HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  lnopaddi Structured version   Unicode version

Theorem lnopaddi 25397
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 9361 . . 3  |-  1  e.  CC
2 lnopl.1 . . . 4  |-  T  e. 
LinOp
32lnopli 25394 . . 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 1301 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  (
( 1  .h  A
)  +h  B ) )  =  ( ( 1  .h  ( T `
 A ) )  +h  ( T `  B ) ) )
5 ax-hvmulid 24430 . . . . 5  |-  ( A  e.  ~H  ->  (
1  .h  A )  =  A )
65oveq1d 6127 . . . 4  |-  ( A  e.  ~H  ->  (
( 1  .h  A
)  +h  B )  =  ( A  +h  B ) )
76fveq2d 5716 . . 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 25395 . . . . . 6  |-  T : ~H
--> ~H
109ffvelrni 5863 . . . . 5  |-  ( A  e.  ~H  ->  ( T `  A )  e.  ~H )
11 ax-hvmulid 24430 . . . . 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 6127 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( 1  .h  ( T `  A
) )  +h  ( T `  B )
)  =  ( ( T `  A )  +h  ( T `  B ) ) )
154, 8, 143eqtr3d 2483 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 1369    e. wcel 1756   ` cfv 5439  (class class class)co 6112   CCcc 9301   1c1 9304   ~Hchil 24343    +h cva 24344    .h csm 24345   LinOpclo 24371
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 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-1cn 9361  ax-hilex 24423  ax-hvmulid 24430
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-map 7237  df-lnop 25267
This theorem is referenced by:  lnopaddmuli  25399  lnophsi  25427  lnopeq0lem1  25431  lnophmlem2  25443  imaelshi  25484  cnlnadjlem2  25494
  Copyright terms: Public domain W3C validator