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Theorem adjeq 23430
Description: A property that determines the adjoint of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
adjeq  |-  ( ( T : ~H --> ~H  /\  S : ~H --> ~H  /\  A. x  e.  ~H  A. y  e.  ~H  (
( T `  x
)  .ih  y )  =  ( x  .ih  ( S `  y ) ) )  ->  ( adjh `  T )  =  S )
Distinct variable groups:    x, y, S    x, T, y

Proof of Theorem adjeq
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funadj 23381 . 2  |-  Fun  adjh
2 df-adjh 23344 . . . . . 6  |-  adjh  =  { <. z ,  w >.  |  ( z : ~H --> ~H  /\  w : ~H --> ~H  /\  A. x  e.  ~H  A. y  e. 
~H  ( ( z `
 x )  .ih  y )  =  ( x  .ih  ( w `
 y ) ) ) }
32eleq2i 2499 . . . . 5  |-  ( <. T ,  S >.  e. 
adjh 
<-> 
<. T ,  S >.  e. 
{ <. z ,  w >.  |  ( z : ~H --> ~H  /\  w : ~H --> ~H  /\  A. x  e.  ~H  A. y  e. 
~H  ( ( z `
 x )  .ih  y )  =  ( x  .ih  ( w `
 y ) ) ) } )
4 ax-hilex 22494 . . . . . . 7  |-  ~H  e.  _V
5 fex 5961 . . . . . . 7  |-  ( ( T : ~H --> ~H  /\  ~H  e.  _V )  ->  T  e.  _V )
64, 5mpan2 653 . . . . . 6  |-  ( T : ~H --> ~H  ->  T  e.  _V )
7 fex 5961 . . . . . . 7  |-  ( ( S : ~H --> ~H  /\  ~H  e.  _V )  ->  S  e.  _V )
84, 7mpan2 653 . . . . . 6  |-  ( S : ~H --> ~H  ->  S  e.  _V )
9 feq1 5568 . . . . . . . 8  |-  ( z  =  T  ->  (
z : ~H --> ~H  <->  T : ~H
--> ~H ) )
10 fveq1 5719 . . . . . . . . . . 11  |-  ( z  =  T  ->  (
z `  x )  =  ( T `  x ) )
1110oveq1d 6088 . . . . . . . . . 10  |-  ( z  =  T  ->  (
( z `  x
)  .ih  y )  =  ( ( T `
 x )  .ih  y ) )
1211eqeq1d 2443 . . . . . . . . 9  |-  ( z  =  T  ->  (
( ( z `  x )  .ih  y
)  =  ( x 
.ih  ( w `  y ) )  <->  ( ( T `  x )  .ih  y )  =  ( x  .ih  ( w `
 y ) ) ) )
13122ralbidv 2739 . . . . . . . 8  |-  ( z  =  T  ->  ( A. x  e.  ~H  A. y  e.  ~H  (
( z `  x
)  .ih  y )  =  ( x  .ih  ( w `  y
) )  <->  A. x  e.  ~H  A. y  e. 
~H  ( ( T `
 x )  .ih  y )  =  ( x  .ih  ( w `
 y ) ) ) )
149, 133anbi13d 1256 . . . . . . 7  |-  ( z  =  T  ->  (
( z : ~H --> ~H  /\  w : ~H --> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( ( z `  x )  .ih  y
)  =  ( x 
.ih  ( w `  y ) ) )  <-> 
( T : ~H --> ~H  /\  w : ~H --> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( ( T `  x )  .ih  y
)  =  ( x 
.ih  ( w `  y ) ) ) ) )
15 feq1 5568 . . . . . . . 8  |-  ( w  =  S  ->  (
w : ~H --> ~H  <->  S : ~H
--> ~H ) )
16 fveq1 5719 . . . . . . . . . . 11  |-  ( w  =  S  ->  (
w `  y )  =  ( S `  y ) )
1716oveq2d 6089 . . . . . . . . . 10  |-  ( w  =  S  ->  (
x  .ih  ( w `  y ) )  =  ( x  .ih  ( S `  y )
) )
1817eqeq2d 2446 . . . . . . . . 9  |-  ( w  =  S  ->  (
( ( T `  x )  .ih  y
)  =  ( x 
.ih  ( w `  y ) )  <->  ( ( T `  x )  .ih  y )  =  ( x  .ih  ( S `
 y ) ) ) )
19182ralbidv 2739 . . . . . . . 8  |-  ( w  =  S  ->  ( A. x  e.  ~H  A. y  e.  ~H  (
( T `  x
)  .ih  y )  =  ( x  .ih  ( w `  y
) )  <->  A. x  e.  ~H  A. y  e. 
