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Theorem hodmval 22317
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 21579 . . 3  |-  ~H  e.  _V
21, 1elmap 6796 . 2  |-  ( S  e.  ( ~H  ^m  ~H )  <->  S : ~H --> ~H )
31, 1elmap 6796 . 2  |-  ( T  e.  ( ~H  ^m  ~H )  <->  T : ~H --> ~H )
4 fveq1 5524 . . . . 5  |-  ( f  =  S  ->  (
f `  x )  =  ( S `  x ) )
54oveq1d 5873 . . . 4  |-  ( f  =  S  ->  (
( f `  x
)  -h  ( g `
 x ) )  =  ( ( S `
 x )  -h  ( g `  x
) ) )
65mpteq2dv 4107 . . 3  |-  ( f  =  S  ->  (
x  e.  ~H  |->  ( ( f `  x
)  -h  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( ( S `  x )  -h  ( g `  x ) ) ) )
7 fveq1 5524 . . . . 5  |-  ( g  =  T  ->  (
g `  x )  =  ( T `  x ) )
87oveq2d 5874 . . . 4  |-  ( g  =  T  ->  (
( S `  x
)  -h  ( g `
 x ) )  =  ( ( S `
 x )  -h  ( T `  x
) ) )
98mpteq2dv 4107 . . 3  |-  ( g  =  T  ->  (
x  e.  ~H  |->  ( ( S `  x
)  -h  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x ) ) ) )
10 df-hodif 22312 . . 3  |-  -op  =  ( f  e.  ( ~H  ^m  ~H ) ,  g  e.  ( ~H  ^m  ~H )  |->  ( x  e.  ~H  |->  ( ( f `  x
)  -h  ( g `
 x ) ) ) )
111mptex 5746 . . 3  |-  ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x ) ) )  e.  _V
126, 9, 10, 11ovmpt2 5983 . 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 466 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 358    = wceq 1623    e. wcel 1684    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   ~Hchil 21499    -h cmv 21505    -op chod 21520
This theorem is referenced by:  hodval  22322  hosubcli  22349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-hodif 22312
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