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Theorem mapdh9a 31147
Description: Lemma for part (9) in [Baer] p. 48. TODO: why is this 50% larger than mapdh9aOLDN 31148? (Contributed by NM, 14-May-2015.)
Hypotheses
Ref Expression
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
Ref Expression
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 >. )
) )
Distinct variable groups:    x, h,  .-    .0. , h, x    C, h    D, h, x    h, F, x    h, I    h, J, x    h, M, x   
h, N, x    ph, h    R, h, x    x, Q    T, h, x    U, h   
h, X, x    x, I    h, V    y, z, D    y, F, z    y, I, z    y, N, z   
y,  .0. , z    y, T, z    z, U    y, V, z    y, X, z    ph, y, z    z, h, x
Allowed substitution hints:    ph( x)    C( x, y, z)    Q( y, z, h)    R( y,
z)    U( x, y)    H( x, y, z, h)    J( y, z)    K( x, y, z, h)    M( y,
z)    .- ( y, z)    V( x)    W( x, y, z, h)

Proof of Theorem mapdh9a
StepHypRef Expression
1 mapdh8a.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
2 mapdh8a.u . . . . . . 7  |-  U  =  ( ( DVecH `  K
) `  W )
3 mapdh8a.v . . . . . . 7  |-  V  =  ( Base `  U
)
4 mapdh8a.s . . . . . . 7  |-  .-  =  ( -g `  U )
5 mapdh8a.o . . . . . . 7  |-  .0.  =  ( 0g `  U )
6 mapdh8a.n . . . . . . 7  |-  N  =  ( LSpan `  U )
7 mapdh8a.c . . . . . . 7  |-  C  =  ( (LCDual `  K
) `  W )
8 mapdh8a.d . . . . . . 7  |-  D  =  ( Base `  C
)
9 mapdh8a.r . . . . . . 7  |-  R  =  ( -g `  C
)
10 mapdh8a.q . . . . . . 7  |-  Q  =  ( 0g `  C
)
11 mapdh8a.j . . . . . . 7  |-  J  =  ( LSpan `  C )
12 mapdh8a.m . . . . . . 7  |-  M  =  ( (mapd `  K
) `  W )
13 mapdh8a.i . . . . . . 7  |-  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 . . . . . . . 8  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
15143ad2ant1 981 . . . . . . 7  |-  ( (
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 ) )
16 mapdh8h.f . . . . . . . 8  |-  ( ph  ->  F  e.  D )
17163ad2ant1 981 . . . . . . 7  |-  ( (
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 )
18 mapdh8h.mn . . . . . . . 8  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { F } ) )
19183ad2ant1 981 . . . . . . 7  |-  ( (
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 }
) )
20 mapdh9a.x . . . . . . . 8  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
21203ad2ant1 981 . . . . . . 7  |-  ( (
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.  } ) )
22 simp3ll 1031 . . . . . . 7  |-  ( (
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.  } ) )
23 simp3rl 1033 . . . . . . 7  |-  ( (
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.  } ) )
24 simplrl 739 . . . . . . . . 9  |-  ( ( ( 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 } ) )
25243ad2ant3 983 . . . . . . . 8  |-  ( (
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 } ) )
2625necomd 2504 . . . . . . 7  |-  ( (
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 } ) )
27 simprrl 743 . . . . . . . . 9  |-  ( ( ( 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 } ) )
28273ad2ant3 983 . . . . . . . 8  |-  ( (
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 } ) )
2928necomd 2504 . . . . . . 7  |-  ( (
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 } ) )
30 simplrr 740 . . . . . . . 8  |-  ( ( ( 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 } ) )
31303ad2ant3 983 . . . . . . 7  |-  ( (
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 } ) )
32 simprrr 744 . . . . . . . 8  |-  ( ( ( 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 } ) )
33323ad2ant3 983 . . . . . . 7  |-  ( (
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 } ) )
