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Theorem lnfnli 22620
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 22511 . . 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 661 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  ~H  /\  C  e.  ~H )
)  ->  ( T `  ( ( A  .h  B )  +h  C
) )  =  ( ( A  x.  ( T `  B )
)  +  ( T `
 C ) ) )
433impb 1147 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735    + caddc 8740    x. cmul 8742   ~Hchil 21499    +h cva 21500    .h csm 21501   LinFnclf 21534
This theorem is referenced by:  lnfn0i  22622  lnfnaddi  22623  lnfnmuli  22624
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-hilex 21579
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-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-lnfn 22428
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