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Theorem hodval 25267
Description: Value of the difference of two Hilbert space operators. (Contributed by NM, 10-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
hodval  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  A  e.  ~H )  ->  ( ( S  -op  T ) `  A )  =  ( ( S `
 A )  -h  ( T `  A
) ) )

Proof of Theorem hodval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 hodmval 25262 . . . 4  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  -op  T
)  =  ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x ) ) ) )
21fveq1d 5777 . . 3  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( ( S  -op  T ) `  A )  =  ( ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x ) ) ) `
 A ) )
3 fveq2 5775 . . . . 5  |-  ( x  =  A  ->  ( S `  x )  =  ( S `  A ) )
4 fveq2 5775 . . . . 5  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
53, 4oveq12d 6194 . . . 4  |-  ( x  =  A  ->  (
( S `  x
)  -h  ( T `
 x ) )  =  ( ( S `
 A )  -h  ( T `  A
) ) )
6 eqid 2450 . . . 4  |-  ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x ) ) )  =  ( x  e. 
~H  |->  ( ( S `
 x )  -h  ( T `  x
) ) )
7 ovex 6201 . . . 4  |-  ( ( S `  A )  -h  ( T `  A ) )  e. 
_V
85, 6, 7fvmpt 5859 . . 3  |-  ( A  e.  ~H  ->  (
( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x )
) ) `  A
)  =  ( ( S `  A )  -h  ( T `  A ) ) )
92, 8sylan9eq 2510 . 2  |-  ( ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  /\  A  e. 
~H )  ->  (
( S  -op  T
) `  A )  =  ( ( S `
 A )  -h  ( T `  A
) ) )
1093impa 1183 1  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  A  e.  ~H )  ->  ( ( S  -op  T ) `  A )  =  ( ( S `
 A )  -h  ( T `  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757    |-> cmpt 4434   -->wf 5498   ` cfv 5502  (class class class)co 6176   ~Hchil 24442    -h cmv 24448    -op chod 24463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-hilex 24522
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-map 7302  df-hodif 25257
This theorem is referenced by:  hodcl  25272  hodsi  25300  hocsubdiri  25305  honegsubi  25321  hoddii  25514  lnopeqi  25533  leop2  25649  pjddii  25681  pjssposi  25697  pjssdif2i  25699
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