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Theorem adj1 22529
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 22482 . . . . . . 7  |-  Fun  adjh
2 funfvop 5653 . . . . . . 7  |-  ( ( Fun  adjh  /\  T  e. 
dom  adjh )  ->  <. T , 
( adjh `  T ) >.  e.  adjh )
31, 2mpan 651 . . . . . 6  |-  ( T  e.  dom  adjh  ->  <. T ,  ( adjh `  T ) >.  e.  adjh )
4 dfadj2 22481 . . . . . 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 2386 . . . . 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 5555 . . . . . 6  |-  ( adjh `  T )  e.  _V
7 feq1 5391 . . . . . . . 8  |-  ( z  =  T  ->  (
z : ~H --> ~H  <->  T : ~H
--> ~H ) )
8 fveq1 5540 . . . . . . . . . . 11  |-  ( z  =  T  ->  (
z `  y )  =  ( T `  y ) )
98oveq2d 5890 . . . . . . . . . 10  |-  ( z  =  T  ->  (
x  .ih  ( z `  y ) )  =  ( x  .ih  ( T `  y )
) )
109eqeq1d 2304 . . . . . . . . 9  |-  ( z  =  T  ->  (
( x  .ih  (
z `  y )
)  =  ( ( w `  x ) 
.ih  y )  <->  ( x  .ih  ( T `  y
) )  =  ( ( w `  x
)  .ih  y )
) )
11102ralbidv 2598 . . . . . . . 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 1254 . . . . . . 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 5391 . . . . . . . 8  |-  ( w  =  ( adjh `  T
)  ->  ( w : ~H --> ~H  <->  ( adjh `  T
) : ~H --> ~H )
)
14 fveq1 5540 . . . . . . . . . . 11  |-  ( w  =  ( adjh `  T
)  ->  ( w `  x )  =  ( ( adjh `  T
) `  x )
)
1514oveq1d 5889 . . . . . . . . . 10  |-  ( w  =  ( adjh `  T
)  ->  ( (
w `  x )  .ih  y )  =  ( ( ( adjh `  T
) `  x )  .ih  y ) )
1615eqeq2d 2307 . . . . . . . . 9  |-  ( w  =  ( adjh `  T
)  ->  ( (
x  .ih  ( T `  y ) )  =  ( ( w `  x )  .ih  y
)  <->  ( x  .ih  ( T `  y ) )  =  ( ( ( adjh `  T
) `  x )  .ih  y ) ) )
17162ralbidv 2598 . . . . . . . 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 1255 . . . . . . 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 4299 . . . . . 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 652 . . . . 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 201 . . . 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 969 . . 3  |-  ( T  e.  dom  adjh  ->  A. x  e.  ~H  A. y  e.  ~H  (
x  .ih  ( T `  y ) )  =  ( ( ( adjh `  T ) `  x
)  .ih  y )
)
23 oveq1 5881 . . . . 5  |-  ( x  =  A  ->  (
x  .ih  ( T `  y ) )  =  ( A  .ih  ( T `  y )
) )
24 fveq2 5541 . . . . . 6  |-  ( x  =  A  ->  (
( adjh `  T ) `  x )  =  ( ( adjh `  T
) `  A )
)
2524oveq1d 5889 . . . . 5  |-  ( x  =  A  ->  (
( ( adjh `  T
) `  x )  .ih  y )  =  ( ( ( adjh `  T
) `  A )  .ih  y ) )
2623, 25eqeq12d 2310 . . . 4  |-  ( x  =  A  ->  (
( x  .ih  ( T `  y )
)  =  ( ( ( adjh `  T
) `  x )  .ih  y )  <->  ( A  .ih  ( T `  y
) )  =  ( ( ( adjh `  T
) `  A )  .ih  y ) ) )
27 fveq2 5541 . . . . . 6  |-  ( y  =  B  ->  ( T `  y )  =  ( T `  B ) )
2827oveq2d 5890 . . . . 5  |-  ( y  =  B  ->  ( A  .ih  ( T `  y ) )  =  ( A  .ih  ( T `  B )
) )
29 oveq2 5882 . . . . 5  |-  ( y  =  B  ->  (
( ( adjh `  T
) `  A )  .ih  y )  =  ( ( ( adjh `  T
) `  A )  .ih  B ) )
3028, 29eqeq12d 2310 . . . 4  |-  ( y  =  B  ->  (
( A  .ih  ( T `  y )
)  =  ( ( ( adjh `  T
) `  A )  .ih  y )  <->  ( A  .ih  ( T `  B
) )  =  ( ( ( adjh `  T
) `  A )  .ih  B ) ) )
3126, 30rspc2v 2903 . . 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 26 . 2  |-  ( T  e.  dom  adjh  ->  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  ( T `  B )
)  =  ( ( ( adjh `  T
) `  A )  .ih  B ) ) )
33323impib 1149 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801   <.cop 3656   {copab 4092   dom cdm 4705   Fun wfun 5265   -->wf 5267   ` cfv 5271  (class class class)co 5874   ~Hchil 21515    .ih csp 21518   adjhcado 21551
This theorem is referenced by:  adj2  22530  adjadj  22532  hmopadj2  22537
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-hfvadd 21596  ax-hvcom 21597  ax-hvass 21598  ax-hv0cl 21599  ax-hvaddid 21600  ax-hfvmul 21601  ax-hvmulid 21602  ax-hvdistr2 21605  ax-hvmul0 21606  ax-hfi 21674  ax-his1 21677  ax-his2 21678  ax-his3 21679  ax-his4 21680
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-2 9820  df-cj 11600  df-re 11601  df-im 11602  df-hvsub 21567  df-adjh 22445
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