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

Theorem hoeqi 22455
Description: Equality 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
hoeqi  |-  ( A. x  e.  ~H  ( S `  x )  =  ( T `  x )  <->  S  =  T )
Distinct variable groups:    x, S    x, T

Proof of Theorem hoeqi
StepHypRef Expression
1 hoeq.1 . 2  |-  S : ~H
--> ~H
2 hoeq.2 . 2  |-  T : ~H
--> ~H
3 hoeq 22454 . 2  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( A. x  e. 
~H  ( S `  x )  =  ( T `  x )  <-> 
S  =  T ) )
41, 2, 3mp2an 653 1  |-  ( A. x  e.  ~H  ( S `  x )  =  ( T `  x )  <->  S  =  T )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1642   A.wral 2619   -->wf 5333   ` cfv 5337   ~Hchil 21613
This theorem is referenced by:  hoaddcomi  22466  hodsi  22469  hoaddassi  22470  hocadddiri  22473  hocsubdiri  22474  hoaddid1i  22480  ho0coi  22482  hoid1i  22483  hoid1ri  22484  honegsubi  22490  hoddii  22683  pjsdii  22849  pjddii  22850  pjss1coi  22857  pjss2coi  22858  pjorthcoi  22863  pjscji  22864  pjtoi  22873  pjclem4  22893  pj3si  22901  pj3cor1i  22903
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-fv 5345
  Copyright terms: Public domain W3C validator