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Theorem hoaddcl 22298
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 5597 . . . . 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 5597 . . . . 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 21552 . . . 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 2258 . . 3  |-  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) )  =  ( x  e. 
~H  |->  ( ( S `
 x )  +h  ( T `  x
) ) )
86, 7fmptd 5618 . 2  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x )
) ) : ~H --> ~H )
9 hosmval 22275 . . 3  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  +op  T
)  =  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) ) )
109feq1d 5317 . 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 4051   -->wf 4669   ` cfv 4673  (class class class)co 5792   ~Hchil 21459    +h cva 21460    +op chos 21478
This theorem is referenced by:  hoaddcli  22308  hoadd4  22324  hoadddi  22343  hoadddir  22344  hosub4  22353  hoaddsubass  22355  ho2times  22359  hmops  22560  adjadd  22633  opsqrlem6  22685
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 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-hilex 21539  ax-hfvadd 21540
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-map 6742  df-hosum 22270
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