Theorem hococli 32095

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 → ((𝑆 ∘ 𝑇)‘𝐴) ∈ ℋ)

Proof of Theorem hococli

Step	Hyp	Ref	Expression
1		hoeq.1	. . 3 ⊢ 𝑆: ℋ⟶ ℋ
2		hoeq.2	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
3	1, 2	hocoi 32094	. 2 ⊢ (𝐴 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝐴) = (𝑆‘(𝑇‘𝐴)))
4	2	ffvelcdmi 7080	. . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ)
5	1	ffvelcdmi 7080	. . 3 ⊢ ((𝑇‘𝐴) ∈ ℋ → (𝑆‘(𝑇‘𝐴)) ∈ ℋ)
6	4, 5	syl 18	. 2 ⊢ (𝐴 ∈ ℋ → (𝑆‘(𝑇‘𝐴)) ∈ ℋ)
7	3, 6	eqeltrd 2863	1 ⊢ (𝐴 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝐴) ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2143   ∘ ccom 5667  ⟶wf 6534  ‘cfv 6538   ℋchba 31249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546

This theorem is referenced by:  nmopcoadji  32431  pjcohcli  32490  pj3si  32537  pj3cor1i  32539