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Theorem mapdpglem25 31887
Description: Lemma for mapdpg 31896. Baer p. 45 line 12: "Then we have Gy' = Gy'' and G(x'-y') = G(x'-y'')." (Contributed by NM, 21-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 } ) )
mapdpgem25.h1  |-  ( ph  ->  ( h  e.  F  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) ) ) )
mapdpgem25.i1  |-  ( ph  ->  ( i  e.  F  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )
Assertion
Ref Expression
mapdpglem25  |-  ( ph  ->  ( ( J `  { h } )  =  ( J `  { i } )  /\  ( J `  { ( G R h ) } )  =  ( J `  { ( G R i ) } ) ) )

Proof of Theorem mapdpglem25
StepHypRef Expression
1 mapdpgem25.h1 . . . . 5  |-  ( ph  ->  ( h  e.  F  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) ) ) )
21simprd 449 . . . 4  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
h } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R h ) } ) ) )
32simpld 445 . . 3  |-  ( ph  ->  ( M `  ( N `  { Y } ) )  =  ( J `  {
h } ) )
4 mapdpgem25.i1 . . . . 5  |-  ( ph  ->  ( i  e.  F  /\  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) ) )
54simprd 449 . . . 4  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) )  =  ( J `  {
i } )  /\  ( M `  ( N `
 { ( X 
.-  Y ) } ) )  =  ( J `  { ( G R i ) } ) ) )
65simpld 445 . . 3  |-  ( ph  ->  ( M `  ( N `  { Y } ) )  =  ( J `  {
i } ) )
73, 6eqtr3d 2317 . 2  |-  ( ph  ->  ( J `  {
h } )  =  ( J `  {
i } ) )
82simprd 449 . . 3  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  { ( G R h ) } ) )
95simprd 449 . . 3  |-  ( ph  ->  ( M `  ( N `  { ( X  .-  Y ) } ) )  =  ( J `  { ( G R i ) } ) )
108, 9eqtr3d 2317 . 2  |-  ( ph  ->  ( J `  {
( G R h ) } )  =  ( J `  {
( G R i ) } ) )
117, 10jca 518 1  |-  ( ph  ->  ( ( J `  { h } )  =  ( J `  { i } )  /\  ( J `  { ( G R h ) } )  =  ( J `  { ( G R i ) } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   {csn 3640   ` cfv 5255  (class class class)co 5858   Basecbs 13148   0gc0g 13400   -gcsg 14365   LSpanclspn 15728   HLchlt 29540   LHypclh 30173   DVecHcdvh 31268  LCDualclcd 31776  mapdcmpd 31814
This theorem is referenced by:  mapdpglem26  31888  mapdpglem27  31889
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-cleq 2276
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