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Theorem mapdpglem32 32505
Description: Lemma for mapdpg 32506. Uniqueness part - consolidate hypotheses in mapdpglem31 32503. (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 979 . . . 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 979 . . . 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 979 . . . 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 979 . . . 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 979 . . . 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 979 . . . 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 984 . . . . 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 986 . . . . 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 520 . . . 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 985 . . . . 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 987 . . . . 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 520 . . . 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 32498 . . 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 32499 . . 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 2877 . . 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 647 . 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 979 . . . . 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 979 . . . . 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 979 . . . . 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 979 . . . . 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 979 . . . . 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 979 . . . . 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 1071 . . . . . 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 1073 . . . . . 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 520 . . . . 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 1072 . . . . . 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 1074 . . . . . 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 520 . . . . 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 3471 . . . . . . 7  |-  ( v  e.  ( ( Base `  (Scalar `  U )
)  \  { ( 0g `  (Scalar `  U
) ) } )  ->  v  e.  (
Base `  (Scalar `  U
) ) )
5150adantl 454 . . . . . 6  |-  ( ( u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  -> 
v  e.  ( Base `  (Scalar `  U )
) )
52513ad2ant2 980 . . . . 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 986 . . . . 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 987 . . . . 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 3471 . . . . . . 7  |-  ( u  e.  ( ( Base `  (Scalar `  U )
)  \  { ( 0g `  (Scalar `  U
) ) } )  ->  u  e.  (
Base `  (Scalar `  U
) ) )
5655adantr 453 . . . . . 6  |-  ( ( u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  ->  u  e.  ( Base `  (Scalar `  U )
) )
57563ad2ant2 980 . . . . 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 32503 . . . 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 1153 . . 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 2838 . 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 set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708    \ cdif 3319   {csn 3816   ` cfv 5456  (class class class)co 6083   Basecbs 13471  Scalarcsca 13534   .scvsca 13535   0gc0g 13725   -gcsg 14690   LSpanclspn 16049   HLchlt 30150   LHypclh 30783   DVecHcdvh 31878  LCDualclcd 32386  mapdcmpd 32424
This theorem is referenced by:  mapdpg  32506
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-fal 1330  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-tpos 6481  df-undef 6545  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-sca 13547  df-vsca 13548  df-0g 13729  df-mre 13813  df-mrc 13814  df-acs 13816  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-mnd 14692  df-submnd 14741  df-grp 14814  df-minusg 14815  df-sbg 14816  df-subg 14943  df-cntz 15118  df-oppg 15144  df-lsm 15272  df-cmn 15416  df-abl 15417  df-mgp 15651  df-rng 15665  df-ur 15667  df-oppr 15730  df-dvdsr 15748  df-unit 15749  df-invr 15779  df-dvr 15790  df-drng 15839  df-lmod 15954  df-lss 16011  df-lsp 16050  df-lvec 16177  df-lsatoms 29776  df-lshyp 29777  df-lcv 29819  df-lfl 29858  df-lkr 29886  df-ldual 29924  df-oposet 29976  df-ol 29978  df-oml 29979  df-covers 30066  df-ats 30067  df-atl 30098  df-cvlat 30122  df-hlat 30151  df-llines 30297  df-lplanes 30298  df-lvols 30299  df-lines 30300  df-psubsp 30302  df-pmap 30303  df-padd 30595  df-lhyp 30787  df-laut 30788  df-ldil 30903  df-ltrn 30904  df-trl 30958  df-tgrp 31542  df-tendo 31554  df-edring 31556  df-dveca 31802  df-disoa 31829  df-dvech 31879  df-dib 31939  df-dic 31973  df-dih 32029  df-doch 32148  df-djh 32195  df-lcdual 32387  df-mapd 32425
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