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Theorem hosmval 27067
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 26330 . . 3  |-  ~H  e.  _V
21, 1elmap 7485 . 2  |-  ( S  e.  ( ~H  ^m  ~H )  <->  S : ~H --> ~H )
31, 1elmap 7485 . 2  |-  ( T  e.  ( ~H  ^m  ~H )  <->  T : ~H --> ~H )
4 fveq1 5848 . . . . 5  |-  ( f  =  S  ->  (
f `  x )  =  ( S `  x ) )
54oveq1d 6293 . . . 4  |-  ( f  =  S  ->  (
( f `  x
)  +h  ( g `
 x ) )  =  ( ( S `
 x )  +h  ( g `  x
) ) )
65mpteq2dv 4482 . . 3  |-  ( f  =  S  ->  (
x  e.  ~H  |->  ( ( f `  x
)  +h  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( g `  x ) ) ) )
7 fveq1 5848 . . . . 5  |-  ( g  =  T  ->  (
g `  x )  =  ( T `  x ) )
87oveq2d 6294 . . . 4  |-  ( g  =  T  ->  (
( S `  x
)  +h  ( g `
 x ) )  =  ( ( S `
 x )  +h  ( T `  x
) ) )
98mpteq2dv 4482 . . 3  |-  ( g  =  T  ->  (
x  e.  ~H  |->  ( ( S `  x
)  +h  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) ) )
10 df-hosum 27062 . . 3  |-  +op  =  ( f  e.  ( ~H  ^m  ~H ) ,  g  e.  ( ~H  ^m  ~H )  |->  ( x  e.  ~H  |->  ( ( f `  x
)  +h  ( g `
 x ) ) ) )
111mptex 6124 . . 3  |-  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) )  e.  _V
126, 9, 10, 11ovmpt2 6419 . 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 478 1  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  +op  T
)  =  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    |-> cmpt 4453   -->wf 5565   ` cfv 5569  (class class class)co 6278    ^m cmap 7457   ~Hchil 26250    +h cva 26251    +op chos 26269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-hilex 26330
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-map 7459  df-hosum 27062
This theorem is referenced by:  hosval  27072  hoaddcl  27090
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