~H  ( ( T `
 x )  .ih  y )  =  ( x  .ih  ( S `
 y ) ) ) )
2015, 193anbi23d 1257 . . . . . . 7  |-  ( w  =  S  ->  (
( T : ~H --> ~H  /\  w : ~H --> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( ( T `  x )  .ih  y
)  =  ( x 
.ih  ( w `  y ) ) )  <-> 
( T : ~H --> ~H  /\  S : ~H --> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( ( T `  x )  .ih  y
)  =  ( x 
.ih  ( S `  y ) ) ) ) )
2114, 20opelopabg 4465 . . . . . 6  |-  ( ( T  e.  _V  /\  S  e.  _V )  ->  ( <. T ,  S >.  e.  { <. z ,  w >.  |  (
z : ~H --> ~H  /\  w : ~H --> ~H  /\  A. x  e.  ~H  A. y  e.  ~H  (
( z `  x
)  .ih  y )  =  ( x  .ih  ( w `  y
) ) ) }  <-> 
( T : ~H --> ~H  /\  S : ~H --> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( ( T `  x )  .ih  y
)  =  ( x 
.ih  ( S `  y ) ) ) ) )
226, 8, 21syl2an 464 . . . . 5  |-  ( ( T : ~H --> ~H  /\  S : ~H --> ~H )  ->  ( <. T ,  S >.  e.  { <. z ,  w >.  |  (
z : ~H --> ~H  /\  w : ~H --> ~H  /\  A. x  e.  ~H  A. y  e.  ~H  (
( z `  x
)  .ih  y )  =  ( x  .ih  ( w `  y
) ) ) }  <-> 
( T : ~H --> ~H  /\  S : ~H --> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( ( T `  x )  .ih  y
)  =  ( x 
.ih  ( S `  y ) ) ) ) )
233, 22syl5bb 249 . . . 4  |-  ( ( T : ~H --> ~H  /\  S : ~H --> ~H )  ->  ( <. T ,  S >.  e.  adjh  <->  ( T : ~H
--> ~H  /\  S : ~H
--> ~H  /\  A. x  e.  ~H  A. y  e. 
~H  ( ( T `
 x )  .ih  y )  =  ( x  .ih  ( S `
 y ) ) ) ) )
24 df-3an 938 . . . . 5  |-  ( ( T : ~H --> ~H  /\  S : ~H --> ~H  /\  A. x  e.  ~H  A. y  e.  ~H  (
( T `  x
)  .ih  y )  =  ( x  .ih  ( S `  y ) ) )  <->  ( ( T : ~H --> ~H  /\  S : ~H --> ~H )  /\  A. x  e.  ~H  A. y  e.  ~H  (
( T `  x
)  .ih  y )  =  ( x  .ih  ( S `  y ) ) ) )
2524baibr 873 . . . 4  |-  ( ( T : ~H --> ~H  /\  S : ~H --> ~H )  ->  ( A. x  e. 
~H  A. y  e.  ~H  ( ( T `  x )  .ih  y
)  =  ( x 
.ih  ( S `  y ) )  <->  ( T : ~H --> ~H  /\  S : ~H
--> ~H  /\  A. x  e.  ~H  A. y  e. 
~H  ( ( T `
 x )  .ih  y )  =  ( x  .ih  ( S `
 y ) ) ) ) )
2623, 25bitr4d 248 . . 3  |-  ( ( T : ~H --> ~H  /\  S : ~H --> ~H )  ->  ( <. T ,  S >.  e.  adjh  <->  A. x  e.  ~H  A. y  e.  ~H  (
( T `  x
)  .ih  y )  =  ( x  .ih  ( S `  y ) ) ) )
2726biimp3ar 1284 . 2  |-  ( ( T : ~H --> ~H  /\  S : ~H --> ~H  /\  A. x  e.  ~H  A. y  e.  ~H  (
( T `  x
)  .ih  y )  =  ( x  .ih  ( S `  y ) ) )  ->  <. T ,  S >.  e.  adjh )
28 funopfv 5758 . 2  |-  ( Fun 
adjh  ->  ( <. T ,  S >.  e.  adjh  ->  (
adjh `  T )  =  S ) )
291, 27, 28mpsyl 61 1  |-  ( ( T : ~H --> ~H  /\  S : ~H --> ~H  /\  A. x  e.  ~H  A. y  e.  ~H  (
( T `  x
)  .ih  y )  =  ( x  .ih  ( S `  y ) ) )  ->  ( adjh `  T )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948   <.cop 3809   {copab 4257   Fun wfun 5440   -->wf 5442   ` cfv 5446  (class class class)co 6073   ~Hchil 22414    .ih csp 22417   adjhcado 22450
This theorem is referenced by:  unopadj2  23433  hmopadj  23434  adj0  23489  adjmul  23587  adjadd  23588
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-hilex 22494  ax-hfvadd 22495  ax-hvcom 22496  ax-hvass 22497  ax-hv0cl 22498  ax-hvaddid 22499  ax-hfvmul 22500  ax-hvmulid 22501  ax-hvdistr2 22504  ax-hvmul0 22505  ax-hfi 22573  ax-his1 22576  ax-his2 22577  ax-his3 22578  ax-his4 22579
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-2 10050  df-cj 11896  df-re 11897  df-im 11898  df-hvsub 22466  df-adjh 23344
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