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Theorem hosval 23245
Description: Value of the sum of two Hilbert space operators. (Contributed by NM, 10-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
hosval  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  A  e.  ~H )  ->  ( ( S  +op  T ) `  A )  =  ( ( S `
 A )  +h  ( T `  A
) ) )

Proof of Theorem hosval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 hosmval 23240 . . . 4  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  +op  T
)  =  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) ) )
21fveq1d 5732 . . 3  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( ( S  +op  T ) `  A )  =  ( ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) ) `
 A ) )
3 fveq2 5730 . . . . 5  |-  ( x  =  A  ->  ( S `  x )  =  ( S `  A ) )
4 fveq2 5730 . . . . 5  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
53, 4oveq12d 6101 . . . 4  |-  ( x  =  A  ->  (
( S `  x
)  +h  ( T `
 x ) )  =  ( ( S `
 A )  +h  ( T `  A
) ) )
6 eqid 2438 . . . 4  |-  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) )  =  ( x  e. 
~H  |->  ( ( S `
 x )  +h  ( T `  x
) ) )
7 ovex 6108 . . . 4  |-  ( ( S `  A )  +h  ( T `  A ) )  e. 
_V
85, 6, 7fvmpt 5808 . . 3  |-  ( A  e.  ~H  ->  (
( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x )
) ) `  A
)  =  ( ( S `  A )  +h  ( T `  A ) ) )
92, 8sylan9eq 2490 . 2  |-  ( ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  /\  A  e. 
~H )  ->  (
( S  +op  T
) `  A )  =  ( ( S `
 A )  +h  ( T `  A
) ) )
1093impa 1149 1  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  A  e.  ~H )  ->  ( ( S  +op  T ) `  A )  =  ( ( S `
 A )  +h  ( T `  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    e. cmpt 4268   -->wf 5452   ` cfv 5456  (class class class)co 6083   ~Hchil 22424    +h cva 22425    +op chos 22443
This theorem is referenced by:  hoscl  23250  hoaddcomi  23277  hodsi  23280  hoaddassi  23281  hocadddiri  23284  hoaddid1i  23291  honegsubi  23301  hoadddi  23308  hoadddir  23309  lnophsi  23506  hmops  23525  adjadd  23598  nmoptrii  23599  leopadd  23637  pjsdii  23660  pjscji  23675  pjtoi  23684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-hilex 22504
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-map 7022  df-hosum 23235
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