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Theorem hodval 26777
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 26772 . . . 4  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  -op  T
)  =  ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x ) ) ) )
21fveq1d 5776 . . 3  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( ( S  -op  T ) `  A )  =  ( ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x ) ) ) `
 A ) )
3 fveq2 5774 . . . . 5  |-  ( x  =  A  ->  ( S `  x )  =  ( S `  A ) )
4 fveq2 5774 . . . . 5  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
53, 4oveq12d 6214 . . . 4  |-  ( x  =  A  ->  (
( S `  x
)  -h  ( T `
 x ) )  =  ( ( S `
 A )  -h  ( T `  A
) ) )
6 eqid 2382 . . . 4  |-  ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x ) ) )  =  ( x  e. 
~H  |->  ( ( S `
 x )  -h  ( T `  x
) ) )
7 ovex 6224 . . . 4  |-  ( ( S `  A )  -h  ( T `  A ) )  e. 
_V
85, 6, 7fvmpt 5857 . . 3  |-  ( A  e.  ~H  ->  (
( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x )
) ) `  A
)  =  ( ( S `  A )  -h  ( T `  A ) ) )
92, 8sylan9eq 2443 . 2  |-  ( ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  /\  A  e. 
~H )  ->  (
( S  -op  T
) `  A )  =  ( ( S `
 A )  -h  ( T `  A
) ) )
1093impa 1189 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826    |-> cmpt 4425   -->wf 5492   ` cfv 5496  (class class class)co 6196   ~Hchil 25953    -h cmv 25959    -op chod 25974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-hilex 26033
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-map 7340  df-hodif 26767
This theorem is referenced by:  hodcl  26782  hodsi  26810  hocsubdiri  26815  honegsubi  26831  hoddii  27024  lnopeqi  27043  leop2  27159  pjddii  27191  pjssposi  27207  pjssdif2i  27209
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