HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  adj1 Structured version   Unicode version

Theorem adj1 23428
Description: Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
adj1  |-  ( ( T  e.  dom  adjh  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  ( T `  B )
)  =  ( ( ( adjh `  T
) `  A )  .ih  B ) )

Proof of Theorem adj1
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funadj 23381 . . . . . . 7  |-  Fun  adjh
2 funfvop 5834 . . . . . . 7  |-  ( ( Fun  adjh  /\  T  e. 
dom  adjh )  ->  <. T , 
( adjh `  T ) >.  e.  adjh )
31, 2mpan 652 . . . . . 6  |-  ( T  e.  dom  adjh  ->  <. T ,  ( adjh `  T ) >.  e.  adjh )
4 dfadj2 23380 . . . . . 6  |-  adjh  =  { <. z ,  w >.  |  ( z : ~H --> ~H  /\  w : ~H --> ~H  /\  A. x  e.  ~H  A. y  e. 
~H  ( x  .ih  ( z `  y
) )  =  ( ( w `  x
)  .ih  y )
) }
53, 4syl6eleq 2525 . . . . 5  |-  ( T  e.  dom  adjh  ->  <. T ,  ( adjh `  T ) >.  e.  { <. z ,  w >.  |  ( z : ~H --> ~H  /\  w : ~H --> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( x  .ih  ( z `
 y ) )  =  ( ( w `
 x )  .ih  y ) ) } )
6 fvex 5734 . . . . . 6  |-  ( adjh `  T )  e.  _V
7 feq1 5568 . . . . . . . 8  |-  ( z  =  T  ->  (
z : ~H --> ~H  <->  T : ~H
--> ~H ) )
8 fveq1 5719 . . . . . . . . . . 11  |-  ( z  =  T  ->  (
z `  y )  =  ( T `  y ) )
98oveq2d 6089 . . . . . . . . . 10  |-  ( z  =  T  ->  (
x  .ih  ( z `  y ) )  =  ( x  .ih  ( T `  y )
) )
109eqeq1d 2443 . . . . . . . . 9  |-  ( z  =  T  ->  (
( x  .ih  (
z `  y )
)  =  ( ( w `  x ) 
.ih  y )  <->  ( x  .ih  ( T `  y
) )  =  ( ( w `  x
)  .ih  y )
) )
11102ralbidv 2739 . . . . . . . 8  |-  ( z  =  T  ->  ( A. x  e.  ~H  A. y  e.  ~H  (
x  .ih  ( z `  y ) )  =  ( ( w `  x )  .ih  y
)  <->  A. x  e.  ~H  A. y  e.  ~H  (
x  .ih  ( T `  y ) )  =  ( ( w `  x )  .ih  y
) ) )
127, 113anbi13d 1256 . . . . . . 7  |-  ( z  =  T  ->  (
( z : ~H --> ~H  /\  w : ~H --> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( x  .ih  ( z `
 y ) )  =  ( ( w `
 x )  .ih  y ) )  <->  ( T : ~H --> ~H  /\  w : ~H --> ~H  /\  A. x  e.  ~H  A. y  e. 
~H  ( x  .ih  ( T `  y ) )  =  ( ( w `  x ) 
.ih  y ) ) ) )
13 feq1 5568 . . . . . . . 8  |-  ( w  =  ( adjh `  T
)  ->  ( w : ~H --> ~H  <->  ( adjh `  T
) : ~H --> ~H )
)
14 fveq1 5719 . . . . . . . . . . 11  |-  ( w  =  ( adjh `  T
)  ->  ( w `  x )  =  ( ( adjh `  T
) `  x )
)
1514oveq1d 6088 . . . . . . . . . 10  |-  ( w  =  ( adjh `  T
)  ->  ( (
w `  x )  .ih  y )  =  ( ( ( adjh `  T
) `  x )  .ih  y ) )
1615eqeq2d 2446 . . . . . . . . 9  |-  ( w  =  ( adjh `  T
)  ->  ( (
x  .ih  ( T `  y ) )  =  ( ( w `  x )  .ih  y
)  <->  ( x  .ih  ( T `  y ) )  =  ( ( ( adjh `  T
) `  x )  .ih  y ) ) )
17162ralbidv 2739 . . . . . . . 8  |-  ( w  =  ( adjh `  T
)  ->  ( A. x  e.  ~H  A. y  e.  ~H  ( x  .ih  ( T `  y ) )  =  ( ( w `  x ) 
.ih  y )  <->  A. x  e.  ~H  A. y  e. 
~H  ( x  .ih  ( T `  y ) )  =  ( ( ( adjh `  T
) `  x )  .ih  y ) ) )
1813, 173anbi23d 1257 . . . . . . 7  |-  ( w  =  ( adjh `  T
)  ->  ( ( T : ~H --> ~H  /\  w : ~H --> ~H  /\  A. x  e.  ~H  A. y  e.  ~H  (
x  .ih  ( T `  y ) )  =  ( ( w `  x )  .ih  y
) )  <->  ( T : ~H --> ~H  /\  ( adjh `  T ) : ~H --> ~H  /\  A. x  e.  ~H  A. y  e. 
