HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hococli Unicode version

Theorem hococli 22361
Description: Closure of composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
Hypotheses
Ref Expression
hoeq.1  |-  S : ~H
--> ~H
hoeq.2  |-  T : ~H
--> ~H
Assertion
Ref Expression
hococli  |-  ( A  e.  ~H  ->  (
( S  o.  T
) `  A )  e.  ~H )

Proof of Theorem hococli
StepHypRef Expression
1 hoeq.1 . . 3  |-  S : ~H
--> ~H
2 hoeq.2 . . 3  |-  T : ~H
--> ~H
31, 2hocoi 22360 . 2  |-  ( A  e.  ~H  ->  (
( S  o.  T
) `  A )  =  ( S `  ( T `  A ) ) )
42ffvelrni 5680 . . 3  |-  ( A  e.  ~H  ->  ( T `  A )  e.  ~H )
51ffvelrni 5680 . . 3  |-  ( ( T `  A )  e.  ~H  ->  ( S `  ( T `  A ) )  e. 
~H )
64, 5syl 15 . 2  |-  ( A  e.  ~H  ->  ( S `  ( T `  A ) )  e. 
~H )
73, 6eqeltrd 2370 1  |-  ( A  e.  ~H  ->  (
( S  o.  T
) `  A )  e.  ~H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696    o. ccom 4709   -->wf 5267   ` cfv 5271   ~Hchil 21515
This theorem is referenced by:  nmopcoadji  22697  pjcohcli  22756  pj3si  22803  pj3cor1i  22805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279
  Copyright terms: Public domain W3C validator