Theorem hocofni 32376

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 32375	. 2 ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ
4		ffn 6712	. 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 5654   Fn wfn 6536  ⟶wf 6537   ℋchba 31528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-fun 6543  df-fn 6544  df-f 6545

This theorem is used by:  pjcofni  32771  pjinvari  32800  pj3si  32816