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Theorem ho2coi 31804
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 31789 . . 3 (𝑅𝑆): ℋ⟶ ℋ
4 hods.3 . . 3 𝑇: ℋ⟶ ℋ
53, 4hocoi 31787 . 2 (𝐴 ∈ ℋ → (((𝑅𝑆) ∘ 𝑇)‘𝐴) = ((𝑅𝑆)‘(𝑇𝐴)))
64ffvelcdmi 7115 . . 3 (𝐴 ∈ ℋ → (𝑇𝐴) ∈ ℋ)
71, 2hocoi 31787 . . 3 ((𝑇𝐴) ∈ ℋ → ((𝑅𝑆)‘(𝑇𝐴)) = (𝑅‘(𝑆‘(𝑇𝐴))))
86, 7syl 17 . 2 (𝐴 ∈ ℋ → ((𝑅𝑆)‘(𝑇𝐴)) = (𝑅‘(𝑆‘(𝑇𝐴))))
95, 8eqtrd 2774 1 (𝐴 ∈ ℋ → (((𝑅𝑆) ∘ 𝑇)‘𝐴) = (𝑅‘(𝑆‘(𝑇𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2103  ccom 5703  wf 6568  cfv 6572  chba 30942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-fv 6580
This theorem is referenced by:  pj2cocli  32228
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