Theorem ho2coi 11344

Description: Double composition of Hilbert space operators.

Hypotheses

Ref	Expression
hods.1	|- R:~H-->~H
hods.2	|- S:~H-->~H
hods.3	|- T:~H-->~H

Assertion

Ref	Expression
ho2coi	|- (A e. ~H -> (((R o. S) o. T)` A) = (R` (S` (T` A))))

Proof of Theorem ho2coi

Step	Hyp	Ref	Expression
1		hods.1	. . . 4 |- R:~H-->~H
2		hods.2	. . . 4 |- S:~H-->~H
3	1, 2	hocofi 11329	. . 3 |- (R o. S):~H-->~H
4		hods.3	. . 3 |- T:~H-->~H
5	3, 4	hocoi 11327	. 2 |- (A e. ~H -> (((R o. S) o. T)` A) = ((R o. S)` (T` A)))
6	4	ffvelrni 4788	. . 3 |- (A e. ~H -> (T` A) e. ~H)
7	1, 2	hocoi 11327	. . 3 |- ((T` A) e. ~H -> ((R o. S)` (T` A)) = (R` (S` (T` A))))
8	6, 7	syl 12	. 2 |- (A e. ~H -> ((R o. S)` (T` A)) = (R` (S` (T` A))))
9	5, 8	eqtrd 1925	1 |- (A e. ~H -> (((R o. S) o. T)` A) = (R` (S` (T` A))))

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300   o. ccom 3990  -->wf 3994  ` cfv 3998  ~Hchil 10420

This theorem is referenced by:  pj2cocli 11778

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014

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