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Theorem hoscl 23097
Description: Closure of the sum of two Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
Assertion
Ref Expression
hoscl  |-  ( ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  /\  A  e. 
~H )  ->  (
( S  +op  T
) `  A )  e.  ~H )

Proof of Theorem hoscl
StepHypRef Expression
1 hosval 23092 . . 3  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  A  e.  ~H )  ->  ( ( S  +op  T ) `  A )  =  ( ( S `
 A )  +h  ( T `  A
) ) )
213expa 1153 . 2  |-  ( ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  /\  A  e. 
~H )  ->  (
( S  +op  T
) `  A )  =  ( ( S `
 A )  +h  ( T `  A
) ) )
3 ffvelrn 5808 . . . . 5  |-  ( ( S : ~H --> ~H  /\  A  e.  ~H )  ->  ( S `  A
)  e.  ~H )
4 ffvelrn 5808 . . . . 5  |-  ( ( T : ~H --> ~H  /\  A  e.  ~H )  ->  ( T `  A
)  e.  ~H )
53, 4anim12i 550 . . . 4  |-  ( ( ( S : ~H --> ~H  /\  A  e.  ~H )  /\  ( T : ~H
--> ~H  /\  A  e. 
~H ) )  -> 
( ( S `  A )  e.  ~H  /\  ( T `  A
)  e.  ~H )
)
65anandirs 805 . . 3  |-  ( ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  /\  A  e. 
~H )  ->  (
( S `  A
)  e.  ~H  /\  ( T `  A )  e.  ~H ) )
7 hvaddcl 22364 . . 3  |-  ( ( ( S `  A
)  e.  ~H  /\  ( T `  A )  e.  ~H )  -> 
( ( S `  A )  +h  ( T `  A )
)  e.  ~H )
86, 7syl 16 . 2  |-  ( ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  /\  A  e. 
~H )  ->  (
( S `  A
)  +h  ( T `
 A ) )  e.  ~H )
92, 8eqeltrd 2462 1  |-  ( ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  /\  A  e. 
~H )  ->  (
( S  +op  T
) `  A )  e.  ~H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   -->wf 5391   ` cfv 5395  (class class class)co 6021   ~Hchil 22271    +h cva 22272    +op chos 22290
This theorem is referenced by:  hoscli  23114
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-hilex 22351  ax-hfvadd 22352
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-map 6957  df-hosum 23082
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