO: why is this 50% larger than mapdh9aOLDN ? (Contributed by NM, 14-May-2015)

Ref Expression
Hypotheses mapdh8a.h 
|- H = ( LHyp ` K )
mapdh8a.u 
|- U = ( ( DVecH ` K ) ` W )
mapdh8a.v 
|- V = ( Base ` U )
mapdh8a.s 
|- .- = ( -g ` U )
mapdh8a.o 
|- .0. = ( 0g ` U )
mapdh8a.n 
|- N = ( LSpan ` U )
mapdh8a.c 
|- C = ( ( LCDual ` K ) ` W )
mapdh8a.d 
|- D = ( Base ` C )
mapdh8a.r 
|- R = ( -g ` C )
mapdh8a.q 
|- Q = ( 0g ` C )
mapdh8a.j 
|- J = ( LSpan ` C )
mapdh8a.m 
|- M = ( ( mapd ` K ) ` W )
mapdh8a.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 ) } ) ) ) ) )
mapdh8a.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
mapdh8h.f 
|- ( ph -> F e. D )
mapdh8h.mn 
|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
mapdh9a.x 
|- ( ph -> X e. ( V \ { .0. } ) )
mapdh9a.t 
|- ( ph -> T e. V )
Assertion mapdh9a 
|- ( 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 mapdh8a.h 
 |-  H = ( LHyp ` K )
2 mapdh8a.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 mapdh8a.v 
 |-  V = ( Base ` U )
4 mapdh8a.s 
 |-  .- = ( -g ` U )
5 mapdh8a.o 
 |-  .0. = ( 0g ` U )
6 mapdh8a.n 
 |-  N = ( LSpan ` U )
7 mapdh8a.c 
 |-  C = ( ( LCDual ` K ) ` W )
8 mapdh8a.d 
 |-  D = ( Base ` C )
9 mapdh8a.r 
 |-  R = ( -g ` C )
10 mapdh8a.q 
 |-  Q = ( 0g ` C )
11 mapdh8a.j 
 |-  J = ( LSpan ` C )
12 mapdh8a.m 
 |-  M = ( ( mapd ` K ) ` W )
13 mapdh8a.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 ) } ) ) ) ) )
14 mapdh8a.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
15 mapdh8h.f 
 |-  ( ph -> F e. D )
16 mapdh8h.mn 
 |-  ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
17 mapdh9a.x 
 |-  ( ph -> X e. ( V \ { .0. } ) )
18 mapdh9a.t 
 |-  ( ph -> T e. V )
19 14 3ad2ant1 
 |-  ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( K e. HL /\ W e. H ) )
20 15 3ad2ant1 
 |-  ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> F e. D )
21 16 3ad2ant1 
 |-  ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
22 17 3ad2ant1 
 |-  ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> X e. ( V \ { .0. } ) )
23 simp3ll 
 |-  ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> z e. ( V \ { .0. } ) )
24 simp3rl 
 |-  ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> w e. ( V \ { .0. } ) )
25 simplrl 
 |-  ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( N ` { z } ) =/= ( N ` { X } ) )
26 25 3ad2ant3 
 |-  ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( N ` { z } ) =/= ( N ` { X } ) )
27 26 necomd 
 |-  ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( N ` { X } ) =/= ( N ` { z } ) )
28 simprrl 
 |-  ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( N ` { w } ) =/= ( N ` { X } ) )
29 28 3ad2ant3 
 |-  ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( N ` { w } ) =/= ( N ` { X } ) )
30 29 necomd 
 |-  ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( N ` { X } ) =/= ( N ` { w } ) )
31 simplrr 
 |-  ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( N ` { z } ) =/= ( N ` { T } ) )
32 31 3ad2ant3 
 |-  ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( N ` { z } ) =/= ( N ` { T } ) )
33 simprrr 
 |-  ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( N ` { w } ) =/= ( N ` { T } ) )
34 33 3ad2ant3 
 |-  ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( N ` { w } ) =/= ( N ` { T } ) )
35 18 3ad2ant1 
 |-  ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> T e. V )
36 1 2 3 4 5 6 7 8 9 10 11 12 13 19 20 21 22 23 24 27 30 32 34 35 mapdh8 
 |-  ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) = ( I ` <. w , ( I ` <. X , F , w >. ) , T >. ) )
37 36 3exp 
 |-  ( ph -> ( ( z e. V /\ w e. V ) -> ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) = ( I ` <. w , ( I ` <. X , F , w >. ) , T >. ) ) ) )
38 37 ralrimivv 
 |-  ( ph -> A. z e. V A. w e. V ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) = ( I ` <. w , ( I ` <. X , F , w >. ) , T >. ) ) )
39 17 eldifad 
 |-  ( ph -> X e. V )
40 1 2 3 6 14 39 18 dvh3dim 
 |-  ( ph -> E. z e. V -. z e. ( N ` { X , T } ) )
41 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
42 1 2 14 dvhlmod 
 |-  ( ph -> U e. LMod )
43 42 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> U e. LMod )
44 3 41 6 42 39 18 lspprcl 
 |-  ( ph -> ( N ` { X , T } ) e. ( LSubSp ` U ) )
45 44 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> ( N ` { X , T } ) e. ( LSubSp ` U ) )
46 simplr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> z e. V )
47 simpr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> -. z e. ( N ` { X , T } ) )
48 5 41 43 45 46 47 lssneln0 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> z e. ( V \ { .0. } ) )
49 1 2 14 dvhlvec 
 |-  ( ph -> U e. LVec )
50 49 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> U e. LVec )
