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Theorem lnfnaddi 25445
Description: Additive 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
lnfnaddi  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  B ) )  =  ( ( T `
 A )  +  ( T `  B
) ) )

Proof of Theorem lnfnaddi
StepHypRef Expression
1 ax-1cn 9338 . . 3  |-  1  e.  CC
2 lnfnl.1 . . . 4  |-  T  e. 
LinFn
32lnfnli 25442 . . 3  |-  ( ( 1  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( (
1  .h  A )  +h  B ) )  =  ( ( 1  x.  ( T `  A ) )  +  ( T `  B
) ) )
41, 3mp3an1 1301 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  (
( 1  .h  A
)  +h  B ) )  =  ( ( 1  x.  ( T `
 A ) )  +  ( T `  B ) ) )
5 ax-hvmulid 24406 . . . . 5  |-  ( A  e.  ~H  ->  (
1  .h  A )  =  A )
65oveq1d 6104 . . . 4  |-  ( A  e.  ~H  ->  (
( 1  .h  A
)  +h  B )  =  ( A  +h  B ) )
76fveq2d 5693 . . 3  |-  ( A  e.  ~H  ->  ( T `  ( (
1  .h  A )  +h  B ) )  =  ( T `  ( A  +h  B
) ) )
87adantr 465 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  (
( 1  .h  A
)  +h  B ) )  =  ( T `
 ( A  +h  B ) ) )
92lnfnfi 25443 . . . . . 6  |-  T : ~H
--> CC
109ffvelrni 5840 . . . . 5  |-  ( A  e.  ~H  ->  ( T `  A )  e.  CC )
1110mulid2d 9402 . . . 4  |-  ( A  e.  ~H  ->  (
1  x.  ( T `
 A ) )  =  ( T `  A ) )
1211adantr 465 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( 1  x.  ( T `  A )
)  =  ( T `
 A ) )
1312oveq1d 6104 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( 1  x.  ( T `  A
) )  +  ( T `  B ) )  =  ( ( T `  A )  +  ( T `  B ) ) )
144, 8, 133eqtr3d 2481 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  B ) )  =  ( ( T `
 A )  +  ( T `  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   ` cfv 5416  (class class class)co 6089   CCcc 9278   1c1 9281    + caddc 9283    x. cmul 9285   ~Hchil 24319    +h cva 24320    .h csm 24321   LinFnclf 24354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-mulcl 9342  ax-mulcom 9344  ax-mulass 9346  ax-distr 9347  ax-1rid 9350  ax-cnre 9353  ax-hilex 24399  ax-hvmulid 24406
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-map 7214  df-lnfn 25250
This theorem is referenced by:  lnfnaddmuli  25447  nlelshi  25462
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