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Theorem hocoi 28928
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 6433 . 2 ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑆𝑇)‘𝐴) = (𝑆‘(𝑇𝐴)))
31, 2mpan 708 1 (𝐴 ∈ ℋ → ((𝑆𝑇)‘𝐴) = (𝑆‘(𝑇𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1628  wcel 2135  ccom 5266  wf 6041  cfv 6045  chil 28081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-sbc 3573  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-br 4801  df-opab 4861  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-fv 6053
This theorem is referenced by:  hococli  28929  hocadddiri  28943  hocsubdiri  28944  ho2coi  28945  ho0coi  28952  hoid1i  28953  hoid1ri  28954  hoddii  29153  lnopcoi  29167  lnopco0i  29168  nmopcoi  29259  adjcoi  29264  nmopcoadji  29265  hmopidmchi  29315  hmopidmpji  29316  pjsdii  29319  pjddii  29320  pjcoi  29322  pjcohocli  29367  pjadj2coi  29368  pj3lem1  29370
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