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Theorem hocoi 29322
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 6588 . 2 ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑆𝑇)‘𝐴) = (𝑆‘(𝑇𝐴)))
31, 2mpan 677 1 (𝐴 ∈ ℋ → ((𝑆𝑇)‘𝐴) = (𝑆‘(𝑇𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wcel 2050  ccom 5411  wf 6184  cfv 6188  chba 28475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-fv 6196
This theorem is referenced by:  hococli  29323  hocadddiri  29337  hocsubdiri  29338  ho2coi  29339  ho0coi  29346  hoid1i  29347  hoid1ri  29348  hoddii  29547  lnopcoi  29561  lnopco0i  29562  nmopcoi  29653  adjcoi  29658  nmopcoadji  29659  hmopidmchi  29709  hmopidmpji  29710  pjsdii  29713  pjddii  29714  pjcoi  29716  pjcohocli  29761  pjadj2coi  29762  pj3lem1  29764
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