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Theorem hoeq1 23181
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 5807 . . . . 5  |-  ( ( S : ~H --> ~H  /\  x  e.  ~H )  ->  ( S `  x
)  e.  ~H )
2 ffvelrn 5807 . . . . 5  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
3 hial2eq 22456 . . . . 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 464 . . . 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 805 . . 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 2665 . 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 5531 . . 3  |-  ( S : ~H --> ~H  ->  S  Fn  ~H )
8 ffn 5531 . . 3  |-  ( T : ~H --> ~H  ->  T  Fn  ~H )
9 eqfnfv 5766 . . 3  |-  ( ( S  Fn  ~H  /\  T  Fn  ~H )  ->  ( S  =  T  <->  A. x  e.  ~H  ( S `  x )  =  ( T `  x ) ) )
107, 8, 9syl2an 464 . 2  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  =  T  <->  A. x  e.  ~H  ( S `  x )  =  ( T `  x ) ) )
116, 10bitr4d 248 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020   ~Hchil 22270    .ih csp 22273
This theorem is referenced by:  hoeq2  23182  adjmo  23183  adjadj  23287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-hfvadd 22351  ax-hvcom 22352  ax-hvass 22353  ax-hv0cl 22354  ax-hvaddid 22355  ax-hfvmul 22356  ax-hvmulid 22357  ax-hvdistr2 22360  ax-hvmul0 22361  ax-hfi 22429  ax-his2 22433  ax-his3 22434  ax-his4 22435
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-ltxr 9058  df-sub 9225  df-neg 9226  df-hvsub 22322
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