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Theorem adj1 25272
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 25225 . . . . . . 7  |-  Fun  adjh
2 funfvop 5812 . . . . . . 7  |-  ( ( Fun  adjh  /\  T  e. 
dom  adjh )  ->  <. T , 
( adjh `  T ) >.  e.  adjh )
31, 2mpan 665 . . . . . 6  |-  ( T  e.  dom  adjh  ->  <. T ,  ( adjh `  T ) >.  e.  adjh )
4 dfadj2 25224 . . . . . 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 2531 . . . . 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 5698 . . . . . 6  |-  ( adjh `  T )  e.  _V
7 feq1 5539 . . . . . . . 8  |-  ( z  =  T  ->  (
z : ~H --> ~H  <->  T : ~H
--> ~H ) )
8 fveq1 5687 . . . . . . . . . . 11  |-  ( z  =  T  ->  (
z `  y )  =  ( T `  y ) )
98oveq2d 6106 . . . . . . . . . 10  |-  ( z  =  T  ->  (
x  .ih  ( z `  y ) )  =  ( x  .ih  ( T `  y )
) )
109eqeq1d 2449 . . . . . . . . 9  |-  ( z  =  T  ->  (
( x  .ih  (
z `  y )
)  =  ( ( w `  x ) 
.ih  y )  <->  ( x  .ih  ( T `  y
) )  =  ( ( w `  x
)  .ih  y )
) )
11102ralbidv 2755 . . . . . . . 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 1286 . . . . . . 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 5539 . . . . . . . 8  |-  ( w  =  ( adjh `  T
)  ->  ( w : ~H --> ~H  <->  ( adjh `  T
) : ~H --> ~H )
)
14 fveq1 5687 . . . . . . . . . . 11  |-  ( w  =  ( adjh `  T
)  ->  ( w `  x )  =  ( ( adjh `  T
) `  x )
)
1514oveq1d 6105 . . . . . . . . . 10  |-  ( w  =  ( adjh `  T
)  ->  ( (
w `  x )  .ih  y )  =  ( ( ( adjh `  T
) `  x )  .ih  y ) )
1615eqeq2d 2452 . . . . . . . . 9  |-  ( w  =  ( adjh `  T
)  ->  ( (
x  .ih  ( T `  y ) )  =  ( ( w `  x )  .ih  y
)  <->  ( x  .ih  ( T `  y ) )  =  ( ( ( adjh `  T
) `  x )  .ih  y ) ) )
17162ralbidv 2755 . . . . . . . 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 1287 . . . . . . 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 4605 . . . . . 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 666 . . . . 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 997 . . 3  |-  ( T  e.  dom  adjh  ->  A. x  e.  ~H  A. y  e.  ~H  (
x  .ih  ( T `  y ) )  =  ( ( ( adjh `  T ) `  x
)  .ih  y )
)
23 oveq1 6097 . . . . 5  |-  ( x  =  A  ->  (
x  .ih  ( T `  y ) )  =  ( A  .ih  ( T `  y )
) )
24 fveq2 5688 . . . . . 6  |-  ( x  =  A  ->  (
( adjh `  T ) `  x )  =  ( ( adjh `  T
) `  A )
)
2524oveq1d 6105 . . . . 5  |-  ( x  =  A  ->  (
( ( adjh `  T
) `  x )  .ih  y )  =  ( ( ( adjh `  T
) `  A )  .ih  y ) )
2623, 25eqeq12d 2455 . . . 4  |-  ( x  =  A  ->  (
( x  .ih  ( T `  y )
)  =  ( ( ( adjh `  T
) `  x )  .ih  y )  <->  ( A  .ih  ( T `  y
) )  =  ( ( ( adjh `  T
) `  A )  .ih  y ) ) )
27 fveq2 5688 . . . . . 6  |-  ( y  =  B  ->  ( T `  y )  =  ( T `  B ) )
2827oveq2d 6106 . . . . 5  |-  ( y  =  B  ->  ( A  .ih  ( T `  y ) )  =  ( A  .ih  ( T `  B )
) )
29 oveq2 6098 . . . . 5  |-  ( y  =  B  ->  (
( ( adjh `  T
) `  A )  .ih  y )  =  ( ( ( adjh `  T
) `  A )  .ih  B ) )
3028, 29eqeq12d 2455 . . . 4  |-  ( y  =  B  ->  (
( A  .ih  ( T `  y )
)  =  ( ( ( adjh `  T
) `  A )  .ih  y )  <->  ( A  .ih  ( T `  B
) )  =  ( ( ( adjh `  T
) `  A )  .ih  B ) ) )
3126, 30rspc2v 3076 . . 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 1180 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 960    = wceq 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970   <.cop 3880   {copab 4346   dom cdm 4836   Fun wfun 5409   -->wf 5411   ` cfv 5415  (class class class)co 6090   ~Hchil 24256    .ih csp 24259   adjhcado 24292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-hfvadd 24337  ax-hvcom 24338  ax-hvass 24339  ax-hv0cl 24340  ax-hvaddid 24341  ax-hfvmul 24342  ax-hvmulid 24343  ax-hvdistr2 24346  ax-hvmul0 24347  ax-hfi 24416  ax-his1 24419  ax-his2 24420  ax-his3 24421  ax-his4 24422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-2 10376  df-cj 12584  df-re 12585  df-im 12586  df-hvsub 24308  df-adjh 25188
This theorem is referenced by:  adj2  25273  adjadj  25275  hmopadj2  25280
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