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Theorem lcfrlem8 32361
Description: Lemma for lcf1o 32363 and lcfr 32397. (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 3664 . . . . . . 7  |-  ( x  =  X  ->  { x }  =  { X } )
32fveq2d 5545 . . . . . 6  |-  ( x  =  X  ->  (  ._|_  `  { x }
)  =  (  ._|_  `  { X } ) )
4 oveq2 5882 . . . . . . . 8  |-  ( x  =  X  ->  (
k  .x.  x )  =  ( k  .x.  X ) )
54oveq2d 5890 . . . . . . 7  |-  ( x  =  X  ->  (
w  .+  ( k  .x.  x ) )  =  ( w  .+  (
k  .x.  X )
) )
65eqeq2d 2307 . . . . . 6  |-  ( x  =  X  ->  (
v  =  ( w 
.+  ( k  .x.  x ) )  <->  v  =  ( w  .+  ( k 
.x.  X ) ) ) )
73, 6rexeqbidv 2762 . . . . 5  |-  ( x  =  X  ->  ( E. w  e.  (  ._|_  `  { x }
) v  =  ( w  .+  ( k 
.x.  x ) )  <->  E. w  e.  (  ._|_  `  { X }
) v  =  ( w  .+  ( k 
.x.  X ) ) ) )
87riotabidv 6322 . . . 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 4123 . . 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 5555 . . . . 5  |-  ( Base `  U )  e.  _V
1311, 12eqeltri 2366 . . . 4  |-  V  e. 
_V
1413mptex 5762 . . 3  |-  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X } ) v  =  ( w  .+  (
k  .x.  X )
) ) )  e. 
_V
159, 10, 14fvmpt 5618 . 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 15 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 358    = wceq 1632    e. wcel 1696   E.wrex 2557   {crab 2560   _Vcvv 2801    \ cdif 3162   {csn 3653    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   iota_crio 6313   Basecbs 13164   +g cplusg 13224  Scalarcsca 13227   .scvsca 13228   0gc0g 13416  LFnlclfn 29869  LKerclk 29897  LDualcld 29935   HLchlt 30162   LHypclh 30795   DVecHcdvh 31890   ocHcoch 32159
This theorem is referenced by:  lcfrlem9  32362  lcfrlem10  32364  lcfrlem11  32365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-riota 6320
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