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Theorem lcfrlem8 31666
Description: Lemma for lcf1o 31668 and lcfr 31702. (Contributed by NM, 21-Feb-2015.)
Hypotheses
Ref Expression
lcf1o.h  |-  H  =  ( LHyp `  K
)
lcf1o.o  |-  ._|_  =  ( ( ocH `  K
) `  W )
lcf1o.u  |-  U  =  ( ( DVecH `  K
) `  W )
lcf1o.v  |-  V  =  ( Base `  U
)
lcf1o.a  |-  .+  =  ( +g  `  U )
lcf1o.t  |-  .x.  =  ( .s `  U )
lcf1o.s  |-  S  =  (Scalar `  U )
lcf1o.r  |-  R  =  ( Base `  S
)
lcf1o.z  |-  .0.  =  ( 0g `  U )
lcf1o.f  |-  F  =  (LFnl `  U )
lcf1o.l  |-  L  =  (LKer `  U )
lcf1o.d  |-  D  =  (LDual `  U )
lcf1o.q  |-  Q  =  ( 0g `  D
)
lcf1o.c  |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }
lcf1o.j  |-  J  =  ( x  e.  ( V  \  {  .0.  } )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
k  .x.  x )
) ) ) )
lcflo.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
lcfrlem8.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
Assertion
Ref Expression
lcfrlem8  |-  ( ph  ->  ( J `  X
)  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X } ) v  =  ( w  .+  (
k  .x.  X )
) ) ) )
Distinct variable groups:    x, w,  ._|_    x,  .0.    x, v, V    x,  .x.    v, k, w, x, X    x,  .+    x, R
Allowed substitution hints:    ph( x, w, v, f, k)    C( x, w, v, f, k)    D( x, w, v, f, k)    .+ ( w, v, f, k)    Q( x, w, v, f, k)    R( w, v, f, k)    S( x, w, v, f, k)    .x. ( w, v, f, k)    U( x, w, v, f, k)    F( x, w, v, f, k)    H( x, w, v, f, k)    J( x, w, v, f, k)    K( x, w, v, f, k)    L( x, w, v, f, k)    ._|_ ( v, f, k)    V( w, f, k)    W( x, w, v, f, k)    X( f)    .0. ( w, v, f, k)

Proof of Theorem lcfrlem8
StepHypRef Expression
1 lcfrlem8.x . 2  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
2 sneq 3770 . . . . . . 7  |-  ( x  =  X  ->  { x }  =  { X } )
32fveq2d 5674 . . . . . 6  |-  ( x  =  X  ->  (  ._|_  `  { x }
)  =  (  ._|_  `  { X } ) )
4 oveq2 6030 . . . . . . . 8  |-  ( x  =  X  ->  (
k  .x.  x )  =  ( k  .x.  X ) )
54oveq2d 6038 . . . . . . 7  |-  ( x  =  X  ->  (
w  .+  ( k  .x.  x ) )  =  ( w  .+  (
k  .x.  X )
) )
65eqeq2d 2400 . . . . . 6  |-  ( x  =  X  ->  (
v  =  ( w 
.+  ( k  .x.  x ) )  <->  v  =  ( w  .+  ( k 
.x.  X ) ) ) )
73, 6rexeqbidv 2862 . . . . 5  |-  ( x  =  X  ->  ( E. w  e.  (  ._|_  `  { x }
) v  =  ( w  .+  ( k 
.x.  x ) )  <->  E. w  e.  (  ._|_  `  { X }
) v  =  ( w  .+  ( k 
.x.  X ) ) ) )
87riotabidv 6489 . . . 4  |-  ( x  =  X  ->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w 
.+  ( k  .x.  x ) ) )  =  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X } ) v  =  ( w  .+  (
k  .x.  X )
) ) )
98mpteq2dv 4239 . . 3  |-  ( x  =  X  ->  (
v  e.  V  |->  (
iota_ k  e.  R E. w  e.  (  ._|_  `  { x }
) v  =  ( w  .+  ( k 
.x.  x ) ) ) )  =  ( v  e.  V  |->  (
iota_ k  e.  R E. w  e.  (  ._|_  `  { X }
) v  =  ( w  .+  ( k 
.x.  X ) ) ) ) )
10 lcf1o.j . . 3  |-  J  =  ( x  e.  ( V  \  {  .0.  } )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
k  .x.  x )
) ) ) )
11 lcf1o.v . . . . 5  |-  V  =  ( Base `  U
)
12 fvex 5684 . . . . 5  |-  ( Base `  U )  e.  _V
1311, 12eqeltri 2459 . . . 4  |-  V  e. 
_V
1413mptex 5907 . . 3  |-  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X } ) v  =  ( w  .+  (
k  .x.  X )
) ) )  e. 
_V
159, 10, 14fvmpt 5747 . 2  |-  ( X  e.  ( V  \  {  .0.  } )  -> 
( J `  X
)  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X } ) v  =  ( w  .+  (
k  .x.  X )
) ) ) )
161, 15syl 16 1  |-  ( ph  ->  ( J `  X
)  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X } ) v  =  ( w  .+  (
k  .x.  X )
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2652   {crab 2655   _Vcvv 2901    \ cdif 3262   {csn 3759    e. cmpt 4209   ` cfv 5396  (class class class)co 6022   iota_crio 6480   Basecbs 13398   +g cplusg 13458  Scalarcsca 13461   .scvsca 13462   0gc0g 13652  LFnlclfn 29174  LKerclk 29202  LDualcld 29240   HLchlt 29467   LHypclh 30100   DVecHcdvh 31195   ocHcoch 31464
This theorem is referenced by:  lcfrlem9  31667  lcfrlem10  31669  lcfrlem11  31670
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-riota 6487
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