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Theorem hocofni 28010
 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 (𝑆𝑇) Fn ℋ

Proof of Theorem hocofni
StepHypRef Expression
1 hoeq.1 . . 3 𝑆: ℋ⟶ ℋ
2 hoeq.2 . . 3 𝑇: ℋ⟶ ℋ
31, 2hocofi 28009 . 2 (𝑆𝑇): ℋ⟶ ℋ
4 ffn 5958 . 2 ((𝑆𝑇): ℋ⟶ ℋ → (𝑆𝑇) Fn ℋ)
53, 4ax-mp 5 1 (𝑆𝑇) Fn ℋ
 Colors of variables: wff setvar class Syntax hints:   ∘ ccom 5042   Fn wfn 5799  ⟶wf 5800   ℋchil 27160 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808 This theorem is referenced by:  pjcofni  28405  pjinvari  28434  pj3si  28450
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