HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hodval Unicode version

Theorem hodval 22756
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 22751 . . . 4  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  -op  T
)  =  ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x ) ) ) )
21fveq1d 5634 . . 3  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( ( S  -op  T ) `  A )  =  ( ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x ) ) ) `
 A ) )
3 fveq2 5632 . . . . 5  |-  ( x  =  A  ->  ( S `  x )  =  ( S `  A ) )
4 fveq2 5632 . . . . 5  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
53, 4oveq12d 5999 . . . 4  |-  ( x  =  A  ->  (
( S `  x
)  -h  ( T `
 x ) )  =  ( ( S `
 A )  -h  ( T `  A
) ) )
6 eqid 2366 . . . 4  |-  ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x ) ) )  =  ( x  e. 
~H  |->  ( ( S `
 x )  -h  ( T `  x
) ) )
7 ovex 6006 . . . 4  |-  ( ( S `  A )  -h  ( T `  A ) )  e. 
_V
85, 6, 7fvmpt 5709 . . 3  |-  ( A  e.  ~H  ->  (
( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x )
) ) `  A
)  =  ( ( S `  A )  -h  ( T `  A ) ) )
92, 8sylan9eq 2418 . 2  |-  ( ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  /\  A  e. 
~H )  ->  (
( S  -op  T
) `  A )  =  ( ( S `
 A )  -h  ( T `  A
) ) )
1093impa 1147 1  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  A  e.  ~H )  ->  ( ( S  -op  T ) `  A )  =  ( ( S `
 A )  -h  ( T `  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    e. cmpt 4179   -->wf 5354   ` cfv 5358  (class class class)co 5981   ~Hchil 21933    -h cmv 21939    -op chod 21954
This theorem is referenced by:  hodcl  22761  hodsi  22789  hocsubdiri  22794  honegsubi  22810  hoddii  23003  lnopeqi  23022  leop2  23138  pjddii  23170  pjssposi  23186  pjssdif2i  23188
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-hilex 22013
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-map 6917  df-hodif 22746
  Copyright terms: Public domain W3C validator