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Theorem hocofi 23117
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 5540 . 2  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  o.  T
) : ~H --> ~H )
41, 2, 3mp2an 654 1  |-  ( S  o.  T ) : ~H --> ~H
Colors of variables: wff set class
Syntax hints:    o. ccom 4822   -->wf 5390   ~Hchil 22270
This theorem is referenced by:  hocofni  23118  hocadddiri  23130  hocsubdiri  23131  ho2coi  23132  ho0coi  23139  hoid1i  23140  hoid1ri  23141  hoddii  23340  lnopcoi  23354  bdopcoi  23449  adjcoi  23451  nmopcoadji  23452  unierri  23455  pjsdii  23506  pjddii  23507  pjsdi2i  23508  pjss1coi  23514  pjss2coi  23515  pjorthcoi  23520  pjinvari  23542  pjclem1  23546  pjclem4  23550  pjadj2coi  23555  pj3lem1  23557  pj3si  23558  pj3cor1i  23560
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-fun 5396  df-fn 5397  df-f 5398
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