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Theorem hosmval 25161
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 24423 . . 3  |-  ~H  e.  _V
21, 1elmap 7262 . 2  |-  ( S  e.  ( ~H  ^m  ~H )  <->  S : ~H --> ~H )
31, 1elmap 7262 . 2  |-  ( T  e.  ( ~H  ^m  ~H )  <->  T : ~H --> ~H )
4 fveq1 5711 . . . . 5  |-  ( f  =  S  ->  (
f `  x )  =  ( S `  x ) )
54oveq1d 6127 . . . 4  |-  ( f  =  S  ->  (
( f `  x
)  +h  ( g `
 x ) )  =  ( ( S `
 x )  +h  ( g `  x
) ) )
65mpteq2dv 4400 . . 3  |-  ( f  =  S  ->  (
x  e.  ~H  |->  ( ( f `  x
)  +h  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( g `  x ) ) ) )
7 fveq1 5711 . . . . 5  |-  ( g  =  T  ->  (
g `  x )  =  ( T `  x ) )
87oveq2d 6128 . . . 4  |-  ( g  =  T  ->  (
( S `  x
)  +h  ( g `
 x ) )  =  ( ( S `
 x )  +h  ( T `  x
) ) )
98mpteq2dv 4400 . . 3  |-  ( g  =  T  ->  (
x  e.  ~H  |->  ( ( S `  x
)  +h  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) ) )
10 df-hosum 25156 . . 3  |-  +op  =  ( f  e.  ( ~H  ^m  ~H ) ,  g  e.  ( ~H  ^m  ~H )  |->  ( x  e.  ~H  |->  ( ( f `  x
)  +h  ( g `
 x ) ) ) )
111mptex 5969 . . 3  |-  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) )  e.  _V
126, 9, 10, 11ovmpt2 6247 . 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 480 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 369    = wceq 1369    e. wcel 1756    e. cmpt 4371   -->wf 5435   ` cfv 5439  (class class class)co 6112    ^m cmap 7235   ~Hchil 24343    +h cva 24344    +op chos 24362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-hilex 24423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-map 7237  df-hosum 25156
This theorem is referenced by:  hosval  25166  hoaddcl  25184
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