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Theorem hocofi 22362
Description: Mapping of composition of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
Hypotheses
Ref Expression
hoeq.1  |-  S : ~H
--> ~H
hoeq.2  |-  T : ~H
--> ~H
Assertion
Ref Expression
hocofi  |-  ( S  o.  T ) : ~H --> ~H

Proof of Theorem hocofi
StepHypRef Expression
1 hoeq.1 . 2  |-  S : ~H
--> ~H
2 hoeq.2 . 2  |-  T : ~H
--> ~H
3 fco 5414 . 2  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  o.  T
) : ~H --> ~H )
41, 2, 3mp2an 653 1  |-  ( S  o.  T ) : ~H --> ~H
Colors of variables: wff set class
Syntax hints:    o. ccom 4709   -->wf 5267   ~Hchil 21515
This theorem is referenced by:  hocofni  22363  hocadddiri  22375  hocsubdiri  22376  ho2coi  22377  ho0coi  22384  hoid1i  22385  hoid1ri  22386  hoddii  22585  lnopcoi  22599  bdopcoi  22694  adjcoi  22696  nmopcoadji  22697  unierri  22700  pjsdii  22751  pjddii  22752  pjsdi2i  22753  pjss1coi  22759  pjss2coi  22760  pjorthcoi  22765  pjinvari  22787  pjclem1  22791  pjclem4  22795  pjadj2coi  22800  pj3lem1  22802  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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  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-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-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-fun 5273  df-fn 5274  df-f 5275
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