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Theorem lnfnf 25409
Description: A linear Hilbert space functional is a functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
Assertion
Ref Expression
lnfnf  |-  ( T  e.  LinFn  ->  T : ~H
--> CC )

Proof of Theorem lnfnf
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ellnfn 25408 . 2  |-  ( T  e.  LinFn 
<->  ( T : ~H --> CC  /\  A. x  e.  CC  A. y  e. 
~H  A. z  e.  ~H  ( T `  ( ( x  .h  y )  +h  z ) )  =  ( ( x  x.  ( T `  y ) )  +  ( T `  z
) ) ) )
21simplbi 460 1  |-  ( T  e.  LinFn  ->  T : ~H
--> CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1757   A.wral 2792   -->wf 5498   ` cfv 5502  (class class class)co 6176   CCcc 9367    + caddc 9372    x. cmul 9374   ~Hchil 24442    +h cva 24443    .h csm 24444   LinFnclf 24477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-hilex 24522
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-sbc 3271  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-br 4377  df-opab 4435  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-fv 5510  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-map 7302  df-lnfn 25373
This theorem is referenced by:  nmfn0  25512  lnfnfi  25566  rnbra  25632
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