Theorem hocofni 32166

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 32165	. 2 ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ
4		ffn 6709	. 2 ⊢ ((𝑆 ∘ 𝑇): ℋ⟶ ℋ → (𝑆 ∘ 𝑇) Fn ℋ)
5	3, 4	ax-mp 5	1 ⊢ (𝑆 ∘ 𝑇) Fn ℋ

This proof depends on syntax axioms:   ∘ ccom 5667   Fn wfn 6535  ⟶wf 6536   ℋchba 31318

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  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-fun 6542  df-fn 6543  df-f 6544

This theorem is used by:  pjcofni  32561  pjinvari  32590  pj3si  32606