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Theorem hodmval 27082
Description: Value of the difference 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
hodmval  |-  ( ( 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 hodmval
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-hilex 26343 . . 3  |-  ~H  e.  _V
21, 1elmap 7487 . 2  |-  ( S  e.  ( ~H  ^m  ~H )  <->  S : ~H --> ~H )
31, 1elmap 7487 . 2  |-  ( T  e.  ( ~H  ^m  ~H )  <->  T : ~H --> ~H )
4 fveq1 5850 . . . . 5  |-  ( f  =  S  ->  (
f `  x )  =  ( S `  x ) )
54oveq1d 6295 . . . 4  |-  ( f  =  S  ->  (
( f `  x
)  -h  ( g `
 x ) )  =  ( ( S `
 x )  -h  ( g `  x
) ) )
65mpteq2dv 4484 . . 3  |-  ( f  =  S  ->  (
x  e.  ~H  |->  ( ( f `  x
)  -h  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( ( S `  x )  -h  ( g `  x ) ) ) )
7 fveq1 5850 . . . . 5  |-  ( g  =  T  ->  (
g `  x )  =  ( T `  x ) )
87oveq2d 6296 . . . 4  |-  ( g  =  T  ->  (
( S `  x
)  -h  ( g `
 x ) )  =  ( ( S `
 x )  -h  ( T `  x
) ) )
98mpteq2dv 4484 . . 3  |-  ( g  =  T  ->  (
x  e.  ~H  |->  ( ( S `  x
)  -h  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x ) ) ) )
10 df-hodif 27077 . . 3  |-  -op  =  ( f  e.  ( ~H  ^m  ~H ) ,  g  e.  ( ~H  ^m  ~H )  |->  ( x  e.  ~H  |->  ( ( f `  x
)  -h  ( g `
 x ) ) ) )
111mptex 6126 . . 3  |-  ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x ) ) )  e.  _V
126, 9, 10, 11ovmpt2 6421 . 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 1407    e. wcel 1844    |-> cmpt 4455   -->wf 5567   ` cfv 5571  (class class class)co 6280    ^m cmap 7459   ~Hchil 26263    -h cmv 26269    -op chod 26284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-hilex 26343
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-map 7461  df-hodif 27077
This theorem is referenced by:  hodval  27087  hosubcli  27114
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