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Theorem ho2coi 31539
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 31524 . . 3 (𝑅𝑆): ℋ⟶ ℋ
4 hods.3 . . 3 𝑇: ℋ⟶ ℋ
53, 4hocoi 31522 . 2 (𝐴 ∈ ℋ → (((𝑅𝑆) ∘ 𝑇)‘𝐴) = ((𝑅𝑆)‘(𝑇𝐴)))
64ffvelcdmi 7078 . . 3 (𝐴 ∈ ℋ → (𝑇𝐴) ∈ ℋ)
71, 2hocoi 31522 . . 3 ((𝑇𝐴) ∈ ℋ → ((𝑅𝑆)‘(𝑇𝐴)) = (𝑅‘(𝑆‘(𝑇𝐴))))
86, 7syl 17 . 2 (𝐴 ∈ ℋ → ((𝑅𝑆)‘(𝑇𝐴)) = (𝑅‘(𝑆‘(𝑇𝐴))))
95, 8eqtrd 2766 1 (𝐴 ∈ ℋ → (((𝑅𝑆) ∘ 𝑇)‘𝐴) = (𝑅‘(𝑆‘(𝑇𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  ccom 5673  wf 6532  cfv 6536  chba 30677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544
This theorem is referenced by:  pj2cocli  31963
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