34 mapdh9a.t . . . . . . . 8  |-  ( ph  ->  T  e.  V )
35343ad2ant1 981 . . . . . . 7  |-  ( (
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 )
361, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 35mapdh8 31146 . . . . . 6  |-  ( (
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 >. ) )
37363exp 1155 . . . . 5  |-  ( 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 >. ) ) ) )
3837ralrimivv 2609 . . . 4  |-  ( 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 eldifi 3273 . . . . . . . . 9  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  e.  V )
4020, 39syl 17 . . . . . . . 8  |-  ( ph  ->  X  e.  V )
411, 2, 3, 6, 14, 40, 34dvh3dim 30803 . . . . . . 7  |-  ( ph  ->  E. z  e.  V  -.  z  e.  ( N `  { X ,  T } ) )
42 eqid 2258 . . . . . . . . . . 11  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
431, 2, 14dvhlmod 30467 . . . . . . . . . . . 12  |-  ( ph  ->  U  e.  LMod )
4443ad2antrr 709 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  V )  /\  -.  z  e.  ( N `  { X ,  T } ) )  ->  U  e.  LMod )
453, 42, 6, 43, 40, 34lspprcl 15697 . . . . . . . . . . . 12  |-  ( ph  ->  ( N `  { X ,  T }
)  e.  ( LSubSp `  U ) )
4645ad2antrr 709 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  V )  /\  -.  z  e.  ( N `  { X ,  T } ) )  -> 
( N `  { X ,  T }
)  e.  ( LSubSp `  U ) )
47 simplr 734 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  V )  /\  -.  z  e.  ( N `  { X ,  T } ) )  -> 
z  e.  V )
48 simpr 449 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  V )  /\  -.  z  e.  ( N `  { X ,  T } ) )  ->  -.  z  e.  ( N `  { X ,  T } ) )
493, 5, 42, 44, 46, 47, 48lssneln0 15671 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  V )  /\  -.  z  e.  ( N `  { X ,  T } ) )  -> 
z  e.  ( V 
\  {  .0.  }
) )
501, 2, 14dvhlvec 30466 . . . . . . . . . . . 12  |-  ( ph  ->  U  e.  LVec )
5150ad2antrr 709 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  V )  /\  -.  z  e.  ( N `  { X ,  T } ) )  ->  U  e.  LVec )
5240ad2antrr 709 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  V )  /\  -.  z  e.  ( N `  { X ,  T } ) )  ->  X  e.  V )
5334ad2antrr 709 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  V )  /\  -.  z  e.  ( N `  { X ,  T } ) )  ->  T  e.  V )
543, 6, 51, 47, 52, 53, 48lspindpi 15847 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  V )  /\  -.  z  e.  ( N `  { X ,  T } ) )  -> 
( ( N `  { z } )  =/=  ( N `  { X } )  /\  ( N `  { z } )  =/=  ( N `  { T } ) ) )
5549, 54jca 520 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  V )  /\  -.  z  e.  ( N `  { X ,  T } ) )  -> 
( z  e.  ( V  \  {  .0.  } )  /\  ( ( N `  { z } )  =/=  ( N `  { X } )  /\  ( N `  { z } )  =/=  ( N `  { T } ) ) ) )
5655ex 425 . . . . . . . 8  |-  ( (
ph  /\  z  e.  V )  ->  ( -.  z  e.  ( N `  { X ,  T } )  -> 
( z  e.  ( V  \  {  .0.  } )  /\  ( ( N `  { z } )  =/=  ( N `  { X } )  /\  ( N `  { z } )  =/=  ( N `  { T } ) ) ) ) )
5756reximdva 2630 . . . . . . 7  |-  ( 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 } ) ) ) ) )
5841, 57mpd 16 . . . . . 6  |-  ( ph  ->  E. z  e.  V  ( z  e.  ( V  \  {  .0.  } )  /\  ( ( N `  { z } )  =/=  ( N `  { X } )  /\  ( N `  { z } )  =/=  ( N `  { T } ) ) ) )
5914ad2antrr 709 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  V )  /\  (
z  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { z } )  =/=  ( N `  { X } )  /\  ( N `  { z } )  =/=  ( N `  { T } ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
6016ad2antrr 709 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  V )  /\  (
