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Theorem hoaddcl 22330
Description: The sum of Hilbert space operators is an operator. (Contributed by NM, 21-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
hoaddcl  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  +op  T
) : ~H --> ~H )
Dummy variable  x is distinct from all other variables.

Proof of Theorem hoaddcl
StepHypRef Expression
1 ffvelrn 5624 . . . . 5  |-  ( ( S : ~H --> ~H  /\  x  e.  ~H )  ->  ( S `  x
)  e.  ~H )
21adantlr 697 . . . 4  |-  ( ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( S `  x )  e.  ~H )
3 ffvelrn 5624 . . . . 5  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
43adantll 696 . . . 4  |-  ( ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( T `  x )  e.  ~H )
5 hvaddcl 21584 . . . 4  |-  ( ( ( S `  x
)  e.  ~H  /\  ( T `  x )  e.  ~H )  -> 
( ( S `  x )  +h  ( T `  x )
)  e.  ~H )
62, 4, 5syl2anc 644 . . 3  |-  ( ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  (
( S `  x
)  +h  ( T `
 x ) )  e.  ~H )
7 eqid 2284 . . 3  |-  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) )  =  ( x  e. 
~H  |->  ( ( S `
 x )  +h  ( T `  x
) ) )
86, 7fmptd 5645 . 2  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x )
) ) : ~H --> ~H )
9 hosmval 22307 . . 3  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  +op  T
)  =  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) ) )
109feq1d 5344 . 2  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( ( S  +op  T ) : ~H --> ~H  <->  ( x  e.  ~H  |->  ( ( S `
 x )  +h  ( T `  x
) ) ) : ~H --> ~H ) )
118, 10mpbird 225 1  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  +op  T
) : ~H --> ~H )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    e. wcel 1685    e. cmpt 4078   -->wf 5217   ` cfv 5221  (class class class)co 5819   ~Hchil 21491    +h cva 21492    +op chos 21510
This theorem is referenced by:  hoaddcli  22340  hoadd4  22356  hoadddi  22375  hoadddir  22376  hosub4  22385  hoaddsubass  22387  ho2times  22391  hmops  22592  adjadd  22665  opsqrlem6  22717
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-hilex 21571  ax-hfvadd 21572
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-map 6769  df-hosum 22302
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