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Theorem lnfnli 27528
Description: Basic property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
lnfnl.1  |-  T  e. 
LinFn
Assertion
Ref Expression
lnfnli  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( T `  ( ( A  .h  B )  +h  C ) )  =  ( ( A  x.  ( T `  B ) )  +  ( T `
 C ) ) )

Proof of Theorem lnfnli
StepHypRef Expression
1 lnfnl.1 . . 3  |-  T  e. 
LinFn
2 lnfnl 27419 . . 3  |-  ( ( ( T  e.  LinFn  /\  A  e.  CC )  /\  ( B  e. 
~H  /\  C  e.  ~H ) )  ->  ( T `  ( ( A  .h  B )  +h  C ) )  =  ( ( A  x.  ( T `  B ) )  +  ( T `
 C ) ) )
31, 2mpanl1 684 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  ~H  /\  C  e.  ~H )
)  ->  ( T `  ( ( A  .h  B )  +h  C
) )  =  ( ( A  x.  ( T `  B )
)  +  ( T `
 C ) ) )
433impb 1201 1  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( T `  ( ( A  .h  B )  +h  C ) )  =  ( ( A  x.  ( T `  B ) )  +  ( T `
 C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   ` cfv 5601  (class class class)co 6305   CCcc 9536    + caddc 9541    x. cmul 9543   ~Hchil 26407    +h cva 26408    .h csm 26409   LinFnclf 26442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-hilex 26487
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-map 7482  df-lnfn 27336
This theorem is referenced by:  lnfn0i  27530  lnfnaddi  27531  lnfnmuli  27532
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