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Theorem hocoi 26809
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 5950 . 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 1395    e. wcel 1819    o. ccom 5012   -->wf 5590   ` cfv 5594   ~Hchil 25962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602
This theorem is referenced by:  hococli  26810  hocadddiri  26824  hocsubdiri  26825  ho2coi  26826  ho0coi  26833  hoid1i  26834  hoid1ri  26835  hoddii  27034  lnopcoi  27048  lnopco0i  27049  nmopcoi  27140  adjcoi  27145  nmopcoadji  27146  hmopidmchi  27196  hmopidmpji  27197  pjsdii  27200  pjddii  27201  pjcoi  27203  pjcohocli  27248  pjadj2coi  27249  pj3lem1  27251
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