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Theorem hocofni 23112
Description: Functionality of 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
hocofni  |-  ( S  o.  T )  Fn 
~H

Proof of Theorem hocofni
StepHypRef Expression
1 hoeq.1 . . 3  |-  S : ~H
--> ~H
2 hoeq.2 . . 3  |-  T : ~H
--> ~H
31, 2hocofi 23111 . 2  |-  ( S  o.  T ) : ~H --> ~H
4 ffn 5525 . 2  |-  ( ( S  o.  T ) : ~H --> ~H  ->  ( S  o.  T )  Fn  ~H )
53, 4ax-mp 8 1  |-  ( S  o.  T )  Fn 
~H
Colors of variables: wff set class
Syntax hints:    o. ccom 4816    Fn wfn 5383   -->wf 5384   ~Hchil 22264
This theorem is referenced by:  pjcofni  23507  pjinvari  23536  pj3si  23552
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 2362  ax-sep 4265  ax-nul 4273  ax-pr 4338
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 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-ral 2648  df-rex 2649  df-rab 2652  df-v 2895  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-sn 3757  df-pr 3758  df-op 3760  df-br 4148  df-opab 4202  df-id 4433  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-fun 5390  df-fn 5391  df-f 5392
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