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Theorem hodmval 22333
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 21595 . . 3  |-  ~H  e.  _V
21, 1elmap 6812 . 2  |-  ( S  e.  ( ~H  ^m  ~H )  <->  S : ~H --> ~H )
31, 1elmap 6812 . 2  |-  ( T  e.  ( ~H  ^m  ~H )  <->  T : ~H --> ~H )
4 fveq1 5540 . . . . 5  |-  ( f  =  S  ->  (
f `  x )  =  ( S `  x ) )
54oveq1d 5889 . . . 4  |-  ( f  =  S  ->  (
( f `  x
)  -h  ( g `
 x ) )  =  ( ( S `
 x )  -h  ( g `  x
) ) )
65mpteq2dv 4123 . . 3  |-  ( f  =  S  ->  (
x  e.  ~H  |->  ( ( f `  x
)  -h  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( ( S `  x )  -h  ( g `  x ) ) ) )
7 fveq1 5540 . . . . 5  |-  ( g  =  T  ->  (
g `  x )  =  ( T `  x ) )
87oveq2d 5890 . . . 4  |-  ( g  =  T  ->  (
( S `  x
)  -h  ( g `
 x ) )  =  ( ( S `
 x )  -h  ( T `  x
) ) )
98mpteq2dv 4123 . . 3  |-  ( g  =  T  ->  (
x  e.  ~H  |->  ( ( S `  x
)  -h  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x ) ) ) )
10 df-hodif 22328 . . 3  |-  -op  =  ( f  e.  ( ~H  ^m  ~H ) ,  g  e.  ( ~H  ^m  ~H )  |->  ( x  e.  ~H  |->  ( ( f `  x
)  -h  ( g `
 x ) ) ) )
111mptex 5762 . . 3  |-  ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x ) ) )  e.  _V
126, 9, 10, 11ovmpt2 5999 . 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 1632    e. wcel 1696    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   ~Hchil 21515    -h cmv 21521    -op chod 21536
This theorem is referenced by:  hodval  22338  hosubcli  22365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-hilex 21595
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-hodif 22328
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