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Theorem hoaddcl 22168
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 )

Proof of Theorem hoaddcl
StepHypRef Expression
1 ffvelrn 5515 . . . . 5  |-  ( ( S : ~H --> ~H  /\  x  e.  ~H )  ->  ( S `  x
)  e.  ~H )
21adantlr 698 . . . 4  |-  ( ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( S `  x )  e.  ~H )
3 ffvelrn 5515 . . . . 5  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
43adantll 697 . . . 4  |-  ( ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( T `  x )  e.  ~H )
5 hvaddcl 21422 . . . 4  |-  ( ( ( S `  x
)  e.  ~H  /\  ( T `  x )  e.  ~H )  -> 
( ( S `  x )  +h  ( T `  x )
)  e.  ~H )
62, 4, 5syl2anc 645 . . 3  |-  ( ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  (
( S `  x
)  +h  ( T `
 x ) )  e.  ~H )
7 eqid 2253 . . 3  |-  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) )  =  ( x  e. 
~H  |->  ( ( S `
 x )  +h  ( T `  x
) ) )
86, 7fmptd 5536 . 2  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x )
) ) : ~H --> ~H )
9 hosmval 21997 . . 3  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  +op  T
)  =  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) ) )
109feq1d 5236 . 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 1621    e. cmpt 3974   -->wf 4588   ` cfv 4592  (class class class)co 5710   ~Hchil 21329    +h cva 21330    +op chos 21348
This theorem is referenced by:  hoaddcli  22178  hoadd4  22194  hoadddi  22213  hoadddir  22214  hosub4  22223  hoaddsubass  22225  ho2times  22229  hmops  22430  adjadd  22503  opsqrlem6  22555
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-hilex 21409  ax-hfvadd 21410
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-map 6660  df-hosum 21992
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