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Theorem hocoi 23228
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 5767 . 2  |-  ( ( T : ~H --> ~H  /\  A  e.  ~H )  ->  ( ( S  o.  T ) `  A
)  =  ( S `
 ( T `  A ) ) )
31, 2mpan 652 1  |-  ( A  e.  ~H  ->  (
( S  o.  T
) `  A )  =  ( S `  ( T `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    o. ccom 4849   -->wf 5417   ` cfv 5421   ~Hchil 22383
This theorem is referenced by:  hococli  23229  hocadddiri  23243  hocsubdiri  23244  ho2coi  23245  ho0coi  23252  hoid1i  23253  hoid1ri  23254  hoddii  23453  lnopcoi  23467  lnopco0i  23468  nmopcoi  23559  adjcoi  23564  nmopcoadji  23565  hmopidmchi  23615  hmopidmpji  23616  pjsdii  23619  pjddii  23620  pjcoi  23622  pjcohocli  23667  pjadj2coi  23668  pj3lem1  23670
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429
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