Theorem hococl 5599

Description: Closure of composition of Hilbert space operators.

Hypotheses

Ref	Expression
hoeq.1	|- S:H~-->H~
hoeq.2	|- T:H~-->H~

Assertion

Ref	Expression
hococl	|- (A e. H~ -> ((S o. T)` A) e. H~)

Proof of Theorem hococl

Step	Hyp	Ref	Expression
1		hoeq.1	. . 3 |- S:H~-->H~
2		hoeq.2	. . 3 |- T:H~-->H~
3	1, 2	hoco 5598	. 2 |- (A e. H~ -> ((S o. T)` A) = (S` (T` A)))
4	2	hocl 5594	. . 3 |- (A e. H~ -> (T` A) e. H~)
5	1	hocl 5594	. . 3 |- ((T` A) e. H~ -> (S` (T` A)) e. H~)
6	4, 5	syl 12	. 2 |- (A e. H~ -> (S` (T` A)) e. H~)
7	3, 6	eqeltrd 1163	1 |- (A e. H~ -> ((S o. T)` A) e. H~)

Syntax hints:   -> wi 2   e. wcel 1092   o. ccom 2414  -->wf 2418  ` cfv 2422  H~chil 4958

This theorem is referenced by:  pjcohcl 5630  pj3s 5659  pj3cor1 5661

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-fv 2438

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