Theorem hococli 31775

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 31774	. 2 ⊢ (𝐴 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝐴) = (𝑆‘(𝑇‘𝐴)))
4	2	ffvelcdmi 7097	. . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ)
5	1	ffvelcdmi 7097	. . 3 ⊢ ((𝑇‘𝐴) ∈ ℋ → (𝑆‘(𝑇‘𝐴)) ∈ ℋ)
6	4, 5	syl 17	. 2 ⊢ (𝐴 ∈ ℋ → (𝑆‘(𝑇‘𝐴)) ∈ ℋ)
7	3, 6	eqeltrd 2837	1 ⊢ (𝐴 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝐴) ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2104   ∘ ccom 5687  ⟶wf 6554  ‘cfv 6558   ℋchba 30929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5430

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-opab 5212  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-fv 6566

This theorem is referenced by:  nmopcoadji  32111  pjcohcli  32170  pj3si  32217  pj3cor1i  32219