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Theorem hocoi 28007
Description: 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
hocoi (𝐴 ∈ ℋ → ((𝑆𝑇)‘𝐴) = (𝑆‘(𝑇𝐴)))

Proof of Theorem hocoi
StepHypRef Expression
1 hoeq.2 . 2 𝑇: ℋ⟶ ℋ
2 fvco3 6185 . 2 ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑆𝑇)‘𝐴) = (𝑆‘(𝑇𝐴)))
31, 2mpan 702 1 (𝐴 ∈ ℋ → ((𝑆𝑇)‘𝐴) = (𝑆‘(𝑇𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  ccom 5042  wf 5800  cfv 5804  chil 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-8 1979  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-pow 4769  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-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  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-uni 4373  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-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812
This theorem is referenced by:  hococli  28008  hocadddiri  28022  hocsubdiri  28023  ho2coi  28024  ho0coi  28031  hoid1i  28032  hoid1ri  28033  hoddii  28232  lnopcoi  28246  lnopco0i  28247  nmopcoi  28338  adjcoi  28343  nmopcoadji  28344  hmopidmchi  28394  hmopidmpji  28395  pjsdii  28398  pjddii  28399  pjcoi  28401  pjcohocli  28446  pjadj2coi  28447  pj3lem1  28449
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