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Theorem mapdpglem32 35673
Description: Lemma for mapdpg 35674. Uniqueness part - consolidate hypotheses in mapdpglem31 35671. (Contributed by NM, 23-Mar-2015.)
Hypotheses
Ref Expression
mapdpg.h  |-  H  =  ( LHyp `  K
)
mapdpg.m  |-  M  =  ( (mapd `  K
) `  W )
mapdpg.u  |-  U  =  ( ( DVecH `  K
) `  W )
mapdpg.v  |-  V  =  ( Base `  U
)
mapdpg.s  |-  .-  =  ( -g `  U )
mapdpg.z  |-  .0.  =  ( 0g `  U )
mapdpg.n  |-  N  =  ( LSpan `  U )
mapdpg.c  |-  C  =  ( (LCDual `  K
) `  W )
mapdpg.f  |-  F  =  ( Base `  C
)
mapdpg.r  |-  R  =  ( -g `  C
)
mapdpg.j  |-  J  =  ( LSpan `  C )
mapdpg.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
mapdpg.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
mapdpg.y  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
mapdpg.g  |-  ( ph  ->  G  e.  F )
mapdpg.ne  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
mapdpg.e  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )
Assertion
Ref Expression
mapdpglem32  |-  ( (
ph  /\  ( h  e.  F  /\  i  e.  F )  /\  (
( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) )  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )  ->  h  =  i )
Distinct variable groups:    C, h    h, F    h, G    h, J    h, M    h, N    R, h    .- , h    U, h    h, X    h, Y    h, i
Allowed substitution hints:    ph( h, i)    C( i)    R( i)    U( i)    F( i)    G( i)    H( h, i)    J( i)    K( h, i)    M( i)    .- ( i)    N( i)    V( h, i)    W( h, i)    X( i)    Y( i)    .0. ( h, i)

Proof of Theorem mapdpglem32
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mapdpg.h . . . 4  |-  H  =  ( LHyp `  K
)
2 mapdpg.m . . . 4  |-  M  =  ( (mapd `  K
) `  W )
3 mapdpg.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
4 mapdpg.v . . . 4  |-  V  =  ( Base `  U
)
5 mapdpg.s . . . 4  |-  .-  =  ( -g `  U )
6 mapdpg.z . . . 4  |-  .0.  =  ( 0g `  U )
7 mapdpg.n . . . 4  |-  N  =  ( LSpan `  U )
8 mapdpg.c . . . 4  |-  C  =  ( (LCDual `  K
) `  W )
9 mapdpg.f . . . 4  |-  F  =  ( Base `  C
)
10 mapdpg.r . . . 4  |-  R  =  ( -g `  C
)
11 mapdpg.j . . . 4  |-  J  =  ( LSpan `  C )
12 mapdpg.k . . . . 5  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
13123ad2ant1 1009 . . . 4  |-  ( (
ph  /\  ( h  e.  F  /\  i  e.  F )  /\  (
( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) )  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
14 mapdpg.x . . . . 5  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
15143ad2ant1 1009 . . . 4  |-  ( (
ph  /\  ( h  e.  F  /\  i  e.  F )  /\  (
( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) )  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )  ->  X  e.  ( V  \  {  .0.  } ) )
16 mapdpg.y . . . . 5  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
17163ad2ant1 1009 . . . 4  |-  ( (
ph  /\  ( h  e.  F  /\  i  e.  F )  /\  (
( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) )  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )  ->  Y  e.  ( V  \  {  .0.  } ) )
18 mapdpg.g . . . . 5  |-  ( ph  ->  G  e.  F )
19183ad2ant1 1009 . . . 4  |-  ( (
ph  /\  ( h  e.  F  /\  i  e.  F )  /\  (
( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) )  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )  ->  G  e.  F )
20 mapdpg.ne . . . . 5  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
21203ad2ant1 1009 . . . 4  |-  ( (
ph  /\  ( h  e.  F  /\  i  e.  F )  /\  (
( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) )  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
22 mapdpg.e . . . . 5  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )
23223ad2ant1 1009 . . . 4  |-  ( (
ph  /\  ( h  e.  F  /\  i  e.  F )  /\  (
( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) )  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )  ->  ( M `  ( N `  { X } ) )  =  ( J `  { G } ) )
