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Theorem mapdpglem32 34725
Description: Lemma for mapdpg 34726. Uniqueness part - consolidate hypotheses in mapdpglem31 34723. (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 1018 . . . 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 1018 . . . 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 1018 . . . 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 1018 . . . 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 1018 . . . 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 1018 . . . 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 1023 . . . . 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 1025 . . . . 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 530 . . . 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 1024 . . . . 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 1026 . . . . 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 530 . . . 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 2402 . . . 4  |-  (Scalar `  U )  =  (Scalar `  U )
31 eqid 2402 . . . 4  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
32 eqid 2402 . . . 4  |-  ( .s
`  C )  =  ( .s `  C
)
33 eqid 2402 . . . 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 34718 . . 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 34719 . . 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 2975 . . 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 662 . 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 1018 . . . . 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 1018 . . . . 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 1018 . . . . 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 1018 . . . . 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 1018 . . . . 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 1018 . . . . 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 1110 . . . . . 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 1112 . . . . . 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 530 . . . . 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 1111 . . . . . 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 1113 . . . . . 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 530 . . . . 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 3565 . . . . . . 7  |-  ( v  e.  ( ( Base `  (Scalar `  U )
)  \  { ( 0g `  (Scalar `  U
) ) } )  ->  v  e.  (
Base `  (Scalar `  U
) ) )
5150adantl 464 . . . . . 6  |-  ( ( u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  -> 
v  e.  ( Base `  (Scalar `  U )
) )
52513ad2ant2 1019 . . . . 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 1025 . . . . 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 1026 . . . . 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 3565 . . . . . . 7  |-  ( u  e.  ( ( Base `  (Scalar `  U )
)  \  { ( 0g `  (Scalar `  U
) ) } )  ->  u  e.  (
Base `  (Scalar `  U
) ) )
5655adantr 463 . . . . . 6  |-  ( ( u  e.  ( (
Base `  (Scalar `  U
) )  \  {
( 0g `  (Scalar `  U ) ) } )  /\  v  e.  ( ( Base `  (Scalar `  U ) )  \  { ( 0g `  (Scalar `  U ) ) } ) )  ->  u  e.  ( Base `  (Scalar `  U )
) )
57563ad2ant2 1019 . . . . 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 34723 . . . 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 1196 . . 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 2902 . 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2755    \ cdif 3411   {csn 3972   ` cfv 5569  (class class class)co 6278   Basecbs 14841  Scalarcsca 14912   .scvsca 14913   0gc0g 15054   -gcsg 16379   LSpanclspn 17937   HLchlt 32368   LHypclh 33001   DVecHcdvh 34098  LCDualclcd 34606  mapdcmpd 34644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-riotaBAD 31977
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-tpos 6958  df-undef 7005  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-sca 14925  df-vsca 14926  df-0g 15056  df-mre 15200  df-mrc 15201  df-acs 15203  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-p1 15994  df-lat 16000  df-clat 16062  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-grp 16381  df-minusg 16382  df-sbg 16383  df-subg 16522  df-cntz 16679  df-oppg 16705  df-lsm 16980  df-cmn 17124  df-abl 17125  df-mgp 17462  df-ur 17474  df-ring 17520  df-oppr 17592  df-dvdsr 17610  df-unit 17611  df-invr 17641  df-dvr 17652  df-drng 17718  df-lmod 17834  df-lss 17899  df-lsp 17938  df-lvec 18069  df-lsatoms 31994  df-lshyp 31995  df-lcv 32037  df-lfl 32076  df-lkr 32104  df-ldual 32142  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-llines 32515  df-lplanes 32516  df-lvols 32517  df-lines 32518  df-psubsp 32520  df-pmap 32521  df-padd 32813  df-lhyp 33005  df-laut 33006  df-ldil 33121  df-ltrn 33122  df-trl 33177  df-tgrp 33762  df-tendo 33774  df-edring 33776  df-dveca 34022  df-disoa 34049  df-dvech 34099  df-dib 34159  df-dic 34193  df-dih 34249  df-doch 34368  df-djh 34415  df-lcdual 34607  df-mapd 34645
This theorem is referenced by:  mapdpg  34726
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