Hilbert Space Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  HSE Home  >  Th. List  >  hococli Structured version   Visualization version   GIF version

Theorem hococli 28008
 Description: Closure 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
hococli (𝐴 ∈ ℋ → ((𝑆𝑇)‘𝐴) ∈ ℋ)

Proof of Theorem hococli
StepHypRef Expression
1 hoeq.1 . . 3 𝑆: ℋ⟶ ℋ
2 hoeq.2 . . 3 𝑇: ℋ⟶ ℋ
31, 2hocoi 28007 . 2 (𝐴 ∈ ℋ → ((𝑆𝑇)‘𝐴) = (𝑆‘(𝑇𝐴)))
42ffvelrni 6266 . . 3 (𝐴 ∈ ℋ → (𝑇𝐴) ∈ ℋ)
51ffvelrni 6266 . . 3 ((𝑇𝐴) ∈ ℋ → (𝑆‘(𝑇𝐴)) ∈ ℋ)
64, 5syl 17 . 2 (𝐴 ∈ ℋ → (𝑆‘(𝑇𝐴)) ∈ ℋ)
73, 6eqeltrd 2688 1 (𝐴 ∈ ℋ → ((𝑆𝑇)‘𝐴) ∈ ℋ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804   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-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:  nmopcoadji  28344  pjcohcli  28403  pj3si  28450  pj3cor1i  28452
 Copyright terms: Public domain W3C validator