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Theorem lcfrlem8 37419
Description: Lemma for lcf1o 37421 and lcfr 37455. (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 4042 . . . . . . 7  |-  ( x  =  X  ->  { x }  =  { X } )
32fveq2d 5876 . . . . . 6  |-  ( x  =  X  ->  (  ._|_  `  { x }
)  =  (  ._|_  `  { X } ) )
4 oveq2 6304 . . . . . . . 8  |-  ( x  =  X  ->  (
k  .x.  x )  =  ( k  .x.  X ) )
54oveq2d 6312 . . . . . . 7  |-  ( x  =  X  ->  (
w  .+  ( k  .x.  x ) )  =  ( w  .+  (
k  .x.  X )
) )
65eqeq2d 2471 . . . . . 6  |-  ( x  =  X  ->  (
v  =  ( w 
.+  ( k  .x.  x ) )  <->  v  =  ( w  .+  ( k 
.x.  X ) ) ) )
73, 6rexeqbidv 3069 . . . . 5  |-  ( x  =  X  ->  ( E. w  e.  (  ._|_  `  { x }
) v  =  ( w  .+  ( k 
.x.  x ) )  <->  E. w  e.  (  ._|_  `  { X }
) v  =  ( w  .+  ( k 
.x.  X ) ) ) )
87riotabidv 6260 . . . 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 4544 . . 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 5882 . . . . 5  |-  ( Base `  U )  e.  _V
1311, 12eqeltri 2541 . . . 4  |-  V  e. 
_V
1413mptex 6144 . . 3  |-  ( v  e.  V  |->  ( iota_ k  e.  R  E. w  e.  (  ._|_  `  { X } ) v  =  ( w  .+  (
k  .x.  X )
) ) )  e. 
_V
159, 10, 14fvmpt 5956 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808   {crab 2811   _Vcvv 3109    \ cdif 3468   {csn 4032    |-> cmpt 4515   ` cfv 5594   iota_crio 6257  (class class class)co 6296   Basecbs 14644   +g cplusg 14712  Scalarcsca 14715   .scvsca 14716   0gc0g 14857  LFnlclfn 34925  LKerclk 34953  LDualcld 34991   HLchlt 35218   LHypclh 35851   DVecHcdvh 36948   ocHcoch 37217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299
This theorem is referenced by:  lcfrlem9  37420  lcfrlem10  37422  lcfrlem11  37423
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