Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lcfrlem8 Unicode version

Theorem lcfrlem8 31739
Description: Lemma for lcf1o 31741 and lcfr 31775. (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 3651 . . . . . . 7  |-  ( x  =  X  ->  { x }  =  { X } )
32fveq2d 5529 . . . . . 6  |-  ( x  =  X  ->  (  ._|_  `  { x }
)  =  (  ._|_  `  { X } ) )
4 oveq2 5866 . . . . . . . 8  |-  ( x  =  X  ->  (
k  .x.  x )  =  ( k  .x.  X ) )
54oveq2d 5874 . . . . . . 7  |-  ( x  =  X  ->  (
w  .+  ( k  .x.  x ) )  =  ( w  .+  (
k  .x.  X )
) )
65eqeq2d 2294 . . . . . 6  |-  ( x  =  X  ->  (
v  =  ( w 
.+  ( k  .x.  x ) )  <->  v  =  ( w  .+  ( k 
.x.  X ) ) ) )
73, 6rexeqbidv 2749 . . . . 5  |-  ( x  =  X  ->  ( E. w  e.  (  ._|_  `  { x }
) v  =  ( w  .+  ( k 
.x.  x ) )  <->  E. w  e.  (  ._|_  `  { X }
) v  =  ( w  .+  ( k 
.x.  X ) ) ) )
87riotabidv 6306 . . . 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 4107 . . 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 5539 . . . . 5  |-  ( Base `  U )  e.  _V
1311, 12eqeltri 2353 . . . 4  |-  V  e. 
_V
1413mptex 5746 . . 3  |-  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X } ) v  =  ( w  .+  (
k  .x.  X )
) ) )  e. 
_V
159, 10, 14fvmpt 5602 . 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 1623    e. wcel 1684   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149   {csn 3640    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .scvsca 13212   0gc0g 13400  LFnlclfn 29247  LKerclk 29275  LDualcld 29313   HLchlt 29540   LHypclh 30173   DVecHcdvh 31268   ocHcoch 31537
This theorem is referenced by:  lcfrlem9  31740  lcfrlem10  31742  lcfrlem11  31743
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304
  Copyright terms: Public domain W3C validator