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Theorem adjeq 26685
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 26636 . 2  |-  Fun  adjh
2 df-adjh 26599 . . . . . 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 2545 . . . . 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 25747 . . . . . . 7  |-  ~H  e.  _V
5 fex 6144 . . . . . . 7  |-  ( ( T : ~H --> ~H  /\  ~H  e.  _V )  ->  T  e.  _V )
64, 5mpan2 671 . . . . . 6  |-  ( T : ~H --> ~H  ->  T  e.  _V )
7 fex 6144 . . . . . . 7  |-  ( ( S : ~H --> ~H  /\  ~H  e.  _V )  ->  S  e.  _V )
84, 7mpan2 671 . . . . . 6  |-  ( S : ~H --> ~H  ->  S  e.  _V )
9 feq1 5719 . . . . . . . 8  |-  ( z  =  T  ->  (
z : ~H --> ~H  <->  T : ~H
--> ~H ) )
10 fveq1 5871 . . . . . . . . . . 11  |-  ( z  =  T  ->  (
z `  x )  =  ( T `  x ) )
1110oveq1d 6310 . . . . . . . . . 10  |-  ( z  =  T  ->  (
( z `  x
)  .ih  y )  =  ( ( T `
 x )  .ih  y ) )
1211eqeq1d 2469 . . . . . . . . 9  |-  ( z  =  T  ->  (
( ( z `  x )  .ih  y
)  =  ( x 
.ih  ( w `  y ) )  <->  ( ( T `  x )  .ih  y )  =  ( x  .ih  ( w `
 y ) ) ) )
13122ralbidv 2911 . . . . . . . 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 1301 . . . . . . 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 5719 . . . . . . . 8  |-  ( w  =  S  ->  (
w : ~H --> ~H  <->  S : ~H
--> ~H ) )
16 fveq1 5871 . . . . . . . . . . 11  |-  ( w  =  S  ->  (
w `  y )  =  ( S `  y ) )
1716oveq2d 6311 . . . . . . . . . 10  |-  ( w  =  S  ->  (
x  .ih  ( w `  y ) )  =  ( x  .ih  ( S `  y )
) )
1817eqeq2d 2481 . . . . . . . . 9  |-  ( w  =  S  ->  (
( ( T `  x )  .ih  y
)  =  ( x 
.ih  ( w `  y ) )  <->  ( ( T `  x )  .ih  y )  =  ( x  .ih  ( S `
 y ) ) ) )
19182ralbidv 2911 . . . . . . . 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 1302 . . . . . . 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 4771 . . . . . 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 477 . . . . 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 257 . . . 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 975 . . . . 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 902 . . . 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 256 . . 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 1329 . 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 5913 . 2  |-  ( Fun 
adjh  ->  ( <. T ,  S >.  e.  adjh  ->  (
adjh `  T )  =  S ) )
291, 27, 28mpsyl 63 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118   <.cop 4039   {copab 4510   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6295   ~Hchil 25667    .ih csp 25670   adjhcado 25703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-hilex 25747  ax-hfvadd 25748  ax-hvcom 25749  ax-hvass 25750  ax-hv0cl 25751  ax-hvaddid 25752  ax-hfvmul 25753  ax-hvmulid 25754  ax-hvdistr2 25757  ax-hvmul0 25758  ax-hfi 25827  ax-his1 25830  ax-his2 25831  ax-his3 25832  ax-his4 25833
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-2 10606  df-cj 12911  df-re 12912  df-im 12913  df-hvsub 25719  df-adjh 26599
This theorem is referenced by:  unopadj2  26688  hmopadj  26689  adj0  26744  adjmul  26842  adjadd  26843
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