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Theorem adj1 26993
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 26946 . . . . . . 7  |-  Fun  adjh
2 funfvop 5918 . . . . . . 7  |-  ( ( Fun  adjh  /\  T  e. 
dom  adjh )  ->  <. T , 
( adjh `  T ) >.  e.  adjh )
31, 2mpan 668 . . . . . 6  |-  ( T  e.  dom  adjh  ->  <. T ,  ( adjh `  T ) >.  e.  adjh )
4 dfadj2 26945 . . . . . 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 2494 . . . . 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 5801 . . . . . 6  |-  ( adjh `  T )  e.  _V
7 feq1 5638 . . . . . . . 8  |-  ( z  =  T  ->  (
z : ~H --> ~H  <->  T : ~H
--> ~H ) )
8 fveq1 5790 . . . . . . . . . . 11  |-  ( z  =  T  ->  (
z `  y )  =  ( T `  y ) )
98oveq2d 6234 . . . . . . . . . 10  |-  ( z  =  T  ->  (
x  .ih  ( z `  y ) )  =  ( x  .ih  ( T `  y )
) )
109eqeq1d 2398 . . . . . . . . 9  |-  ( z  =  T  ->  (
( x  .ih  (
z `  y )
)  =  ( ( w `  x ) 
.ih  y )  <->  ( x  .ih  ( T `  y
) )  =  ( ( w `  x
)  .ih  y )
) )
11102ralbidv 2840 . . . . . . . 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 1299 . . . . . . 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 5638 . . . . . . . 8  |-  ( w  =  ( adjh `  T
)  ->  ( w : ~H --> ~H  <->  ( adjh `  T
) : ~H --> ~H )
)
14 fveq1 5790 . . . . . . . . . . 11  |-  ( w  =  ( adjh `  T
)  ->  ( w `  x )  =  ( ( adjh `  T
) `  x )
)
1514oveq1d 6233 . . . . . . . . . 10  |-  ( w  =  ( adjh `  T
)  ->  ( (
w `  x )  .ih  y )  =  ( ( ( adjh `  T
) `  x )  .ih  y ) )
1615eqeq2d 2410 . . . . . . . . 9  |-  ( w  =  ( adjh `  T
)  ->  ( (
x  .ih  ( T `  y ) )  =  ( ( w `  x )  .ih  y
)  <->  ( x  .ih  ( T `  y ) )  =  ( ( ( adjh `  T
) `  x )  .ih  y ) ) )
17162ralbidv 2840 . . . . . . . 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 1300 . . . . . . 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 4696 . . . . . 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 669 . . . . 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 1008 . . 3  |-  ( T  e.  dom  adjh  ->  A. x  e.  ~H  A. y  e.  ~H  (
x  .ih  ( T `  y ) )  =  ( ( ( adjh `  T ) `  x
)  .ih  y )
)
23 oveq1 6225 . . . . 5  |-  ( x  =  A  ->  (
x  .ih  ( T `  y ) )  =  ( A  .ih  ( T `  y )
) )
24 fveq2 5791 . . . . . 6  |-  ( x  =  A  ->  (
( adjh `  T ) `  x )  =  ( ( adjh `  T
) `  A )
)
2524oveq1d 6233 . . . . 5  |-  ( x  =  A  ->  (
( ( adjh `  T
) `  x )  .ih  y )  =  ( ( ( adjh `  T
) `  A )  .ih  y ) )
2623, 25eqeq12d 2418 . . . 4  |-  ( x  =  A  ->  (
( x  .ih  ( T `  y )
)  =  ( ( ( adjh `  T
) `  x )  .ih  y )  <->  ( A  .ih  ( T `  y
) )  =  ( ( ( adjh `  T
) `  A )  .ih  y ) ) )
27 fveq2 5791 . . . . . 6  |-  ( y  =  B  ->  ( T `  y )  =  ( T `  B ) )
2827oveq2d 6234 . . . . 5  |-  ( y  =  B  ->  ( A  .ih  ( T `  y ) )  =  ( A  .ih  ( T `  B )
) )
29 oveq2 6226 . . . . 5  |-  ( y  =  B  ->  (
( ( adjh `  T
) `  A )  .ih  y )  =  ( ( ( adjh `  T
) `  A )  .ih  B ) )
3028, 29eqeq12d 2418 . . . 4  |-  ( y  =  B  ->  (
( A  .ih  ( T `  y )
)  =  ( ( ( adjh `  T
) `  A )  .ih  y )  <->  ( A  .ih  ( T `  B
) )  =  ( ( ( adjh `  T
) `  A )  .ih  B ) ) )
3126, 30rspc2v 3161 . . 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 1192 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 367    /\ w3a 971    = wceq 1399    e. wcel 1836   A.wral 2746   _Vcvv 3051   <.cop 3967   {copab 4441   dom cdm 4930   Fun wfun 5507   -->wf 5509   ` cfv 5513  (class class class)co 6218   ~Hchil 25978    .ih csp 25981   adjhcado 26014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-addrcl 9486  ax-mulcl 9487  ax-mulrcl 9488  ax-mulcom 9489  ax-addass 9490  ax-mulass 9491  ax-distr 9492  ax-i2m1 9493  ax-1ne0 9494  ax-1rid 9495  ax-rnegex 9496  ax-rrecex 9497  ax-cnre 9498  ax-pre-lttri 9499  ax-pre-lttrn 9500  ax-pre-ltadd 9501  ax-pre-mulgt0 9502  ax-hfvadd 26059  ax-hvcom 26060  ax-hvass 26061  ax-hv0cl 26062  ax-hvaddid 26063  ax-hfvmul 26064  ax-hvmulid 26065  ax-hvdistr2 26068  ax-hvmul0 26069  ax-hfi 26138  ax-his1 26141  ax-his2 26142  ax-his3 26143  ax-his4 26144
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-po 4731  df-so 4732  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-er 7251  df-en 7458  df-dom 7459  df-sdom 7460  df-pnf 9563  df-mnf 9564  df-xr 9565  df-ltxr 9566  df-le 9567  df-sub 9742  df-neg 9743  df-div 10146  df-2 10533  df-cj 12957  df-re 12958  df-im 12959  df-hvsub 26030  df-adjh 26909
This theorem is referenced by:  adj2  26994  adjadj  26996  hmopadj2  27001
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