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Theorem hocoi 25168
Description: Composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
Hypotheses
Ref Expression
hoeq.1  |-  S : ~H
--> ~H
hoeq.2  |-  T : ~H
--> ~H
Assertion
Ref Expression
hocoi  |-  ( A  e.  ~H  ->  (
( S  o.  T
) `  A )  =  ( S `  ( T `  A ) ) )

Proof of Theorem hocoi
StepHypRef Expression
1 hoeq.2 . 2  |-  T : ~H
--> ~H
2 fvco3 5768 . 2  |-  ( ( T : ~H --> ~H  /\  A  e.  ~H )  ->  ( ( S  o.  T ) `  A
)  =  ( S `
 ( T `  A ) ) )
31, 2mpan 670 1  |-  ( A  e.  ~H  ->  (
( S  o.  T
) `  A )  =  ( S `  ( T `  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756    o. ccom 4844   -->wf 5414   ` cfv 5418   ~Hchil 24321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fv 5426
This theorem is referenced by:  hococli  25169  hocadddiri  25183  hocsubdiri  25184  ho2coi  25185  ho0coi  25192  hoid1i  25193  hoid1ri  25194  hoddii  25393  lnopcoi  25407  lnopco0i  25408  nmopcoi  25499  adjcoi  25504  nmopcoadji  25505  hmopidmchi  25555  hmopidmpji  25556  pjsdii  25559  pjddii  25560  pjcoi  25562  pjcohocli  25607  pjadj2coi  25608  pj3lem1  25610
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