z  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { z } )  =/=  ( N `  { X } )  /\  ( N `  { z } )  =/=  ( N `  { T } ) ) ) )  ->  F  e.  D )
6118ad2antrr 709 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  V )  /\  (
z  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { z } )  =/=  ( N `  { X } )  /\  ( N `  { z } )  =/=  ( N `  { T } ) ) ) )  ->  ( M `  ( N `  { X } ) )  =  ( J `  { F } ) )
6220ad2antrr 709 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  V )  /\  (
z  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { z } )  =/=  ( N `  { X } )  /\  ( N `  { z } )  =/=  ( N `  { T } ) ) ) )  ->  X  e.  ( V  \  {  .0.  } ) )
63 simplr 734 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  V )  /\  (
z  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { z } )  =/=  ( N `  { X } )  /\  ( N `  { z } )  =/=  ( N `  { T } ) ) ) )  ->  z  e.  V )
64 simprrl 743 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  V )  /\  (
z  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { z } )  =/=  ( N `  { X } )  /\  ( N `  { z } )  =/=  ( N `  { T } ) ) ) )  ->  ( N `  { z } )  =/=  ( N `  { X } ) )
6564necomd 2504 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  V )  /\  (
z  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { z } )  =/=  ( N `  { X } )  /\  ( N `  { z } )  =/=  ( N `  { T } ) ) ) )  ->  ( N `  { X } )  =/=  ( N `  { z } ) )
6610, 13, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 59, 60, 61, 62, 63, 65mapdhcl 31084 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  V )  /\  (
z  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { z } )  =/=  ( N `  { X } )  /\  ( N `  { z } )  =/=  ( N `  { T } ) ) ) )  ->  ( I `  <. X ,  F ,  z >. )  e.  D )
67 eqidd 2259 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  V )  /\  (
z  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { z } )  =/=  ( N `  { X } )  /\  ( N `  { z } )  =/=  ( N `  { T } ) ) ) )  ->  ( I `  <. X ,  F ,  z >. )  =  ( I `  <. X ,  F , 
z >. ) )
68 simprl 735 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  V )  /\  (
z  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { z } )  =/=  ( N `  { X } )  /\  ( N `  { z } )  =/=  ( N `  { T } ) ) ) )  ->  z  e.  ( V  \  {  .0.  } ) )
6910, 13, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 59, 60, 61, 62, 68, 66, 65mapdheq 31085 . . . . . . . . . . . 12  |-  ( ( ( 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 >. ) ) } ) ) ) )
7067, 69mpbid 203 . . . . . . . . . . 11  |-  ( ( ( 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 >. ) ) } ) ) )
7170simpld 447 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  V )  /\  (
z  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { z } )  =/=  ( N `  { X } )  /\  ( N `  { z } )  =/=  ( N `  { T } ) ) ) )  ->  ( M `  ( N `  {
z } ) )  =  ( J `  { ( I `  <. X ,  F , 
z >. ) } ) )
7234ad2antrr 709 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  V )  /\  (
z  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { z } )  =/=  ( N `  { X } )  /\  ( N `  { z } )  =/=  ( N `  { T } ) ) ) )  ->  T  e.  V )
73 simprrr 744 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  V )  /\  (
z  e.  ( V 
\  {  .0.  }
)  /\  ( ( N `  { z } )  =/=  ( N `  { X } )  /\  ( N `  { z } )  =/=  ( N `  { T } ) ) ) )  ->  ( N `  { z } )  =/=  ( N `  { T } ) )
7410, 13, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 59, 66, 71, 68, 72, 73mapdhcl 31084 . . . . . . . . 9  |-  ( ( ( 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 )
7574ex 425 . . . . . . . 8  |-  ( (
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
) )