24 simp2l 1014 . . . . 5  |-  ( (
ph  /\  ( h  e.  F  /\  i  e.  F )  /\  (
( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) )  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )  ->  h  e.  F )
25 simp3l 1016 . . . . 5  |-  ( (
ph  /\  ( h  e.  F  /\  i  e.  F )  /\  (
( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) )  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )  ->  ( ( M `  ( N `  { Y } ) )  =  ( J `
 { h }
)  /\  ( M `  ( N `  {
( X  .-  Y
) } ) )  =  ( J `  { ( G R h ) } ) ) )
2624, 25jca 532 . . . 4  |-  ( (
ph  /\  ( h  e.  F  /\  i  e.  F )  /\  (
( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) )  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )  ->  ( h  e.  F  /\  (
( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) ) ) )
27 simp2r 1015 . . . . 5  |-  ( (
ph  /\  ( h  e.  F  /\  i  e.  F )  /\  (
( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) )  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )  ->  i  e.  F )
28 simp3r 1017 . . . . 5  |-  ( (
ph  /\  ( h  e.  F  /\  i  e.  F )  /\  (
( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) )  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )  ->  ( ( M `  ( N `  { Y } ) )  =  ( J `
 { i } )  /\  ( M `
 ( N `  { ( X  .-  Y ) } ) )  =  ( J `
 { ( G R i ) } ) ) )
2927, 28jca 532 . . . 4  |-  ( (
ph  /\  ( h  e.  F  /\  i  e.  F )  /\  (
( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) )  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )  ->  ( i  e.  F  /\  (
( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )
30 eqid 2454 . . . 4  |-  (Scalar `  U )  =  (Scalar `  U )
31 eqid 2454 . . . 4  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
32 eqid 2454 . . . 4  |-  ( .s
`  C )  =  ( .s `  C
)
33 eqid 2454 . . . 4  |-  ( 0g
`  (Scalar `  U )
)  =  ( 0g
`  (Scalar `  U )
)
341, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 21, 23, 26, 29, 30, 31, 32, 33mapdpglem26 35666 . . 3  |-  ( (
ph  /\  ( h  e.  F  /\  i  e.  F )  /\  (
( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) )  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )  ->  E. u  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) h  =  ( u ( .s
`  C ) i ) )
351, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 21, 23, 26, 29, 30, 31, 32, 33mapdpglem27 35667 . . 3  |-  ( (
ph  /\  ( h  e.  F  /\  i  e.  F )  /\  (
( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) )  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )  ->  E. v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) ( G R h )  =  ( v ( .s
`  C ) ( G R i ) ) )
36 reeanv 2992 . . 3  |-  ( E. u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } ) E. v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) ( h  =  ( u ( .s `  C ) i )  /\  ( G R h )  =  ( v ( .s
`  C ) ( G R i ) ) )  <->  ( E. u  e.  ( ( Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } ) h  =  ( u ( .s `  C ) i )  /\  E. v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) ( G R h )  =  ( v ( .s
`  C ) ( G R i ) ) ) )
3734, 35, 36sylanbrc 664 . 2  |-  ( (
ph  /\  ( h  e.  F  /\  i  e.  F )  /\  (
( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) )  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )  ->  E. u  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) E. v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) ( h  =  ( u ( .s `  C ) i )  /\  ( G R h )  =  ( v ( .s
`  C ) ( G R i ) ) ) )
38133ad2ant1 1009 . . . . 5  |-  ( ( ( ph  /\  (
h  e.  F  /\  i  e.  F )  /\  ( ( ( M `
 ( N `  { Y } ) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R h ) } ) )  /\  ( ( M `
 ( N `  { Y } ) )  =  ( J `  { i } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R i ) } ) ) ) )  /\  (
