Theorem hocofn 9693

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 9692	. 2 |- (S o. T):H~-->H~
4		ffn 3627	. 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 3174   Fn wfn 3177  -->wf 3178  H~chil 8788

This theorem is referenced by:  pjcofn 10090  pjinvar 10119  pj3s 10135

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-fun 3192  df-fn 3193  df-f 3194

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