HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  ho2coi Structured version   Visualization version   GIF version

Theorem ho2coi 30765
Description: Double composition of Hilbert space operators. (Contributed by NM, 1-Dec-2000.) (New usage is discouraged.)
Hypotheses
Ref Expression
hods.1 𝑅: ℋ⟶ ℋ
hods.2 𝑆: ℋ⟶ ℋ
hods.3 𝑇: ℋ⟶ ℋ
Assertion
Ref Expression
ho2coi (𝐴 ∈ ℋ → (((𝑅𝑆) ∘ 𝑇)‘𝐴) = (𝑅‘(𝑆‘(𝑇𝐴))))

Proof of Theorem ho2coi
StepHypRef Expression
1 hods.1 . . . 4 𝑅: ℋ⟶ ℋ
2 hods.2 . . . 4 𝑆: ℋ⟶ ℋ
31, 2hocofi 30750 . . 3 (𝑅𝑆): ℋ⟶ ℋ
4 hods.3 . . 3 𝑇: ℋ⟶ ℋ
53, 4hocoi 30748 . 2 (𝐴 ∈ ℋ → (((𝑅𝑆) ∘ 𝑇)‘𝐴) = ((𝑅𝑆)‘(𝑇𝐴)))
64ffvelcdmi 7035 . . 3 (𝐴 ∈ ℋ → (𝑇𝐴) ∈ ℋ)
71, 2hocoi 30748 . . 3 ((𝑇𝐴) ∈ ℋ → ((𝑅𝑆)‘(𝑇𝐴)) = (𝑅‘(𝑆‘(𝑇𝐴))))
86, 7syl 17 . 2 (𝐴 ∈ ℋ → ((𝑅𝑆)‘(𝑇𝐴)) = (𝑅‘(𝑆‘(𝑇𝐴))))
95, 8eqtrd 2773 1 (𝐴 ∈ ℋ → (((𝑅𝑆) ∘ 𝑇)‘𝐴) = (𝑅‘(𝑆‘(𝑇𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  ccom 5638  wf 6493  cfv 6497  chba 29903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505
This theorem is referenced by:  pj2cocli  31189
  Copyright terms: Public domain W3C validator