Theorem ho2co 9702

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
ho2co	|- (A e. H~ -> (((R o. S) o. T)` A) = (R` (S` (T` A))))

Proof of Theorem ho2co

Step	Hyp	Ref	Expression
1		hods.1	. . . 4 |- R:H~-->H~
2		hods.2	. . . 4 |- S:H~-->H~
3	1, 2	hocof 9687	. . 3 |- (R o. S):H~-->H~
4		hods.3	. . 3 |- T:H~-->H~
5	3, 4	hoco 9685	. 2 |- (A e. H~ -> (((R o. S) o. T)` A) = ((R o. S)` (T` A)))
6	4	ffvelrni 3821	. . 3 |- (A e. H~ -> (T` A) e. H~)
7	1, 2	hoco 9685	. . 3 |- ((T` A) e. H~ -> ((R o. S)` (T` A)) = (R` (S` (T` A))))
8	6, 7	syl 10	. 2 |- (A e. H~ -> ((R o. S)` (T` A)) = (R` (S` (T` A))))
9	5, 8	eqtrd 1510	1 |- (A e. H~ -> (((R o. S) o. T)` A) = (R` (S` (T` A))))

Syntax hints:   -> wi 3   = wceq 958   e. wcel 960   o. ccom 3180  -->wf 3184  ` cfv 3188  H~chil 8783

This theorem is referenced by:  pj2cocl 10128

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  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  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-ral 1652  df-rex 1653  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-uni 2508  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-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204

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