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Theorem adj1 25352
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 25305 . . . . . . 7  |-  Fun  adjh
2 funfvop 5830 . . . . . . 7  |-  ( ( Fun  adjh  /\  T  e. 
dom  adjh )  ->  <. T , 
( adjh `  T ) >.  e.  adjh )
31, 2mpan 670 . . . . . 6  |-  ( T  e.  dom  adjh  ->  <. T ,  ( adjh `  T ) >.  e.  adjh )
4 dfadj2 25304 . . . . . 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 2533 . . . . 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 5716 . . . . . 6  |-  ( adjh `  T )  e.  _V
7 feq1 5557 . . . . . . . 8  |-  ( z  =  T  ->  (
z : ~H --> ~H  <->  T : ~H
--> ~H ) )
8 fveq1 5705 . . . . . . . . . . 11  |-  ( z  =  T  ->  (
z `  y )  =  ( T `  y ) )
98oveq2d 6122 . . . . . . . . . 10  |-  ( z  =  T  ->  (
x  .ih  ( z `  y ) )  =  ( x  .ih  ( T `  y )
) )
109eqeq1d 2451 . . . . . . . . 9  |-  ( z  =  T  ->  (
( x  .ih  (
z `  y )
)  =  ( ( w `  x ) 
.ih  y )  <->  ( x  .ih  ( T `  y
) )  =  ( ( w `  x
)  .ih  y )
) )
11102ralbidv 2772 . . . . . . . 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 1291 . . . . . . 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 5557 . . . . . . . 8  |-  ( w  =  ( adjh `  T
)  ->  ( w : ~H --> ~H  <->  ( adjh `  T
) : ~H --> ~H )
)
14 fveq1 5705 . . . . . . . . . . 11  |-  ( w  =  ( adjh `  T
)  ->  ( w `  x )  =  ( ( adjh `  T
) `  x )
)
1514oveq1d 6121 . . . . . . . . . 10  |-  ( w  =  ( adjh `  T
)  ->  ( (
w `  x )  .ih  y )  =  ( ( ( adjh `  T
) `  x )  .ih  y ) )
1615eqeq2d 2454 . . . . . . . . 9  |-  ( w  =  ( adjh `  T
)  ->  ( (
x  .ih  ( T `  y ) )  =  ( ( w `  x )  .ih  y
)  <->  ( x  .ih  ( T `  y ) )  =  ( ( ( adjh `  T
) `  x )  .ih  y ) ) )
17162ralbidv 2772 . . . . . . . 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 1292 . . . . . . 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 4622 . . . . . 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 671 . . . . 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 210 . . . 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 1002 . . 3  |-  ( T  e.  dom  adjh  ->  A. x  e.  ~H  A. y  e.  ~H  (
x  .ih  ( T `  y ) )  =  ( ( ( adjh `  T ) `  x
)  .ih  y )
)
23 oveq1 6113 . . . . 5  |-  ( x  =  A  ->  (
x  .ih  ( T `  y ) )  =  ( A  .ih  ( T `  y )
) )
24 fveq2 5706 . . . . . 6  |-  ( x  =  A  ->  (
( adjh `  T ) `  x )  =  ( ( adjh `  T
) `  A )
)
2524oveq1d 6121 . . . . 5  |-  ( x  =  A  ->  (
( ( adjh `  T
) `  x )  .ih  y )  =  ( ( ( adjh `  T
) `  A )  .ih  y ) )
2623, 25eqeq12d 2457 . . . 4  |-  ( x  =  A  ->  (
( x  .ih  ( T `  y )
)  =  ( ( ( adjh `  T
) `  x )  .ih  y )  <->  ( A  .ih  ( T `  y
) )  =  ( ( ( adjh `  T
) `  A )  .ih  y ) ) )
27 fveq2 5706 . . . . . 6  |-  ( y  =  B  ->  ( T `  y )  =  ( T `  B ) )
2827oveq2d 6122 . . . . 5  |-  ( y  =  B  ->  ( A  .ih  ( T `  y ) )  =  ( A  .ih  ( T `  B )
) )
29 oveq2 6114 . . . . 5  |-  ( y  =  B  ->  (
( ( adjh `  T
) `  A )  .ih  y )  =  ( ( ( adjh `  T
) `  A )  .ih  B ) )
3028, 29eqeq12d 2457 . . . 4  |-  ( y  =  B  ->  (
( A  .ih  ( T `  y )
)  =  ( ( ( adjh `  T
) `  A )  .ih  y )  <->  ( A  .ih  ( T `  B
) )  =  ( ( ( adjh `  T
) `  A )  .ih  B ) ) )
3126, 30rspc2v 3094 . . 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 30 . 2  |-  ( T  e.  dom  adjh  ->  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  ( T `  B )
)  =  ( ( ( adjh `  T
) `  A )  .ih  B ) ) )
33323impib 1185 1  |-  ( ( T  e.  dom  adjh  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  ( T `  B )
)  =  ( ( ( adjh `  T
) `  A )  .ih  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2730   _Vcvv 2987   <.cop 3898   {copab 4364   dom cdm 4855   Fun wfun 5427   -->wf 5429   ` cfv 5433  (class class class)co 6106   ~Hchil 24336    .ih csp 24339   adjhcado 24372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-hfvadd 24417  ax-hvcom 24418  ax-hvass 24419  ax-hv0cl 24420  ax-hvaddid 24421  ax-hfvmul 24422  ax-hvmulid 24423  ax-hvdistr2 24426  ax-hvmul0 24427  ax-hfi 24496  ax-his1 24499  ax-his2 24500  ax-his3 24501  ax-his4 24502
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-po 4656  df-so 4657  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-2 10395  df-cj 12603  df-re 12604  df-im 12605  df-hvsub 24388  df-adjh 25268
This theorem is referenced by:  adj2  25353  adjadj  25355  hmopadj2  25360
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