51 39 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> X e. V )
52 18 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> T e. V )
53 3 6 50 46 51 52 47 lspindpi 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) )
54 48 53 jca 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) )
55 54 ex 
 |-  ( ( ph /\ z e. V ) -> ( -. z e. ( N ` { X , T } ) -> ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) )
56 55 reximdva 
 |-  ( ph -> ( E. z e. V -. z e. ( N ` { X , T } ) -> E. z e. V ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) )
57 40 56 mpd 
 |-  ( ph -> E. z e. V ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) )
58 14 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( K e. HL /\ W e. H ) )
59 15 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> F e. D )
60 16 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
61 17 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> X e. ( V \ { .0. } ) )
62 simplr 
 |-  ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> z e. V )
63 simprrl 
 |-  ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( N ` { z } ) =/= ( N ` { X } ) )
64 63 necomd 
 |-  ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( N ` { X } ) =/= ( N ` { z } ) )
65 10 13 1 12 2 3 4 5 6 7 8 9 11 58 59 60 61 62 64 mapdhcl 
 |-  ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( I ` <. X , F , z >. ) e. D )
66 eqidd 
 |-  ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( I ` <. X , F , z >. ) = ( I ` <. X , F , z >. ) )
67 simprl 
 |-  ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> z e. ( V \ { .0. } ) )
68 10 13 1 12 2 3 4 5 6 7 8 9 11 58 59 60 61 67 65 64 mapdheq 
 |-  ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( ( I ` <. X , F , z >. ) = ( I ` <. X , F , z >. ) <-> ( ( M ` ( N ` { z } ) ) = ( J ` { ( I ` <. X , F , z >. ) } ) /\ ( M ` ( N ` { ( X .- z ) } ) ) = ( J ` { ( F R ( I ` <. X , F , z >. ) ) } ) ) ) )
69 66 68 mpbid 
 |-  ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( ( M ` ( N ` { z } ) ) = ( J ` { ( I ` <. X , F , z >. ) } ) /\ ( M ` ( N ` { ( X .- z ) } ) ) = ( J ` { ( F R ( I ` <. X , F , z >. ) ) } ) ) )
70 69 simpld 
 |-  ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( M ` ( N ` { z } ) ) = ( J ` { ( I ` <. X , F , z >. ) } ) )
71 18 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> T e. V )
72 simprrr 
 |-  ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( N ` { z } ) =/= ( N ` { T } ) )
73 10 13 1 12 2 3 4 5 6 7 8 9 11 58 65 70 67 71 72 mapdhcl 
 |-  ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) e. D )
74 73 ex 
 |-  ( ( ph /\ z e. V ) -> ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) e. D ) )
75 74 ancld 
 |-  ( ( ph /\ z e. V ) -> ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) e. D ) ) )
76 75 reximdva 
 |-  ( ph -> ( E. z e. V ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> E. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) e. D ) ) )
77 57 76 mpd 
 |-  ( ph -> E. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) e. D ) )
78 eleq1w 
 |-  ( z = w -> ( z e. ( V \ { .0. } ) <-> w e. ( V \ { .0. } ) ) )
79 sneq 
 |-  ( z = w -> { z } = { w } )
80 79 fveq2d 
 |-  ( z = w -> ( N ` { z } ) = ( N ` { w } ) )
81 80 neeq1d 
 |-  ( z = w -> ( ( N ` { z } ) =/= ( N ` { X } ) <-> ( N ` { w } ) =/= ( N ` { X } ) ) )
82 80 neeq1d 
 |-  ( z = w -> ( ( N ` { z } ) =/= ( N ` { T } ) <-> ( N ` { w } ) =/= ( N ` { T } ) ) )
83 81 82 anbi12d 
 |-  ( z = w -> ( ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) <-> ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) )
84 78 83 anbi12d 
 |-  ( z = w -> ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) <-> ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) )
85 oteq1 
 |-  ( z = w -> <. z , ( I ` <. X , F , z >. ) , T >. = <. w , ( I ` <. X , F , z >. ) , T >. )
86 oteq3 
 |-  ( z = w -> <. X , F , z >. = <. X , F , w >. )
87 86 fveq2d 
 |-  ( z = w -> ( I ` <. X , F , z >. ) = ( I ` <. X , F , w >. ) )
88 87 oteq2d 
 |-  ( z = w -> <. w , ( I ` <. X , F , z >. ) , T >. = <. w , ( I ` <. X , F , w >. ) , T >. )
89 85 88 eqtrd 
 |-  ( z = w -> <. z , ( I ` <. X , F , z >. ) , T >. = <. w , ( I ` <. X , F , w >. ) , T >. )
90 89 fveq2d 
 |-  ( z = w -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) = ( I ` <. w , ( I ` <. X , F , w >. ) , T >. ) )
91 84 90 reusv3 
 |-  ( E. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) e. D ) -> ( A. z e. V A. w e. V ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) = ( I ` <. w , ( I ` <. X , F , w >. ) , T >. ) ) <-> E. y e. D A. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) )
92 77 91 syl 
 |-  ( ph -> ( A. z e. V A. w e. V ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) = ( I ` <. w , ( I ` <. X , F , w >. ) , T >. ) ) <-> E. y e. D A. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) )
