Theorem ho2co 5611

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 5600	. . 3 |- (R o. S):H~-->H~
4		hods.3	. . 3 |- T:H~-->H~
5	3, 4	hoco 5598	. 2 |- (A e. H~ -> (((R o. S) o. T)` A) = ((R o. S)` (T` A)))
6	4	hocl 5594	. . 3 |- (A e. H~ -> (T` A) e. H~)
7	1, 2	hoco 5598	. . 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 1128	1 |- (A e. H~ -> (((R o. S) o. T)` A) = (R` (S` (T` A))))

Syntax hints:   -> wi 2   = wceq 1091   e. wcel 1092   o. ccom 2414  -->wf 2418  ` cfv 2422  H~chil 4958

This theorem is referenced by:  pj2cocl 5657

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

metamath.org