u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  /\  ( h  =  (
u ( .s `  C ) i )  /\  ( G R h )  =  ( v ( .s `  C ) ( G R i ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
39153ad2ant1 1009 . . . . 5  |-  ( ( ( ph  /\  (
h  e.  F  /\  i  e.  F )  /\  ( ( ( M `
 ( N `  { Y } ) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R h ) } ) )  /\  ( ( M `
 ( N `  { Y } ) )  =  ( J `  { i } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R i ) } ) ) ) )  /\  (
u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  /\  ( h  =  (
u ( .s `  C ) i )  /\  ( G R h )  =  ( v ( .s `  C ) ( G R i ) ) ) )  ->  X  e.  ( V  \  {  .0.  } ) )
40173ad2ant1 1009 . . . . 5  |-  ( ( ( ph  /\  (
h  e.  F  /\  i  e.  F )  /\  ( ( ( M `
 ( N `  { Y } ) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R h ) } ) )  /\  ( ( M `
 ( N `  { Y } ) )  =  ( J `  { i } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R i ) } ) ) ) )  /\  (
u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  /\  ( h  =  (
u ( .s `  C ) i )  /\  ( G R h )  =  ( v ( .s `  C ) ( G R i ) ) ) )  ->  Y  e.  ( V  \  {  .0.  } ) )
41193ad2ant1 1009 . . . . 5  |-  ( ( ( ph  /\  (
h  e.  F  /\  i  e.  F )  /\  ( ( ( M `
 ( N `  { Y } ) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R h ) } ) )  /\  ( ( M `
 ( N `  { Y } ) )  =  ( J `  { i } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R i ) } ) ) ) )  /\  (
u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  /\  ( h  =  (
u ( .s `  C ) i )  /\  ( G R h )  =  ( v ( .s `  C ) ( G R i ) ) ) )  ->  G  e.  F )
42213ad2ant1 1009 . . . . 5  |-  ( ( ( ph  /\  (
h  e.  F  /\  i  e.  F )  /\  ( ( ( M `
 ( N `  { Y } ) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R h ) } ) )  /\  ( ( M `
 ( N `  { Y } ) )  =  ( J `  { i } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R i ) } ) ) ) )  /\  (
u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  /\  ( h  =  (
u ( .s `  C ) i )  /\  ( G R h )  =  ( v ( .s `  C ) ( G R i ) ) ) )  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
43233ad2ant1 1009 . . . . 5  |-  ( ( ( ph  /\  (
h  e.  F  /\  i  e.  F )  /\  ( ( ( M `
 ( N `  { Y } ) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R h ) } ) )  /\  ( ( M `
 ( N `  { Y } ) )  =  ( J `  { i } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R i ) } ) ) ) )  /\  (
u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  /\  ( h  =  (
u ( .s `  C ) i )  /\  ( G R h )  =  ( v ( .s `  C ) ( G R i ) ) ) )  ->  ( M `  ( N `  { X } ) )  =  ( J `
 { G }
) )
44 simp12l 1101 . . . . . 6  |-  ( ( ( ph  /\  (
h  e.  F  /\  i  e.  F )  /\  ( ( ( M `
 ( N `  { Y } ) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R h ) } ) )  /\  ( ( M `
 ( N `  { Y } ) )  =  ( J `  { i } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R i ) } ) ) ) )  /\  (
u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  /\  ( h  =  (
u ( .s `  C ) i )  /\  ( G R h )  =  ( v ( .s `  C ) ( G R i ) ) ) )  ->  h  e.  F )
45 simp13l 1103 . . . . . 6  |-  ( ( ( ph  /\  (
h  e.  F  /\  i  e.  F )  /\  ( ( ( M `
 ( N `  { Y } ) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R h ) } ) )  /\  ( ( M `
 ( N `  { Y } ) )  =  ( J `  { i } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R i ) } ) ) ) )  /\  (
u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  /\  ( h  =  (
u ( .s `  C ) i )  /\  ( G R h )  =  ( v ( .s `  C ) ( G R i ) ) ) )  ->  (
( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) ) )
4644, 45jca 532 . . . . 5  |-  ( ( ( ph  /\  (
h  e.  F  /\  i  e.  F )  /\  ( ( ( M `
 ( N `  { Y } ) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R h ) } ) )  /\  ( ( M `
 ( N `  { Y } ) )  =  ( J `  { i } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R i ) } ) ) ) )  /\  (
u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  /\  ( h  =  (