93 38 92 mpbid 
 |-  ( ph -> E. y e. D A. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) )
94 ioran 
 |-  ( -. ( z e. ( N ` { X } ) \/ z e. ( N ` { T } ) ) <-> ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) )
95 elun 
 |-  ( z e. ( ( N ` { X } ) u. ( N ` { T } ) ) <-> ( z e. ( N ` { X } ) \/ z e. ( N ` { T } ) ) )
96 94 95 xchnxbir 
 |-  ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) <-> ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) )
97 42 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) ) -> U e. LMod )
98 3 41 6 lspsncl 
 |-  ( ( U e. LMod /\ X e. V ) -> ( N ` { X } ) e. ( LSubSp ` U ) )
99 42 39 98 syl2anc 
 |-  ( ph -> ( N ` { X } ) e. ( LSubSp ` U ) )
100 99 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) ) -> ( N ` { X } ) e. ( LSubSp ` U ) )
101 simplr 
 |-  ( ( ( ph /\ z e. V ) /\ ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) ) -> z e. V )
102 simprl 
 |-  ( ( ( ph /\ z e. V ) /\ ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) ) -> -. z e. ( N ` { X } ) )
103 5 41 97 100 101 102 lssneln0 
 |-  ( ( ( ph /\ z e. V ) /\ ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) ) -> z e. ( V \ { .0. } ) )
104 103 ex 
 |-  ( ( ph /\ z e. V ) -> ( ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) -> z e. ( V \ { .0. } ) ) )
105 42 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X } ) ) -> U e. LMod )
106 simplr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X } ) ) -> z e. V )
107 39 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X } ) ) -> X e. V )
108 simpr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X } ) ) -> -. z e. ( N ` { X } ) )
109 3 6 105 106 107 108 lspsnne2 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X } ) ) -> ( N ` { z } ) =/= ( N ` { X } ) )
110 109 ex 
 |-  ( ( ph /\ z e. V ) -> ( -. z e. ( N ` { X } ) -> ( N ` { z } ) =/= ( N ` { X } ) ) )
111 42 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { T } ) ) -> U e. LMod )
112 simplr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { T } ) ) -> z e. V )
113 18 ad2antrr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { T } ) ) -> T e. V )
114 simpr 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { T } ) ) -> -. z e. ( N ` { T } ) )
115 3 6 111 112 113 114 lspsnne2 
 |-  ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { T } ) ) -> ( N ` { z } ) =/= ( N ` { T } ) )
116 115 ex 
 |-  ( ( ph /\ z e. V ) -> ( -. z e. ( N ` { T } ) -> ( N ` { z } ) =/= ( N ` { T } ) ) )
117 110 116 anim12d 
 |-  ( ( ph /\ z e. V ) -> ( ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) -> ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) )
118 104 117 jcad 
 |-  ( ( ph /\ z e. V ) -> ( ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) -> ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) )
119 96 118 biimtrid 
 |-  ( ( ph /\ z e. V ) -> ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) )
120 119 imim1d 
 |-  ( ( ph /\ z e. V ) -> ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) -> ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) )
121 120 ralimdva 
 |-  ( ph -> ( A. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) -> A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) )
122 121 reximdv 
 |-  ( ph -> ( E. y e. D A. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) -> E. y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) )
123 93 122 mpd 
 |-  ( ph -> E. y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) )
124 3 6 42 39 18 lspprid1 
 |-  ( ph -> X e. ( N ` { X , T } ) )
125 41 6 42 44 124 ellspsn5 
 |-  ( ph -> ( N ` { X } ) C_ ( N ` { X , T } ) )
126 3 6 42 39 18 lspprid2 
 |-  ( ph -> T e. ( N ` { X , T } ) )
127 41 6 42 44 126 ellspsn5 
 |-  ( ph -> ( N ` { T } ) C_ ( N ` { X , T } ) )
128 125 127 unssd 
 |-  ( ph -> ( ( N ` { X } ) u. ( N ` { T } ) ) C_ ( N ` { X , T } ) )
129 128 ssneld 
 |-  ( ph -> ( -. z e. ( N ` { X , T } ) -> -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) ) )
130 129 reximdv 
 |-  ( ph -> ( E. z e. V -. z e. ( N ` { X , T } ) -> E. z e. V -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) ) )
131 40 130 mpd 
 |-  ( ph -> E. z e. V -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) )
132 reusv1 
 |-  ( E. z e. V -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> ( E! y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) <-> E. y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) )
133 131 132 syl 
 |-  ( ph -> ( E! y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) <-> E. y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) )
134 123 133 mpbird 
 |-  ( ph -> E! y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) )