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Theorem ho2coi 22361
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 22346 . . 3  |-  ( R  o.  S ) : ~H --> ~H
4 hods.3 . . 3  |-  T : ~H
--> ~H
53, 4hocoi 22344 . 2  |-  ( A  e.  ~H  ->  (
( ( R  o.  S )  o.  T
) `  A )  =  ( ( R  o.  S ) `  ( T `  A ) ) )
64ffvelrni 5664 . . 3  |-  ( A  e.  ~H  ->  ( T `  A )  e.  ~H )
71, 2hocoi 22344 . . 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 2315 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 1623    e. wcel 1684    o. ccom 4693   -->wf 5251   ` cfv 5255   ~Hchil 21499
This theorem is referenced by:  pj2cocli  22785
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263
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