Theorem hocofni 31753

Description: Functionality 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
hocofni	⊢ (𝑆 ∘ 𝑇) Fn ℋ

Proof of Theorem hocofni

Step	Hyp	Ref	Expression
1		hoeq.1	. . 3 ⊢ 𝑆: ℋ⟶ ℋ
2		hoeq.2	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
3	1, 2	hocofi 31752	. 2 ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ
4		ffn 6711	. 2 ⊢ ((𝑆 ∘ 𝑇): ℋ⟶ ℋ → (𝑆 ∘ 𝑇) Fn ℋ)
5	3, 4	ax-mp 5	1 ⊢ (𝑆 ∘ 𝑇) Fn ℋ

Syntax hints:   ∘ ccom 5663   Fn wfn 6531  ⟶wf 6532   ℋchba 30905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-fun 6538  df-fn 6539  df-f 6540

This theorem is referenced by:  pjcofni  32148  pjinvari  32177  pj3si  32193