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Theorem hoeq1 22406
Description: A condition implying that two Hilbert space operators are equal. Lemma 3.2(S9) of [Beran] p. 95. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hoeq1  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( A. x  e. 
~H  A. y  e.  ~H  ( ( S `  x )  .ih  y
)  =  ( ( T `  x ) 
.ih  y )  <->  S  =  T ) )
Distinct variable groups:    x, y, S    x, T, y

Proof of Theorem hoeq1
StepHypRef Expression
1 ffvelrn 5625 . . . . 5  |-  ( ( S : ~H --> ~H  /\  x  e.  ~H )  ->  ( S `  x
)  e.  ~H )
2 ffvelrn 5625 . . . . 5  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
3 hial2eq 21681 . . . . 5  |-  ( ( ( S `  x
)  e.  ~H  /\  ( T `  x )  e.  ~H )  -> 
( A. y  e. 
~H  ( ( S `
 x )  .ih  y )  =  ( ( T `  x
)  .ih  y )  <->  ( S `  x )  =  ( T `  x ) ) )
41, 2, 3syl2an 463 . . . 4  |-  ( ( ( S : ~H --> ~H  /\  x  e.  ~H )  /\  ( T : ~H
--> ~H  /\  x  e. 
~H ) )  -> 
( A. y  e. 
~H  ( ( S `
 x )  .ih  y )  =  ( ( T `  x
)  .ih  y )  <->  ( S `  x )  =  ( T `  x ) ) )
54anandirs 804 . . 3  |-  ( ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( A. y  e.  ~H  ( ( S `  x )  .ih  y
)  =  ( ( T `  x ) 
.ih  y )  <->  ( S `  x )  =  ( T `  x ) ) )
65ralbidva 2560 . 2  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( A. x  e. 
~H  A. y  e.  ~H  ( ( S `  x )  .ih  y
)  =  ( ( T `  x ) 
.ih  y )  <->  A. x  e.  ~H  ( S `  x )  =  ( T `  x ) ) )
7 ffn 5355 . . 3  |-  ( S : ~H --> ~H  ->  S  Fn  ~H )
8 ffn 5355 . . 3  |-  ( T : ~H --> ~H  ->  T  Fn  ~H )
9 eqfnfv 5584 . . 3  |-  ( ( S  Fn  ~H  /\  T  Fn  ~H )  ->  ( S  =  T  <->  A. x  e.  ~H  ( S `  x )  =  ( T `  x ) ) )
107, 8, 9syl2an 463 . 2  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  =  T  <->  A. x  e.  ~H  ( S `  x )  =  ( T `  x ) ) )
116, 10bitr4d 247 1  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( A. x  e. 
~H  A. y  e.  ~H  ( ( S `  x )  .ih  y
)  =  ( ( T `  x ) 
.ih  y )  <->  S  =  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1685   A.wral 2544    Fn wfn 5216   -->wf 5217   ` cfv 5221  (class class class)co 5820   ~Hchil 21495    .ih csp 21498
This theorem is referenced by:  hoeq2  22407  adjmo  22408  adjadj  22512
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-hfvadd 21576  ax-hvcom 21577  ax-hvass 21578  ax-hv0cl 21579  ax-hvaddid 21580  ax-hfvmul 21581  ax-hvmulid 21582  ax-hvdistr2 21585  ax-hvmul0 21586  ax-hfi 21654  ax-his2 21658  ax-his3 21659  ax-his4 21660
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-iota 6253  df-riota 6300  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-ltxr 8868  df-sub 9035  df-neg 9036  df-hvsub 21547
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