7675ancld 538 . . . . . . 7  |-  ( (
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
) ) )
7776reximdva 2630 . . . . . 6  |-  ( 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
) ) )
7858, 77mpd 16 . . . . 5  |-  ( 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
) )
79 eleq1 2318 . . . . . . 7  |-  ( z  =  w  ->  (
z  e.  ( V 
\  {  .0.  }
)  <->  w  e.  ( V  \  {  .0.  }
) ) )
80 sneq 3625 . . . . . . . . . 10  |-  ( z  =  w  ->  { z }  =  { w } )
8180fveq2d 5462 . . . . . . . . 9  |-  ( z  =  w  ->  ( N `  { z } )  =  ( N `  { w } ) )
8281neeq1d 2434 . . . . . . . 8  |-  ( z  =  w  ->  (
( N `  {
z } )  =/=  ( N `  { X } )  <->  ( N `  { w } )  =/=  ( N `  { X } ) ) )
8381neeq1d 2434 . . . . . . . 8  |-  ( z  =  w  ->  (
( N `  {
z } )  =/=  ( N `  { T } )  <->  ( N `  { w } )  =/=  ( N `  { T } ) ) )
8482, 83anbi12d 694 . . . . . . 7  |-  ( z  =  w  ->  (
( ( N `  { z } )  =/=  ( N `  { X } )  /\  ( N `  { z } )  =/=  ( N `  { T } ) )  <->  ( ( N `  { w } )  =/=  ( N `  { X } )  /\  ( N `  { w } )  =/=  ( N `  { T } ) ) ) )
8579, 84anbi12d 694 . . . . . 6  |-  ( 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 } ) ) ) ) )
86 oteq1 3779 . . . . . . . 8  |-  ( z  =  w  ->  <. z ,  ( I `  <. X ,  F , 
z >. ) ,  T >.  =  <. w ,  ( I `  <. X ,  F ,  z >. ) ,  T >. )
87 oteq3 3781 . . . . . . . . . 10  |-  ( z  =  w  ->  <. X ,  F ,  z >.  = 
<. X ,  F ,  w >. )
8887fveq2d 5462 . . . . . . . . 9  |-  ( z  =  w  ->  (
I `  <. X ,  F ,  z >. )  =  ( I `  <. X ,  F ,  w >. ) )
89 oteq2 3780 . . . . . . . . 9  |-  ( ( I `  <. X ,  F ,  z >. )  =  ( I `  <. X ,  F ,  w >. )  ->  <. w ,  ( I `  <. X ,  F , 
z >. ) ,  T >.  =  <. w ,  ( I `  <. X ,  F ,  w >. ) ,  T >. )
9088, 89syl 17 . . . . . . . 8  |-  ( z  =  w  ->  <. w ,  ( I `  <. X ,  F , 
z >. ) ,  T >.  =  <. w ,  ( I `  <. X ,  F ,  w >. ) ,  T >. )
9186, 90eqtrd 2290 . . . . . . 7  |-  ( z  =  w  ->  <. z ,  ( I `  <. X ,  F , 
z >. ) ,  T >.  =  <. w ,  ( I `  <. X ,  F ,  w >. ) ,  T >. )
9291fveq2d 5462 . . . . . 6  |-  ( z  =  w  ->  (
I `  <. z ,  ( I `  <. X ,  F ,  z
>. ) ,  T >. )  =  ( I `  <. w ,  ( I `
 <. X ,  F ,  w >. ) ,  T >. ) )
9385, 92reusv3 4514 . . . . 5  |-  ( 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 >. ) ) ) )
9478, 93syl 17 . . . 4  |-  ( 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 >. ) ) ) )
9538, 94mpbid 203 . . 3  |-  ( 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 >. ) ) )
96 ioran 478 . . . . . . . 8  |-  ( -.  ( z  e.  ( N `  { X } )  \/  z  e.  ( N `  { T } ) )  <->  ( -.  z  e.  ( N `  { X } )  /\  -.  z  e.  ( N `  { T } ) ) )
97 elun 3291 . . . . . . . 8  |-  ( z  e.  ( ( N `
 { X }
)  u.  ( N `
 { T }
) )  <->  ( z  e.  ( N `  { X } )  \/  z  e.  ( N `  { T } ) ) )
9896, 97xchnxbir 302 . . . . . . 7  |-  ( -.  z  e.  ( ( N `  { X } )  u.  ( N `  { T } ) )  <->  ( -.  z  e.  ( N `  { X } )  /\  -.  z  e.  ( N `  { T } ) ) )
9943ad2antrr 709 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  V )  /\  ( -.  z  e.  ( N `  { X } )  /\  -.  z  e.  ( N `  { T } ) ) )  ->  U  e.  LMod )
1003, 42, 6lspsncl 15696 . . . . . . . . . . . 12  |-  ( ( U  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (
LSubSp `  U ) )
10143, 40, 100syl2anc 645 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { X } )  e.  (
LSubSp `  U ) )
102101ad2antrr 709 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  V )  /\  ( -.  z  e.  ( N `  { X } )  /\  -.  z  e.  ( N `  { T } ) ) )  ->  ( N `  { X } )  e.  (
LSubSp `  U ) )
103 simplr 734 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  V )  /\  ( -.  z  e.  ( N `  { X } )  /\  -.  z  e.  ( N `  { T } ) ) )  ->  z  e.  V )
