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Theorem adjeq 23288
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 23239 . 2  |-  Fun  adjh
2 df-adjh 23202 . . . . . 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 2453 . . . . 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 22352 . . . . . . 7  |-  ~H  e.  _V
5 fex 5910 . . . . . . 7  |-  ( ( T : ~H --> ~H  /\  ~H  e.  _V )  ->  T  e.  _V )
64, 5mpan2 653 . . . . . 6  |-  ( T : ~H --> ~H  ->  T  e.  _V )
7 fex 5910 . . . . . . 7  |-  ( ( S : ~H --> ~H  /\  ~H  e.  _V )  ->  S  e.  _V )
84, 7mpan2 653 . . . . . 6  |-  ( S : ~H --> ~H  ->  S  e.  _V )
9 feq1 5518 . . . . . . . 8  |-  ( z  =  T  ->  (
z : ~H --> ~H  <->  T : ~H
--> ~H ) )
10 fveq1 5669 . . . . . . . . . . 11  |-  ( z  =  T  ->  (
z `  x )  =  ( T `  x ) )
1110oveq1d 6037 . . . . . . . . . 10  |-  ( z  =  T  ->  (
( z `  x
)  .ih  y )  =  ( ( T `
 x )  .ih  y ) )
1211eqeq1d 2397 . . . . . . . . 9  |-  ( z  =  T  ->  (
( ( z `  x )  .ih  y
)  =  ( x 
.ih  ( w `  y ) )  <->  ( ( T `  x )  .ih  y )  =  ( x  .ih  ( w `
 y ) ) ) )
13122ralbidv 2693 . . . . . . . 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 5518 . . . . . . . 8  |-  ( w  =  S  ->  (
w : ~H --> ~H  <->  S : ~H
--> ~H ) )
16 fveq1 5669 . . . . . . . . . . 11  |-  ( w  =  S  ->  (
w `  y )  =  ( S `  y ) )
1716oveq2d 6038 . . . . . . . . . 10  |-  ( w  =  S  ->  (
x  .ih  ( w `  y ) )  =  ( x  .ih  ( S `  y )
) )
1817eqeq2d 2400 . . . . . . . . 9  |-  ( w  =  S  ->  (
( ( T `  x )  .ih  y
)  =  ( x 
.ih  ( w `  y ) )  <->  ( ( T `  x )  .ih  y )  =  ( x  .ih  ( S `
 y ) ) ) )
19182ralbidv 2693 . . . . . . . 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 4416 . . . . . 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 5707 . 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 1649    e. wcel 1717   A.wral 2651   _Vcvv 2901   <.cop 3762   {copab 4208   Fun wfun 5390   -->wf 5392   ` cfv 5396  (class class class)co 6022   ~Hchil 22272    .ih csp 22275   adjhcado 22308
This theorem is referenced by:  unopadj2  23291  hmopadj  23292  adj0  23347  adjmul  23445  adjadd  23446
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 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-hilex 22352  ax-hfvadd 22353  ax-hvcom 22354  ax-hvass 22355  ax-hv0cl 22356  ax-hvaddid 22357  ax-hfvmul 22358  ax-hvmulid 22359  ax-hvdistr2 22362  ax-hvmul0 22363  ax-hfi 22431  ax-his1 22434  ax-his2 22435  ax-his3 22436  ax-his4 22437
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-po 4446  df-so 4447  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-riota 6487  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-2 9992  df-cj 11833  df-re 11834  df-im 11835  df-hvsub 22324  df-adjh 23202
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