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Theorem ho2coi 28970
 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 28955 . . 3 (𝑅𝑆): ℋ⟶ ℋ
4 hods.3 . . 3 𝑇: ℋ⟶ ℋ
53, 4hocoi 28953 . 2 (𝐴 ∈ ℋ → (((𝑅𝑆) ∘ 𝑇)‘𝐴) = ((𝑅𝑆)‘(𝑇𝐴)))
64ffvelrni 6522 . . 3 (𝐴 ∈ ℋ → (𝑇𝐴) ∈ ℋ)
71, 2hocoi 28953 . . 3 ((𝑇𝐴) ∈ ℋ → ((𝑅𝑆)‘(𝑇𝐴)) = (𝑅‘(𝑆‘(𝑇𝐴))))
86, 7syl 17 . 2 (𝐴 ∈ ℋ → ((𝑅𝑆)‘(𝑇𝐴)) = (𝑅‘(𝑆‘(𝑇𝐴))))
95, 8eqtrd 2794 1 (𝐴 ∈ ℋ → (((𝑅𝑆) ∘ 𝑇)‘𝐴) = (𝑅‘(𝑆‘(𝑇𝐴))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139   ∘ ccom 5270  ⟶wf 6045  ‘cfv 6049   ℋchil 28106 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057 This theorem is referenced by:  pj2cocli  29394
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