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Theorem ho2coi 22416
Description: Double composition of Hilbert space operators. (Contributed by NM, 1-Dec-2000.) (New usage is discouraged.)
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
StepHypRef Expression
1 hods.1 . . . 4  |-  R : ~H
--> ~H
2 hods.2 . . . 4  |-  S : ~H
--> ~H
31, 2hocofi 22401 . . 3  |-  ( R  o.  S ) : ~H --> ~H
4 hods.3 . . 3  |-  T : ~H
--> ~H
53, 4hocoi 22399 . 2  |-  ( A  e.  ~H  ->  (
( ( R  o.  S )  o.  T
) `  A )  =  ( ( R  o.  S ) `  ( T `  A ) ) )
64ffvelrni 5702 . . 3  |-  ( A  e.  ~H  ->  ( T `  A )  e.  ~H )
71, 2hocoi 22399 . . 3  |-  ( ( T `  A )  e.  ~H  ->  (
( R  o.  S
) `  ( T `  A ) )  =  ( R `  ( S `  ( T `  A ) ) ) )
86, 7syl 15 . 2  |-  ( A  e.  ~H  ->  (
( R  o.  S
) `  ( T `  A ) )  =  ( R `  ( S `  ( T `  A ) ) ) )
95, 8eqtrd 2348 1  |-  ( A  e.  ~H  ->  (
( ( R  o.  S )  o.  T
) `  A )  =  ( R `  ( S `  ( T `
 A ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633    e. wcel 1701    o. ccom 4730   -->wf 5288   ` cfv 5292   ~Hchil 21554
This theorem is referenced by:  pj2cocli  22840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fv 5300
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