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Theorem mapdpglem32 35243
Description: Lemma for mapdpg 35244. Uniqueness part - consolidate hypotheses in mapdpglem31 35241. (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 2438 . . . 4  |-  (Scalar `  U )  =  (Scalar `  U )
31 eqid 2438 . . . 4  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
32 eqid 2438 . . . 4  |-  ( .s
`  C )  =  ( .s `  C
)
33 eqid 2438 . . . 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 35236 . . 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 35237 . . 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 2883 . . 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 3473 . . . . . . 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 3473 . . . . . . 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 35241 . . . 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 1186 . . 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 2842 . 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 1369    e. wcel 1756    =/= wne 2601   E.wrex 2711    \ cdif 3320   {csn 3872   ` cfv 5413  (class class class)co 6086   Basecbs 14166  Scalarcsca 14233   .scvsca 14234   0gc0g 14370   -gcsg 15405   LSpanclspn 17032   HLchlt 32888   LHypclh 33521   DVecHcdvh 34616  LCDualclcd 35124  mapdcmpd 35162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-riotaBAD 32497
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-tpos 6740  df-undef 6784  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-sca 14246  df-vsca 14247  df-0g 14372  df-mre 14516  df-mrc 14517  df-acs 14519  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-p1 15202  df-lat 15208  df-clat 15270  df-mnd 15407  df-submnd 15457  df-grp 15536  df-minusg 15537  df-sbg 15538  df-subg 15669  df-cntz 15826  df-oppg 15852  df-lsm 16126  df-cmn 16270  df-abl 16271  df-mgp 16582  df-ur 16594  df-rng 16637  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-dvr 16765  df-drng 16814  df-lmod 16930  df-lss 16994  df-lsp 17033  df-lvec 17164  df-lsatoms 32514  df-lshyp 32515  df-lcv 32557  df-lfl 32596  df-lkr 32624  df-ldual 32662  df-oposet 32714  df-ol 32716  df-oml 32717  df-covers 32804  df-ats 32805  df-atl 32836  df-cvlat 32860  df-hlat 32889  df-llines 33035  df-lplanes 33036  df-lvols 33037  df-lines 33038  df-psubsp 33040  df-pmap 33041  df-padd 33333  df-lhyp 33525  df-laut 33526  df-ldil 33641  df-ltrn 33642  df-trl 33696  df-tgrp 34280  df-tendo 34292  df-edring 34294  df-dveca 34540  df-disoa 34567  df-dvech 34617  df-dib 34677  df-dic 34711  df-dih 34767  df-doch 34886  df-djh 34933  df-lcdual 35125  df-mapd 35163
This theorem is referenced by:  mapdpg  35244
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