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Theorem adjeq 22623
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 22574 . 2  |-  Fun  adjh
2 df-adjh 22537 . . . . . 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 2422 . . . . 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 21687 . . . . . . 7  |-  ~H  e.  _V
5 fex 5832 . . . . . . 7  |-  ( ( T : ~H --> ~H  /\  ~H  e.  _V )  ->  T  e.  _V )
64, 5mpan2 652 . . . . . 6  |-  ( T : ~H --> ~H  ->  T  e.  _V )
7 fex 5832 . . . . . . 7  |-  ( ( S : ~H --> ~H  /\  ~H  e.  _V )  ->  S  e.  _V )
84, 7mpan2 652 . . . . . 6  |-  ( S : ~H --> ~H  ->  S  e.  _V )
9 feq1 5454 . . . . . . . 8  |-  ( z  =  T  ->  (
z : ~H --> ~H  <->  T : ~H
--> ~H ) )
10 fveq1 5604 . . . . . . . . . . 11  |-  ( z  =  T  ->  (
z `  x )  =  ( T `  x ) )
1110oveq1d 5957 . . . . . . . . . 10  |-  ( z  =  T  ->  (
( z `  x
)  .ih  y )  =  ( ( T `
 x )  .ih  y ) )
1211eqeq1d 2366 . . . . . . . . 9  |-  ( z  =  T  ->  (
( ( z `  x )  .ih  y
)  =  ( x 
.ih  ( w `  y ) )  <->  ( ( T `  x )  .ih  y )  =  ( x  .ih  ( w `
 y ) ) ) )
13122ralbidv 2661 . . . . . . . 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 1254 . . . . . . 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 5454 . . . . . . . 8  |-  ( w  =  S  ->  (
w : ~H --> ~H  <->  S : ~H
--> ~H ) )
16 fveq1 5604 . . . . . . . . . . 11  |-  ( w  =  S  ->  (
w `  y )  =  ( S `  y ) )
1716oveq2d 5958 . . . . . . . . . 10  |-  ( w  =  S  ->  (
x  .ih  ( w `  y ) )  =  ( x  .ih  ( S `  y )
) )
1817eqeq2d 2369 . . . . . . . . 9  |-  ( w  =  S  ->  (
( ( T `  x )  .ih  y
)  =  ( x 
.ih  ( w `  y ) )  <->  ( ( T `  x )  .ih  y )  =  ( x  .ih  ( S `
 y ) ) ) )
19182ralbidv 2661 . . . . . . . 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 1255 . . . . . . 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 4362 . . . . . 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 463 . . . . 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 248 . . . 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 936 . . . . 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 872 . . . 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 247 . . 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 1282 . 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 5642 . 2  |-  ( Fun 
adjh  ->  ( <. T ,  S >.  e.  adjh  ->  (
adjh `  T )  =  S ) )
291, 27, 28mpsyl 59 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 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   A.wral 2619   _Vcvv 2864   <.cop 3719   {copab 4155   Fun wfun 5328   -->wf 5330   ` cfv 5334  (class class class)co 5942   ~Hchil 21607    .ih csp 21610   adjhcado 21643
This theorem is referenced by:  unopadj2  22626  hmopadj  22627  adj0  22682  adjmul  22780  adjadd  22781
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-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-hilex 21687  ax-hfvadd 21688  ax-hvcom 21689  ax-hvass 21690  ax-hv0cl 21691  ax-hvaddid 21692  ax-hfvmul 21693  ax-hvmulid 21694  ax-hvdistr2 21697  ax-hvmul0 21698  ax-hfi 21766  ax-his1 21769  ax-his2 21770  ax-his3 21771  ax-his4 21772
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  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-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-po 4393  df-so 4394  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-riota 6388  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-2 9891  df-cj 11674  df-re 11675  df-im 11676  df-hvsub 21659  df-adjh 22537
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