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Theorem lnfnli 26750
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 26641 . . 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 680 . 2  |-  ( ( A  e.  CC  /\  ( B  e.  ~H  /\  C  e.  ~H )
)  ->  ( T `  ( ( A  .h  B )  +h  C
) )  =  ( ( A  x.  ( T `  B )
)  +  ( T `
 C ) ) )
433impb 1192 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767   ` cfv 5593  (class class class)co 6294   CCcc 9500    + caddc 9505    x. cmul 9507   ~Hchil 25627    +h cva 25628    .h csm 25629   LinFnclf 25662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-hilex 25707
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fv 5601  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-map 7432  df-lnfn 26558
This theorem is referenced by:  lnfn0i  26752  lnfnaddi  26753  lnfnmuli  26754
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