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Theorem hocofi 22346
Description: Mapping of composition of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
Hypotheses
Ref Expression
hoeq.1  |-  S : ~H
--> ~H
hoeq.2  |-  T : ~H
--> ~H
Assertion
Ref Expression
hocofi  |-  ( S  o.  T ) : ~H --> ~H

Proof of Theorem hocofi
StepHypRef Expression
1 hoeq.1 . 2  |-  S : ~H
--> ~H
2 hoeq.2 . 2  |-  T : ~H
--> ~H
3 fco 5398 . 2  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  o.  T
) : ~H --> ~H )
41, 2, 3mp2an 653 1  |-  ( S  o.  T ) : ~H --> ~H
Colors of variables: wff set class
Syntax hints:    o. ccom 4693   -->wf 5251   ~Hchil 21499
This theorem is referenced by:  hocofni  22347  hocadddiri  22359  hocsubdiri  22360  ho2coi  22361  ho0coi  22368  hoid1i  22369  hoid1ri  22370  hoddii  22569  lnopcoi  22583  bdopcoi  22678  adjcoi  22680  nmopcoadji  22681  unierri  22684  pjsdii  22735  pjddii  22736  pjsdi2i  22737  pjss1coi  22743  pjss2coi  22744  pjorthcoi  22749  pjinvari  22771  pjclem1  22775  pjclem4  22779  pjadj2coi  22784  pj3lem1  22786  pj3si  22787  pj3cor1i  22789
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259
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