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

Theorem lflf 29875
Description: A linear functional is a function from vectors to scalars. (lnfnfi 22637 analog.) (Contributed by NM, 15-Apr-2014.)
Hypotheses
Ref Expression
lflf.d  |-  D  =  (Scalar `  W )
lflf.k  |-  K  =  ( Base `  D
)
lflf.v  |-  V  =  ( Base `  W
)
lflf.f  |-  F  =  (LFnl `  W )
Assertion
Ref Expression
lflf  |-  ( ( W  e.  X  /\  G  e.  F )  ->  G : V --> K )

Proof of Theorem lflf
Dummy variables  x  r  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lflf.v . . 3  |-  V  =  ( Base `  W
)
2 eqid 2296 . . 3  |-  ( +g  `  W )  =  ( +g  `  W )
3 lflf.d . . 3  |-  D  =  (Scalar `  W )
4 eqid 2296 . . 3  |-  ( .s
`  W )  =  ( .s `  W
)
5 lflf.k . . 3  |-  K  =  ( Base `  D
)
6 eqid 2296 . . 3  |-  ( +g  `  D )  =  ( +g  `  D )
7 eqid 2296 . . 3  |-  ( .r
`  D )  =  ( .r `  D
)
8 lflf.f . . 3  |-  F  =  (LFnl `  W )
91, 2, 3, 4, 5, 6, 7, 8islfl 29872 . 2  |-  ( W  e.  X  ->  ( G  e.  F  <->  ( G : V --> K  /\  A. r  e.  K  A. x  e.  V  A. y  e.  V  ( G `  ( (
r ( .s `  W ) x ) ( +g  `  W
) y ) )  =  ( ( r ( .r `  D
) ( G `  x ) ) ( +g  `  D ) ( G `  y
) ) ) ) )
109simprbda 606 1  |-  ( ( W  e.  X  /\  G  e.  F )  ->  G : V --> K )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   .rcmulr 13225  Scalarcsca 13227   .scvsca 13228  LFnlclfn 29869
This theorem is referenced by:  lflcl  29876  lfl1  29882  lfladdcl  29883  lfladdcom  29884  lfladdass  29885  lfladd0l  29886  lflnegl  29888  lflvscl  29889  lflvsdi1  29890  lflvsdi2  29891  lflvsass  29893  lfl0sc  29894  lfl1sc  29896  ellkr  29901  lkr0f  29906  lkrsc  29909  eqlkr2  29912  eqlkr3  29913  ldualvaddval  29943  ldualvsval  29950
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  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-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-lfl 29870
  Copyright terms: Public domain W3C validator