Theorem hococli 31740

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 31739	. 2 ⊢ (𝐴 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝐴) = (𝑆‘(𝑇‘𝐴)))
4	2	ffvelcdmi 7016	. . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ)
5	1	ffvelcdmi 7016	. . 3 ⊢ ((𝑇‘𝐴) ∈ ℋ → (𝑆‘(𝑇‘𝐴)) ∈ ℋ)
6	4, 5	syl 17	. 2 ⊢ (𝐴 ∈ ℋ → (𝑆‘(𝑇‘𝐴)) ∈ ℋ)
7	3, 6	eqeltrd 2831	1 ⊢ (𝐴 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝐴) ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2111   ∘ ccom 5620  ⟶wf 6477  ‘cfv 6481   ℋchba 30894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489

This theorem is referenced by:  nmopcoadji  32076  pjcohcli  32135  pj3si  32182  pj3cor1i  32184