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Theorem lnopaddi 26713
Description: Additive property of a linear Hilbert space operator. (Contributed by NM, 11-May-2005.) (New usage is discouraged.)
Hypothesis
Ref Expression
lnopl.1  |-  T  e. 
LinOp
Assertion
Ref Expression
lnopaddi  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  B ) )  =  ( ( T `
 A )  +h  ( T `  B
) ) )

Proof of Theorem lnopaddi
StepHypRef Expression
1 ax-1cn 9562 . . 3  |-  1  e.  CC
2 lnopl.1 . . . 4  |-  T  e. 
LinOp
32lnopli 26710 . . 3  |-  ( ( 1  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( (
1  .h  A )  +h  B ) )  =  ( ( 1  .h  ( T `  A ) )  +h  ( T `  B
) ) )
41, 3mp3an1 1311 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  (
( 1  .h  A
)  +h  B ) )  =  ( ( 1  .h  ( T `
 A ) )  +h  ( T `  B ) ) )
5 ax-hvmulid 25746 . . . . 5  |-  ( A  e.  ~H  ->  (
1  .h  A )  =  A )
65oveq1d 6310 . . . 4  |-  ( A  e.  ~H  ->  (
( 1  .h  A
)  +h  B )  =  ( A  +h  B ) )
76fveq2d 5876 . . 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 ) ) )
92lnopfi 26711 . . . . . 6  |-  T : ~H
--> ~H
109ffvelrni 6031 . . . . 5  |-  ( A  e.  ~H  ->  ( T `  A )  e.  ~H )
11 ax-hvmulid 25746 . . . . 5  |-  ( ( T `  A )  e.  ~H  ->  (
1  .h  ( T `
 A ) )  =  ( T `  A ) )
1210, 11syl 16 . . . 4  |-  ( A  e.  ~H  ->  (
1  .h  ( T `
 A ) )  =  ( T `  A ) )
1312adantr 465 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( 1  .h  ( T `  A )
)  =  ( T `
 A ) )
1413oveq1d 6310 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( 1  .h  ( T `  A
) )  +h  ( T `  B )
)  =  ( ( T `  A )  +h  ( T `  B ) ) )
154, 8, 143eqtr3d 2516 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  B ) )  =  ( ( T `
 A )  +h  ( T `  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   ` cfv 5594  (class class class)co 6295   CCcc 9502   1c1 9505   ~Hchil 25659    +h cva 25660    .h csm 25661   LinOpclo 25687
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-1cn 9562  ax-hilex 25739  ax-hvmulid 25746
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 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7434  df-lnop 26583
This theorem is referenced by:  lnopaddmuli  26715  lnophsi  26743  lnopeq0lem1  26747  lnophmlem2  26759  imaelshi  26800  cnlnadjlem2  26810
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