Theorem hocofn 9688

Description: Functionality of composition of Hilbert space operators.

Hypotheses

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

Assertion

Ref	Expression
hocofn	|- (S o. T) Fn H~

Proof of Theorem hocofn

Step	Hyp	Ref	Expression
1		hoeq.1	. . 3 |- S:H~-->H~
2		hoeq.2	. . 3 |- T:H~-->H~
3	1, 2	hocof 9687	. 2 |- (S o. T):H~-->H~
4		ffn 3633	. 2 |- ((S o. T):H~-->H~ -> (S o. T) Fn H~)
5	3, 4	ax-mp 7	1 |- (S o. T) Fn H~

Syntax hints:   o. ccom 3180   Fn wfn 3183  -->wf 3184  H~chil 8783

This theorem is referenced by:  pjcofn 10085  pjinvar 10114  pj3s 10130

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-fun 3198  df-fn 3199  df-f 3200

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