u ( .s `  C ) i )  /\  ( G R h )  =  ( v ( .s `  C ) ( G R i ) ) ) )  ->  (
h  e.  F  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) ) ) )
47 simp12r 1102 . . . . . 6  |-  ( ( ( ph  /\  (
h  e.  F  /\  i  e.  F )  /\  ( ( ( M `
 ( N `  { Y } ) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R h ) } ) )  /\  ( ( M `
 ( N `  { Y } ) )  =  ( J `  { i } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R i ) } ) ) ) )  /\  (
u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  /\  ( h  =  (
u ( .s `  C ) i )  /\  ( G R h )  =  ( v ( .s `  C ) ( G R i ) ) ) )  ->  i  e.  F )
48 simp13r 1104 . . . . . 6  |-  ( ( ( ph  /\  (
h  e.  F  /\  i  e.  F )  /\  ( ( ( M `
 ( N `  { Y } ) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R h ) } ) )  /\  ( ( M `
 ( N `  { Y } ) )  =  ( J `  { i } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R i ) } ) ) ) )  /\  (
u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  /\  ( h  =  (
u ( .s `  C ) i )  /\  ( G R h )  =  ( v ( .s `  C ) ( G R i ) ) ) )  ->  (
( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) )
4947, 48jca 532 . . . . 5  |-  ( ( ( ph  /\  (
h  e.  F  /\  i  e.  F )  /\  ( ( ( M `
 ( N `  { Y } ) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R h ) } ) )  /\  ( ( M `
 ( N `  { Y } ) )  =  ( J `  { i } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R i ) } ) ) ) )  /\  (
u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  /\  ( h  =  (
u ( .s `  C ) i )  /\  ( G R h )  =  ( v ( .s `  C ) ( G R i ) ) ) )  ->  (
i  e.  F  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )
50 eldifi 3585 . . . . . . 7  |-  ( v  e.  ( ( Base `  (Scalar `  U )
)  \  { ( 0g `  (Scalar `  U
) ) } )  ->  v  e.  (
Base `  (Scalar `  U
) ) )
5150adantl 466 . . . . . 6  |-  ( ( u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  -> 
v  e.  ( Base `  (Scalar `  U )
) )
52513ad2ant2 1010 . . . . 5  |-  ( ( ( ph  /\  (
h  e.  F  /\  i  e.  F )  /\  ( ( ( M `
 ( N `  { Y } ) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R h ) } ) )  /\  ( ( M `
 ( N `  { Y } ) )  =  ( J `  { i } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R i ) } ) ) ) )  /\  (
u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  /\  ( h  =  (
u ( .s `  C ) i )  /\  ( G R h )  =  ( v ( .s `  C ) ( G R i ) ) ) )  ->  v  e.  ( Base `  (Scalar `  U ) ) )
53 simp3l 1016 . . . . 5  |-  ( ( ( ph  /\  (
h  e.  F  /\  i  e.  F )  /\  ( ( ( M `
 ( N `  { Y } ) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R h ) } ) )  /\  ( ( M `
 ( N `  { Y } ) )  =  ( J `  { i } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R i ) } ) ) ) )  /\  (
u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  /\  ( h  =  (
u ( .s `  C ) i )  /\  ( G R h )  =  ( v ( .s `  C ) ( G R i ) ) ) )  ->  h  =  ( u ( .s `  C ) i ) )
54 simp3r 1017 . . . . 5  |-  ( ( ( ph  /\  (
h  e.  F  /\  i  e.  F )  /\  ( ( ( M `
 ( N `  { Y } ) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R h ) } ) )  /\  ( ( M `
 ( N `  { Y } ) )  =  ( J `  { i } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R i ) } ) ) ) )  /\  (
u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  /\  ( h  =  (
u ( .s `  C ) i )  /\  ( G R h )  =  ( v ( .s `  C ) ( G R i ) ) ) )  ->  ( G R h )  =  ( v ( .s
`  C ) ( G R i ) ) )
55 eldifi 3585 . . . . . . 7  |-  ( u  e.  ( ( Base `  (Scalar `  U )
)  \  { ( 0g `  (Scalar `  U
) ) } )  ->  u  e.  (
Base `  (Scalar `  U
) ) )
5655adantr 465 . . . . . 6  |-  ( ( u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  ->  u  e.  ( Base `  (Scalar `  U )
) )
57563ad2ant2 1010 . . . . 5  |-  ( ( ( ph  /\  (
h  e.  F  /\  i  e.  F )  /\  ( ( ( M `
 ( N `  { Y } ) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R h ) } ) )  /\  ( ( M `
