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Theorem hosmval 23079
Description: Value of the sum of two Hilbert space operators. (Contributed by NM, 9-Nov-2000.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
Assertion
Ref Expression
hosmval  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  +op  T
)  =  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) ) )
Distinct variable groups:    x, S    x, T

Proof of Theorem hosmval
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-hilex 22343 . . 3  |-  ~H  e.  _V
21, 1elmap 6971 . 2  |-  ( S  e.  ( ~H  ^m  ~H )  <->  S : ~H --> ~H )
31, 1elmap 6971 . 2  |-  ( T  e.  ( ~H  ^m  ~H )  <->  T : ~H --> ~H )
4 fveq1 5660 . . . . 5  |-  ( f  =  S  ->  (
f `  x )  =  ( S `  x ) )
54oveq1d 6028 . . . 4  |-  ( f  =  S  ->  (
( f `  x
)  +h  ( g `
 x ) )  =  ( ( S `
 x )  +h  ( g `  x
) ) )
65mpteq2dv 4230 . . 3  |-  ( f  =  S  ->  (
x  e.  ~H  |->  ( ( f `  x
)  +h  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( g `  x ) ) ) )
7 fveq1 5660 . . . . 5  |-  ( g  =  T  ->  (
g `  x )  =  ( T `  x ) )
87oveq2d 6029 . . . 4  |-  ( g  =  T  ->  (
( S `  x
)  +h  ( g `
 x ) )  =  ( ( S `
 x )  +h  ( T `  x
) ) )
98mpteq2dv 4230 . . 3  |-  ( g  =  T  ->  (
x  e.  ~H  |->  ( ( S `  x
)  +h  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) ) )
10 df-hosum 23074 . . 3  |-  +op  =  ( f  e.  ( ~H  ^m  ~H ) ,  g  e.  ( ~H  ^m  ~H )  |->  ( x  e.  ~H  |->  ( ( f `  x
)  +h  ( g `
 x ) ) ) )
111mptex 5898 . . 3  |-  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) )  e.  _V
126, 9, 10, 11ovmpt2 6141 . 2  |-  ( ( S  e.  ( ~H 
^m  ~H )  /\  T  e.  ( ~H  ^m  ~H ) )  ->  ( S  +op  T )  =  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x )
) ) )
132, 3, 12syl2anbr 467 1  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  +op  T
)  =  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    e. cmpt 4200   -->wf 5383   ` cfv 5387  (class class class)co 6013    ^m cmap 6947   ~Hchil 22263    +h cva 22264    +op chos 22282
This theorem is referenced by:  hosval  23084  hoaddcl  23102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-hilex 22343
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-map 6949  df-hosum 23074
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