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Theorem hoeq1 23175
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 5801 . . . . 5  |-  ( ( S : ~H --> ~H  /\  x  e.  ~H )  ->  ( S `  x
)  e.  ~H )
2 ffvelrn 5801 . . . . 5  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
3 hial2eq 22450 . . . . 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 2659 . 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 5525 . . 3  |-  ( S : ~H --> ~H  ->  S  Fn  ~H )
8 ffn 5525 . . 3  |-  ( T : ~H --> ~H  ->  T  Fn  ~H )
9 eqfnfv 5760 . . 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 2643    Fn wfn 5383   -->wf 5384   ` cfv 5388  (class class class)co 6014   ~Hchil 22264    .ih csp 22267
This theorem is referenced by:  hoeq2  23176  adjmo  23177  adjadj  23281
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 2362  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635  ax-resscn 8974  ax-1cn 8975  ax-icn 8976  ax-addcl 8977  ax-addrcl 8978  ax-mulcl 8979  ax-mulrcl 8980  ax-mulcom 8981  ax-addass 8982  ax-mulass 8983  ax-distr 8984  ax-i2m1 8985  ax-1ne0 8986  ax-1rid 8987  ax-rnegex 8988  ax-rrecex 8989  ax-cnre 8990  ax-pre-lttri 8991  ax-pre-lttrn 8992  ax-pre-ltadd 8993  ax-hfvadd 22345  ax-hvcom 22346  ax-hvass 22347  ax-hv0cl 22348  ax-hvaddid 22349  ax-hfvmul 22350  ax-hvmulid 22351  ax-hvdistr2 22354  ax-hvmul0 22355  ax-hfi 22423  ax-his2 22427  ax-his3 22428  ax-his4 22429
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 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-nel 2547  df-ral 2648  df-rex 2649  df-reu 2650  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-op 3760  df-uni 3952  df-iun 4031  df-br 4148  df-opab 4202  df-mpt 4203  df-id 4433  df-po 4438  df-so 4439  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-ov 6017  df-oprab 6018  df-mpt2 6019  df-riota 6479  df-er 6835  df-en 7040  df-dom 7041  df-sdom 7042  df-pnf 9049  df-mnf 9050  df-ltxr 9052  df-sub 9219  df-neg 9220  df-hvsub 22316
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