 ( N `  { Y } ) )  =  ( J `  { i } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R i ) } ) ) ) )  /\  (
u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  /\  ( h  =  (
u ( .s `  C ) i )  /\  ( G R h )  =  ( v ( .s `  C ) ( G R i ) ) ) )  ->  u  e.  ( Base `  (Scalar `  U ) ) )
581, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 38, 39, 40, 41, 42, 43, 46, 49, 30, 31, 32, 33, 52, 53, 54, 57mapdpglem31 35671 . . . 4  |-  ( ( ( ph  /\  (
h  e.  F  /\  i  e.  F )  /\  ( ( ( M `
 ( N `  { Y } ) )  =  ( J `  { h } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R h ) } ) )  /\  ( ( M `
 ( N `  { Y } ) )  =  ( J `  { i } )  /\  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  {
( G R i ) } ) ) ) )  /\  (
u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  /\  ( h  =  (
u ( .s `  C ) i )  /\  ( G R h )  =  ( v ( .s `  C ) ( G R i ) ) ) )  ->  h  =  i )
59583exp 1187 . . 3  |-  ( (
ph  /\  ( h  e.  F  /\  i  e.  F )  /\  (
( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) )  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )  ->  ( (
u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  -> 
( ( h  =  ( u ( .s
`  C ) i )  /\  ( G R h )  =  ( v ( .s
`  C ) ( G R i ) ) )  ->  h  =  i ) ) )
6059rexlimdvv 2951 . 2  |-  ( (
ph  /\  ( h  e.  F  /\  i  e.  F )  /\  (
( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) )  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )  ->  ( E. u  e.  ( ( Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } ) E. v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) ( h  =  ( u ( .s `  C ) i )  /\  ( G R h )  =  ( v ( .s
`  C ) ( G R i ) ) )  ->  h  =  i ) )
6137, 60mpd 15 1  |-  ( (
ph  /\  ( h  e.  F  /\  i  e.  F )  /\  (
( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) )  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )  ->  h  =  i )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2647   E.wrex 2799    \ cdif 3432   {csn 3984   ` cfv 5525  (class class class)co 6199   Basecbs 14291  Scalarcsca 14359   .scvsca 14360   0gc0g 14496   -gcsg 15531   LSpanclspn 17174   HLchlt 33318   LHypclh 33951   DVecHcdvh 35046  LCDualclcd 35554  mapdcmpd 35592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-riotaBAD 32927
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-tpos 6854  df-undef 6901  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-map 7325  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-n0 10690  df-z 10757  df-uz 10972  df-fz 11554  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-sca 14372  df-vsca 14373  df-0g 14498  df-mre 14642  df-mrc 14643  df-acs 14645  df-poset 15234  df-plt 15246  df-lub 15262  df-glb 15263  df-join 15264  df-meet 15265  df-p0 15327  df-p1 15328  df-lat 15334  df-clat 15396  df-mnd 15533  df-submnd 15583  df-grp 15663  df-minusg 15664  df-sbg 15665  df-subg 15796  df-cntz 15953  df-oppg 15979  df-lsm 16255  df-cmn 16399  df-abl 16400  df-mgp 16713  df-ur 16725  df-rng 16769  df-oppr 16837  df-dvdsr 16855  df-unit 16856  df-invr 16886  df-dvr 16897  df-drng 16956  df-lmod 17072  df-lss 17136  df-lsp 17175  df-lvec 17306  df-lsatoms 32944  df-lshyp 32945  df-lcv 32987  df-lfl 33026  df-lkr 33054  df-ldual 33092  df-oposet 33144  df-ol 33146  df-oml 33147  df-covers 33234  df-ats 33235  df-atl 33266  df-cvlat 33290  df-hlat 33319  df-llines 33465  df-lplanes 33466  df-lvols 33467  df-lines 33468  df-psubsp 33470  df-pmap 33471  df-padd 33763  df-lhyp 33955  df-laut 33956  df-ldil 34071  df-ltrn 34072  df-trl 34126  df-tgrp 34710  df-tendo 34722  df-edring 34724  df-dveca 34970  df-disoa 34997  df-dvech 35047  df-dib 35107  df-dic 35141  df-dih 35197  df-doch 35316  df-djh 35363  df-lcdual 35555  df-mapd 35593
This theorem is referenced by:  mapdpg  35674
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