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Theorem hodval 23250
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 23245 . . . 4  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  -op  T
)  =  ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x ) ) ) )
21fveq1d 5733 . . 3  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( ( S  -op  T ) `  A )  =  ( ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x ) ) ) `
 A ) )
3 fveq2 5731 . . . . 5  |-  ( x  =  A  ->  ( S `  x )  =  ( S `  A ) )
4 fveq2 5731 . . . . 5  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
53, 4oveq12d 6102 . . . 4  |-  ( x  =  A  ->  (
( S `  x
)  -h  ( T `
 x ) )  =  ( ( S `
 A )  -h  ( T `  A
) ) )
6 eqid 2438 . . . 4  |-  ( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x ) ) )  =  ( x  e. 
~H  |->  ( ( S `
 x )  -h  ( T `  x
) ) )
7 ovex 6109 . . . 4  |-  ( ( S `  A )  -h  ( T `  A ) )  e. 
_V
85, 6, 7fvmpt 5809 . . 3  |-  ( A  e.  ~H  ->  (
( x  e.  ~H  |->  ( ( S `  x )  -h  ( T `  x )
) ) `  A
)  =  ( ( S `  A )  -h  ( T `  A ) ) )
92, 8sylan9eq 2490 . 2  |-  ( ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  /\  A  e. 
~H )  ->  (
( S  -op  T
) `  A )  =  ( ( S `
 A )  -h  ( T `  A
) ) )
1093impa 1149 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726    e. cmpt 4269   -->wf 5453   ` cfv 5457  (class class class)co 6084   ~Hchil 22427    -h cmv 22433    -op chod 22448
This theorem is referenced by:  hodcl  23255  hodsi  23283  hocsubdiri  23288  honegsubi  23304  hoddii  23497  lnopeqi  23516  leop2  23632  pjddii  23664  pjssposi  23680  pjssdif2i  23682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-hilex 22507
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-map 7023  df-hodif 23240
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