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Theorem hocofi 26347
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 5732 . 2  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  o.  T
) : ~H --> ~H )
41, 2, 3mp2an 672 1  |-  ( S  o.  T ) : ~H --> ~H
Colors of variables: wff setvar class
Syntax hints:    o. ccom 4996   -->wf 5575   ~Hchil 25498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-fun 5581  df-fn 5582  df-f 5583
This theorem is referenced by:  hocofni  26348  hocadddiri  26360  hocsubdiri  26361  ho2coi  26362  ho0coi  26369  hoid1i  26370  hoid1ri  26371  hoddii  26570  lnopcoi  26584  bdopcoi  26679  adjcoi  26681  nmopcoadji  26682  unierri  26685  pjsdii  26736  pjddii  26737  pjsdi2i  26738  pjss1coi  26744  pjss2coi  26745  pjorthcoi  26750  pjinvari  26772  pjclem1  26776  pjclem4  26780  pjadj2coi  26785  pj3lem1  26787  pj3si  26788  pj3cor1i  26790
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