104 simprl 735 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  V )  /\  ( -.  z  e.  ( N `  { X } )  /\  -.  z  e.  ( N `  { T } ) ) )  ->  -.  z  e.  ( N `  { X } ) )
1053, 5, 42, 99, 102, 103, 104lssneln0 15671 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  V )  /\  ( -.  z  e.  ( N `  { X } )  /\  -.  z  e.  ( N `  { T } ) ) )  ->  z  e.  ( V  \  {  .0.  } ) )
106105ex 425 . . . . . . . 8  |-  ( (
ph  /\  z  e.  V )  ->  (
( -.  z  e.  ( N `  { X } )  /\  -.  z  e.  ( N `  { T } ) )  ->  z  e.  ( V  \  {  .0.  } ) ) )
10743ad2antrr 709 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  V )  /\  -.  z  e.  ( N `  { X } ) )  ->  U  e.  LMod )
108 simplr 734 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  V )  /\  -.  z  e.  ( N `  { X } ) )  ->  z  e.  V )
10940ad2antrr 709 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  V )  /\  -.  z  e.  ( N `  { X } ) )  ->  X  e.  V )
110 simpr 449 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  V )  /\  -.  z  e.  ( N `  { X } ) )  ->  -.  z  e.  ( N `  { X } ) )
1113, 6, 107, 108, 109, 110lspsnne2 15833 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  V )  /\  -.  z  e.  ( N `  { X } ) )  ->  ( N `  { z } )  =/=  ( N `  { X } ) )
112111ex 425 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  V )  ->  ( -.  z  e.  ( N `  { X } )  ->  ( N `  { z } )  =/=  ( N `  { X } ) ) )
11343ad2antrr 709 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  V )  /\  -.  z  e.  ( N `  { T } ) )  ->  U  e.  LMod )
114 simplr 734 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  V )  /\  -.  z  e.  ( N `  { T } ) )  ->  z  e.  V )
11534ad2antrr 709 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  V )  /\  -.  z  e.  ( N `  { T } ) )  ->  T  e.  V )
116 simpr 449 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  V )  /\  -.  z  e.  ( N `  { T } ) )  ->  -.  z  e.  ( N `  { T } ) )
1173, 6, 113, 114, 115, 116lspsnne2 15833 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  V )  /\  -.  z  e.  ( N `  { T } ) )  ->  ( N `  { z } )  =/=  ( N `  { T } ) )
118117ex 425 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  V )  ->  ( -.  z  e.  ( N `  { T } )  ->  ( N `  { z } )  =/=  ( N `  { T } ) ) )
119112, 118anim12d 548 . . . . . . . 8  |-  ( (
ph  /\  z  e.  V )  ->  (
( -.  z  e.  ( N `  { X } )  /\  -.  z  e.  ( N `  { T } ) )  ->  ( ( N `  { z } )  =/=  ( N `  { X } )  /\  ( N `  { z } )  =/=  ( N `  { T } ) ) ) )
120106, 119jcad 521 . . . . . . 7  |-  ( (
ph  /\  z  e.  V )  ->  (
( -.  z  e.  ( N `  { X } )  /\  -.  z  e.  ( N `  { T } ) )  ->  ( z  e.  ( V  \  {  .0.  } )  /\  (
( N `  {
z } )  =/=  ( N `  { X } )  /\  ( N `  { z } )  =/=  ( N `  { T } ) ) ) ) )
12198, 120syl5bi 210 . . . . . 6  |-  ( (
ph  /\  z  e.  V )  ->  ( -.  z  e.  (
( N `  { X } )  u.  ( N `  { T } ) )  -> 
( z  e.  ( V  \  {  .0.  } )  /\  ( ( N `  { z } )  =/=  ( N `  { X } )  /\  ( N `  { z } )  =/=  ( N `  { T } ) ) ) ) )
122121imim1d 71 . . . . 5  |-  ( (
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 >. )
) ) )
123122ralimdva 2596 . . . 4  |-  ( 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 >. )
) ) )
124123reximdv 2629 . . 3  |-  ( 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 >. )
) ) )
12595, 124mpd 16 . 2  |-  ( ph  ->  E. y  e.  D  A. z  e.  V  ( -.  z  e.  ( ( N `  { X } )  u.  ( N `  { T } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. X ,  F ,  z >. ) ,  T >. )
) )
1263, 6, 43, 40, 34lspprid1 15716 . . . . . . . . 9  |-  ( ph  ->  X  e.  ( N `