~H  ( x  .ih  ( T `  y ) )  =  ( ( ( adjh `  T
) `  x )  .ih  y ) ) ) )
1912, 18opelopabg 4465 . . . . . 6  |-  ( ( T  e.  dom  adjh  /\  ( adjh `  T
)  e.  _V )  ->  ( <. T ,  (
adjh `  T ) >.  e.  { <. z ,  w >.  |  (
z : ~H --> ~H  /\  w : ~H --> ~H  /\  A. x  e.  ~H  A. y  e.  ~H  (
x  .ih  ( z `  y ) )  =  ( ( w `  x )  .ih  y
) ) }  <->  ( T : ~H --> ~H  /\  ( adjh `  T ) : ~H --> ~H  /\  A. x  e.  ~H  A. y  e. 
~H  ( x  .ih  ( T `  y ) )  =  ( ( ( adjh `  T
) `  x )  .ih  y ) ) ) )
206, 19mpan2 653 . . . . 5  |-  ( T  e.  dom  adjh  ->  (
<. T ,  ( adjh `  T ) >.  e.  { <. z ,  w >.  |  ( z : ~H --> ~H  /\  w : ~H --> ~H  /\  A. x  e. 
~H  A. y  e.  ~H  ( x  .ih  ( z `
 y ) )  =  ( ( w `
 x )  .ih  y ) ) }  <-> 
( T : ~H --> ~H  /\  ( adjh `  T
) : ~H --> ~H  /\  A. x  e.  ~H  A. y  e.  ~H  (
x  .ih  ( T `  y ) )  =  ( ( ( adjh `  T ) `  x
)  .ih  y )
) ) )
215, 20mpbid 202 . . . 4  |-  ( T  e.  dom  adjh  ->  ( T : ~H --> ~H  /\  ( adjh `  T ) : ~H --> ~H  /\  A. x  e.  ~H  A. y  e. 
~H  ( x  .ih  ( T `  y ) )  =  ( ( ( adjh `  T
) `  x )  .ih  y ) ) )
2221simp3d 971 . . 3  |-  ( T  e.  dom  adjh  ->  A. x  e.  ~H  A. y  e.  ~H  (
x  .ih  ( T `  y ) )  =  ( ( ( adjh `  T ) `  x
)  .ih  y )
)
23 oveq1 6080 . . . . 5  |-  ( x  =  A  ->  (
x  .ih  ( T `  y ) )  =  ( A  .ih  ( T `  y )
) )
24 fveq2 5720 . . . . . 6  |-  ( x  =  A  ->  (
( adjh `  T ) `  x )  =  ( ( adjh `  T
) `  A )
)
2524oveq1d 6088 . . . . 5  |-  ( x  =  A  ->  (
( ( adjh `  T
) `  x )  .ih  y )  =  ( ( ( adjh `  T
) `  A )  .ih  y ) )
2623, 25eqeq12d 2449 . . . 4  |-  ( x  =  A  ->  (
( x  .ih  ( T `  y )
)  =  ( ( ( adjh `  T
) `  x )  .ih  y )  <->  ( A  .ih  ( T `  y
) )  =  ( ( ( adjh `  T
) `  A )  .ih  y ) ) )
27 fveq2 5720 . . . . . 6  |-  ( y  =  B  ->  ( T `  y )  =  ( T `  B ) )
2827oveq2d 6089 . . . . 5  |-  ( y  =  B  ->  ( A  .ih  ( T `  y ) )  =  ( A  .ih  ( T `  B )
) )
29 oveq2 6081 . . . . 5  |-  ( y  =  B  ->  (
( ( adjh `  T
) `  A )  .ih  y )  =  ( ( ( adjh `  T
) `  A )  .ih  B ) )
3028, 29eqeq12d 2449 . . . 4  |-  ( y  =  B  ->  (
( A  .ih  ( T `  y )
)  =  ( ( ( adjh `  T
) `  A )  .ih  y )  <->  ( A  .ih  ( T `  B
) )  =  ( ( ( adjh `  T
) `  A )  .ih  B ) ) )
3126, 30rspc2v 3050 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A. x  e. 
~H  A. y  e.  ~H  ( x  .ih  ( T `
 y ) )  =  ( ( (
adjh `  T ) `  x )  .ih  y
)  ->  ( A  .ih  ( T `  B
) )  =  ( ( ( adjh `  T
) `  A )  .ih  B ) ) )
3222, 31syl5com 28 . 2  |-  ( T  e.  dom  adjh  ->  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  ( T `  B )
)  =  ( ( ( adjh `  T
) `  A )  .ih  B ) ) )
33323impib 1151 1  |-  ( ( T  e.  dom  adjh  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  ( T `  B )
)  =  ( ( ( adjh `  T
) `  A )  .ih  B ) )
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   dom cdm 4870   Fun wfun 5440   -->wf 5442   ` cfv 5446  (class class class)co 6073   ~Hchil 22414    .ih csp 22417   adjhcado 22450
This theorem is referenced by:  adj2  23429  adjadj  23431  hmopadj2  23436
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-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-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
  Copyright terms: Public domain W3C validator