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Theorem hoscl 23238
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 23233 . . 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 5860 . . . . 5  |-  ( ( S : ~H --> ~H  /\  A  e.  ~H )  ->  ( S `  A
)  e.  ~H )
4 ffvelrn 5860 . . . . 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 22505 . . 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 2509 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 1652    e. wcel 1725   -->wf 5442   ` cfv 5446  (class class class)co 6073   ~Hchil 22412    +h cva 22413    +op chos 22431
This theorem is referenced by:  hoscli  23255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-hilex 22492  ax-hfvadd 22493
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-hosum 23223
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