 { X ,  T } ) )
12742, 6, 43, 45, 126lspsnel5a 15715 . . . . . . . 8  |-  ( ph  ->  ( N `  { X } )  C_  ( N `  { X ,  T } ) )
1283, 6, 43, 40, 34lspprid2 15717 . . . . . . . . 9  |-  ( ph  ->  T  e.  ( N `
 { X ,  T } ) )
12942, 6, 43, 45, 128lspsnel5a 15715 . . . . . . . 8  |-  ( ph  ->  ( N `  { T } )  C_  ( N `  { X ,  T } ) )
130127, 129unssd 3326 . . . . . . 7  |-  ( ph  ->  ( ( N `  { X } )  u.  ( N `  { T } ) )  C_  ( N `  { X ,  T } ) )
131130sseld 3154 . . . . . 6  |-  ( ph  ->  ( z  e.  ( ( N `  { X } )  u.  ( N `  { T } ) )  -> 
z  e.  ( N `
 { X ,  T } ) ) )
132131con3d 127 . . . . 5  |-  ( ph  ->  ( -.  z  e.  ( N `  { X ,  T }
)  ->  -.  z  e.  ( ( N `  { X } )  u.  ( N `  { T } ) ) ) )
133132reximdv 2629 . . . 4  |-  ( ph  ->  ( E. z  e.  V  -.  z  e.  ( N `  { X ,  T }
)  ->  E. z  e.  V  -.  z  e.  ( ( N `  { X } )  u.  ( N `  { T } ) ) ) )
13441, 133mpd 16 . . 3  |-  ( ph  ->  E. z  e.  V  -.  z  e.  (
( N `  { X } )  u.  ( N `  { T } ) ) )
135 reusv1 4506 . . 3  |-  ( 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 >. )
) ) )
136134, 135syl 17 . 2  |-  ( 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 >. )
) ) )
137125, 136mpbird 225 1  |-  ( ph  ->  E! y  e.  D  A. z  e.  V  ( -.  z  e.  ( ( N `  { X } )  u.  ( N `  { T } ) )  -> 
y  =  ( I `
 <. z ,  ( I `  <. X ,  F ,  z >. ) ,  T >. )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   A.wral 2518   E.wrex 2519   E!wreu 2520   _Vcvv 2763    \ cdif 3124    u. cun 3125   ifcif 3539   {csn 3614   {cpr 3615   <.cotp 3618    e. cmpt 4051   ` cfv 4673  (class class class)co 5792   1stc1st 6054   2ndc2nd 6055   iota_crio 6263   Basecbs 13110   0gc0g 13362   -gcsg 14327   LModclmod 15589   LSubSpclss 15651   LSpanclspn 15690   LVecclvec 15817   HLchlt 28707   LHypclh 29340   DVecHcdvh 30435  LCDualclcd 30943  mapdcmpd 30981
This theorem is referenced by:  hdmap1eulem  31181
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-fal 1316  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-ot 3624  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-1st 6056  df-2nd 6057  df-tpos 6168  df-iota 6225  df-undef 6264  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-n0 9933  df-z 9992  df-uz 10198  df-fz 10749  df-struct 13112  df-ndx 13113  df-slot 13114  df-base 13115  df-sets 13116  df-ress 13117  df-plusg 13183  df-mulr 13184  df-sca 13186  df-vsca 13187  df-0g 13366  df-mre 13450  df-mrc 13451  df-acs 13453  df-poset 14042  df-plt 14054  df-lub 14070  df-glb 14071  df-join 14072  df-meet 14073  df-p0 14107  df-p1 14108  df-lat 14114  df-clat 14176  df-mnd 14329  df-submnd 14378  df-grp 14451  df-minusg 14452  df-sbg 14453  df-subg 14580  df-cntz 14755  df-oppg 14781  df-lsm 14909  df-cmn 15053  df-abl 15054  df-mgp 15288  df-ring 15302  df-ur 15304  df-oppr 15367  df-dvdsr 15385  df-unit 15386  df-invr 15416  df-dvr 15427  df-drng 15476  df-lmod 15591  df-lss 15652  df-lsp 15691  df-lvec 15818  df-lsatoms 28333  df-lshyp 28334  df-lcv 28376  df-lfl 28415  df-lkr 28443  df-ldual 28481  df-oposet 28533  df-ol 28535  df-oml 28536  df-covers 28623  df-ats 28624  df-atl 28655  df-cvlat 28679  df-hlat 28708  df-llines 28854  df-lplanes 28855  df-lvols 28856  df-lines 28857  df-psubsp 28859  df-pmap 28860  df-padd 29152  df-lhyp 29344  df-laut 29345  df-ldil 29460  df-ltrn 29461  df-trl 29515  df-tgrp 30099  df-tendo 30111  df-edring 30113  df-dveca 30359  df-disoa 30386  df-dvech 30436  df-dib 30496  df-dic 30530  df-dih 30586  df-doch 30705  df-djh 30752  df-lcdual 30944  df-mapd 30982
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